text: 2 networks with scale-free degree distributions exhibit dif- (a) (b) (c) ferent macroscopic dynamics due to the emergence of Spiking Threshold degree dependent behavior of diﬀerent oscillators. Spike The remainder of this paper is organized as follows. In Sec. we describe the model of pulse coupled theta Rest state neurons used on an arbitrary network. In Sec. setup a mean ﬁeld description of the behavior on the network, and then (Sec. ) show how the methods developed by Ott and Antonsen14–16 can be used to write a low di- mensional set of equations describing the dynamics of the FIG. 1. The dynamics of the theta neuron undergo an SNIC (Saddle node on an Invariant Cycle) bifurcation at η = 0. For mean ﬁeld order parameter. In Sec. we then use this negative η the neuron lies in a rest state, with a threshold for low dimensional system to describe the behavior of the excitation, and for positive η the oscillator undergoes periodic system under diﬀerent parameters and network topolo- spiking. gies. Section concludes the paper with further discussion and summary of the main result. II. THE MODEL where A is the adjacency matrix of a network; A = (cid:80) ij ij The theta neuron model encodes the dynamics of a 1 if there is a directed edge from node j to node i, and single neuron in isolation as follows, A = 0 otherwise. The average degree is then given by (cid:104) i(cid:105)j − k = A /N. P (θ) = d (1 cos θ)n represents the i,j ij (cid:82)n n − pulse-like synapse, whose sharpness is controlled by the θ˙ = (1 cos θ) + (1 + cos θ)η, (1) integer parameter n. The normalization constant d is n 2π determined so that P dθ = 2π. Note that in the case 0 n of a fully connected network, where A = 1 for all i and where θ represents the neuron’s state and the parame- ij j, this model reduces to that of Luke et al.8 ter η speciﬁes its excitability. The dynamics can be visu- III. MEAN FIELD FORMULATION alized as a point traveling around the unit circle (Fig. 1). A neuronal spike is said to occur each time the phase an- (cid:29)(cid:80) We consider the limit of many neurons, N 1, and as- gle of the neuron, θ, crosses the leftmost point at θ = π. sume the network is randomly generated from a given de- When η < 0, there are two zeros of the right hand side gree distribution P(k) (normalized such that P(k) = of Eq. (1), representing a stable rest state (solid circle k N), where k, the node degree, represents a two-vector of in Fig. 1(a)) and an unstable equilibrium (open circle in the in-degree and the out-degree, (k , k ). Addition- Fig. 1(a)). Thus, starting from a typical initial condition, in out (cid:48) → ally, we consider an assortativity function a(k k), the state of the neuron goes towards the stable equilib- which speciﬁes the probability of a link from a node of rium at the rest state represented by the ﬁlled circle. A (cid:48) → ∞ degree k to one of degree k. In this N limit, we resting neuron will spike if an external force pushes its assume that the state of the neurons can be represented state (i.e. the angle θ) from the rest state past the unsta- | by a continuous probability distribution, f(θ, η k, t), such ble equilibrium (termed as the ‘spiking threshold’). As η | that f(θ, η k, t)dθdη is the probability that a node of de- is increased above 0, the neuron exhibits a Saddle Node gree k has an excitability parameter in the range [η, η + bifurcation on an Invariant Cycle (SNIC). In this case (cid:82) dη] and a phase angle in the range [θ, θ + dθ] at time t. there are no ﬁxed points (i.e. no zeros of the right hand Since we are assuming that the excitability parameters side of Eq. (1)), and the neuron now ﬁres periodically, | do not vary with time, we deﬁne g(η k) = fdθ, which is as shown in Fig. 1(c). Note that the neuron does not the time independent distribution of the excitability pa- move at the same rate along the entire circle, and may rameters η in the network for a randomly chosen node go faster or slower around θ = π dependent on whether i of degree k. η is less than or greater than 1, respectively (eq. see the − In order to describe the synchronization behavior of plot of (1 cos θ) versus time in Fig. 1(c)). this system, we deﬁne the order parameter to be1, The theta neuron model can be extended from a single (cid:88) neuron in isolation to networks of neurons. We consider a system of N theta neurons coupled together in a general N 1 network via pulse-like synaptic signals, Ii, to each neuron R(t) = eiθj. (4) i: N j=1 −(cid:88) θ˙ = (1 cos θ ) + (1 + cos θ )[η + I ], (2) i i i i i N 1 Some authors, such as (cid:80)Restre(cid:80)po and Ott17 d(cid:16)e(cid:80)ﬁne t(cid:80)he order p(cid:17)a- K I = (cid:104) (cid:105) A P (θ ), (3) rameter diﬀerently so as to be weighted with the out-degree at i k j=1 ij n j each node, i.e., R(t) = Ni=1 Nj=1 Aijeiθj/ Ni=1 Nj=1 Aij ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: networks

text: with

text: scale-free

text: degree

text: distributions

text: exhibit

text: dif-

text: (a)

font: default

text: (b)

text: (c)

text: ferent

text: macroscopic

text: dynamics

text: due

text: to

text: the

text: emergence

text: of

text: Spiking

text: Threshold

text: degree

text: dependent

text: behavior

text: of

text: diﬀerent

text: oscillators.

text: Spike

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: we

text: describe

text: the

text: model

text: of

text: pulse

text: coupled

text: theta

text: Rest

text: state

text: neurons

text: used

text: on

text: an

text: arbitrary

text: network.

text: In

text: Sec.

text: setup

text: a

text: mean

text: ﬁeld

text: description

text: of

text: the

text: behavior

text: on

text: the

text: network,

text: and

text: then

text: (Sec.

text: show

text: how

text: the

text: methods

text: developed

text: by

text: Ott

text: and

text: can

text: be

text: used

text: to

text: write

text: a

text: low

text: di-

text: mensional

text: set

text: of

text: equations

text: describing

text: the

text: dynamics

text: of

text: the

text: FIG.

text: The

text: dynamics

text: of

text: the

text: theta

text: neuron

text: undergo

text: an

text: SNIC

text: (Saddle

text: node

text: on

text: an

text: Invariant

text: Cycle)

text: bifurcation

text: at

text: η

text: For

text: mean

text: ﬁeld

text: order

text: parameter.

text: In

text: Sec.

text: we

text: then

text: use

text: this

text: negative

text: η

text: the

text: neuron

text: lies

text: in

text: a

text: rest

text: state,

text: with

text: a

text: threshold

text: for

text: low

text: dimensional

text: system

text: to

text: describe

text: the

text: behavior

text: of

text: the

text: excitation,

text: and

text: for

text: positive

text: η

text: the

text: oscillator

text: undergoes

text: periodic

text: system

text: under

text: diﬀerent

text: parameters

text: and

text: network

text: topolo-

text: spiking.

text: gies.

text: Section

text: concludes

text: the

text: paper

text: with

text: further

text: discussion

text: and

text: summary

text: of

text: the

text: main

text: result.

text: II.

text: THE

text: MODEL

text: where

text: A

text: is

text: the

text: adjacency

text: matrix

text: of

text: a

text: network;

text: A

text: ij

text: ij

text: The

text: theta

text: neuron

text: model

text: encodes

text: the

text: dynamics

text: of

text: a

text: if

text: there

text: is

text: a

text: directed

text: edge

text: from

text: node

text: j

text: to

text: node

text: i,

text: and

text: single

text: neuron

text: in

text: isolation

text: as

text: follows,

text: A

text: otherwise.

text: The

text: average

text: degree

text: is

text: then

text: given

text: by

text: k

text: A

text: /N.

text: P

text: (θ)

text: d

text: cos

text: θ)n

text: represents

text: the

text: i,j

text: ij

text: n

text: pulse-like

text: synapse,

text: whose

text: sharpness

text: is

text: controlled

text: by

text: the

text: θ˙

text: cos

text: θ)

text: cos

text: θ)η,

text: integer

text: parameter

text: n.

text: The

text: normalization

text: constant

text: d

text: is

text: n

text: 2π

text: determined

text: so

text: that

text: P

text: dθ

text: Note

text: that

text: in

text: the

text: case

text: n

text: of

text: a

text: fully

text: connected

text: network,

text: where

text: A

text: for

text: all

text: i

text: and

text: where

text: θ

text: represents

text: the

text: neuron’s

text: state

text: and

text: the

text: parame-

text: ij

text: j,

text: this

text: model

text: reduces

text: to

text: that

text: of

text: Luke

text: et

text: ter

text: η

text: speciﬁes

text: its

text: excitability.

text: The

text: dynamics

text: can

text: be

text: visu-

text: III.

text: MEAN

text: FIELD

text: FORMULATION

text: alized

text: as

text: a

text: point

text: traveling

text: around

text: the

text: unit

text: circle

text: (Fig.

text: A

text: neuronal

text: spike

text: is

text: said

text: to

text: occur

text: each

text: time

text: the

text: phase

text: an-

text: We

text: consider

text: the

text: limit

text: of

text: many

text: neurons,

text: N

text: and

text: as-

text: gle

text: of

text: the

text: neuron,

text: θ,

text: crosses

text: the

text: leftmost

text: point

text: at

text: θ

text: π.

text: sume

text: the

text: network

text: is

text: randomly

text: generated

text: from

text: a

text: given

text: de-

text: When

text: η

text: there

text: are

text: two

text: zeros

text: of

text: the

text: right

text: hand

text: side

text: gree

text: distribution

text: P(k)

text: (normalized

text: such

text: that

text: P(k)

text: of

text: Eq.

text: representing

text: a

text: stable

text: rest

text: state

text: (solid

text: circle

text: k

text: N),

text: where

text: k,

text: the

text: node

text: degree,

text: represents

text: a

text: two-vector

text: of

text: in

text: Fig.

text: and

text: an

text: unstable

text: equilibrium

text: (open

text: circle

text: in

text: the

text: in-degree

text: and

text: the

text: out-degree,

text: (k

text: k

text: Addition-

text: Fig.

text: Thus,

text: starting

text: from

text: a

text: typical

text: initial

text: condition,

text: in

text: out

text: ally,

text: we

text: consider

text: an

text: assortativity

text: function

text: a(k

text: k),

text: the

text: state

text: of

text: the

text: neuron

text: goes

text: towards

text: the

text: stable

text: equilib-

text: which

text: speciﬁes

text: the

text: probability

text: of

text: a

text: link

text: from

text: a

text: node

text: of

text: rium

text: at

text: the

text: rest

text: state

text: represented

text: by

text: the

text: ﬁlled

text: circle.

text: A

text: degree

text: k

text: to

text: one

text: of

text: degree

text: k.

text: In

text: this

text: N

text: limit,

text: we

text: resting

text: neuron

text: will

text: spike

text: if

text: an

text: external

text: force

text: pushes

text: its

text: assume

text: that

text: the

text: state

text: of

text: the

text: neurons

text: can

text: be

text: represented

text: state

text: (i.e.

text: the

text: angle

text: θ)

text: from

text: the

text: rest

text: state

text: past

text: the

text: unsta-

text: by

text: a

text: continuous

text: probability

text: distribution,

text: f(θ,

text: η

text: k,

text: t),

text: such

text: ble

text: equilibrium

text: (termed

text: as

text: the

text: ‘spiking

text: threshold’).

text: As

text: η

text: that

text: f(θ,

text: η

text: k,

text: t)dθdη

text: is

text: the

text: probability

text: that

text: a

text: node

text: of

text: de-

text: is

text: increased

text: above

text: the

text: neuron

text: exhibits

text: a

text: Saddle

text: Node

text: gree

text: k

text: has

text: an

text: excitability

text: parameter

text: in

text: the

text: range

text: [η,

text: η

text: bifurcation

text: on

text: an

text: Invariant

text: Cycle

text: (SNIC).

text: In

text: this

text: case

text: dη]

text: and

text: a

text: phase

text: angle

text: in

text: the

text: range

text: [θ,

text: θ

text: dθ]

text: at

text: time

text: t.

text: there

text: are

text: no

text: ﬁxed

text: points

text: (i.e.

text: no

text: zeros

text: of

text: the

text: right

text: hand

text: Since

text: we

text: are

text: assuming

text: that

text: the

text: excitability

text: parameters

text: side

text: of

text: Eq.

text: and

text: the

text: neuron

text: now

text: ﬁres

text: periodically,

text: do

text: not

text: vary

text: with

text: time,

text: we

text: deﬁne

text: g(η

text: k)

text: fdθ,

text: which

text: is

text: as

text: shown

text: in

text: Fig.

text: Note

text: that

text: the

text: neuron

text: does

text: not

text: the

text: time

text: independent

text: distribution

text: of

text: the

text: excitability

text: pa-

text: move

text: at

text: the

text: same

text: rate

text: along

text: the

text: entire

text: circle,

text: and

text: may

text: rameters

text: η

text: in

text: the

text: network

text: for

text: a

text: randomly

text: chosen

text: node

text: go

text: faster

text: or

text: slower

text: around

text: θ

text: π

text: dependent

text: on

text: whether

text: i

text: of

text: degree

text: k.

text: η

text: is

text: less

text: than

text: or

text: greater

text: than

text: respectively

text: (eq.

text: see

text: the

text: In

text: order

text: to

text: describe

text: the

text: synchronization

text: behavior

text: of

text: plot

text: of

text: cos

text: θ)

text: versus

text: time

text: in

text: Fig.

text: this

text: system,

text: we

text: deﬁne

text: the

text: order

text: parameter

text: to

text: The

text: theta

text: neuron

text: model

text: can

text: be

text: extended

text: from

text: a

text: single

text: neuron

text: in

text: isolation

text: to

text: networks

text: of

text: neurons.

text: We

text: consider

text: a

text: system

text: of

text: N

text: theta

text: neurons

text: coupled

text: together

text: in

text: a

text: general

text: N

text: network

text: via

text: pulse-like

text: synaptic

text: signals,

text: Ii,

text: to

text: each

text: neuron

text: R(t)

text: eiθj.

text: i:

text: N

text: θ˙

text: cos

text: θ

text: cos

text: θ

text: )[η

text: I

text: i

text: i

text: i

text: i

text: i

text: N

text: Some

text: authors,

text: such

text: as

text: and

text: order

text: K

text: I

text: A

text: P

text: (θ

text: rameter

text: diﬀerently

text: so

text: as

text: to

text: be

text: weighted

text: with

text: the

text: out-degree

text: at

text: i

text: k

text: ij

text: n

text: j

text: each

text: node,

text: i.e.,

text: R(t)

text: Aijeiθj/

text: Aij

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Neptune was predicted because the calculated orbit of Uranus did not match its observed behavior. Dirac predicted anti-particles as a second solution to his equations before they were observed. Physicists love those stories. We all know them well, as great examples of the power of our calculations. Cautionary tales, such as the belief that light needed an aether to propagate, are often told as steps to a better theory and the lesson in caution, lost. And it is not in the precision of our theories that physicists can claim success. In most cases, our theories have provided us with predictions that, given the ac- curacy of our experiments, were also very discriminating. Newtonian Mechanics and General relativity provide almost identical predictions for many systems. But there are plenty of cases where the expected behavior is very diﬀerent. And we can make experiments that show General Relativity is a better theory. This can be said with certainty or, more exactly, we can be so close to certainty that the diﬀerence is pure technicality. Chances smaller than 1 in 10100 can be called zero, without much abuse of language, after all. We have had success both in obtaining very precise theories, as well as very discriminating ones. Under these circumstances, epistemological considerations were not necessary. Everything works ﬁne if you know epistemology well. But things also work ﬁne if you know nothing about epistemological problems. We have been living in a epistemically safe world. But that is not always the case. As we look at the status of our current theories and how the community works with them, it is not hard to notice that there is something odd going on. We have never-ending discussions about the status of string theory. Those discussion even include the question of whether it is an acceptable theory [2]. We currently can not make observations that would provide evidence that could allow us to choose between string theory and the standard model. Based on that, some physicists claim string theory is not scientiﬁc. And science, we were taught, must be testable. And it must provide one single answer. Or must it not? Inductive reasoning, such as Bayesian methods, can help us answer that ques- tion [3]. Indeed, recent eﬀorts to bring some modern, non dated epistemology to the question [4] have point out that the underdetermination problem is real and something we might have to learn to accept. Underdetermination happens when the predictions of two diﬀerent models are identical. As a consequence, the probability ratio between the two models can not be altered by observing any data. And the initial, often subjective prior guesses remain. Under these circumstances, attempts to determine how to choose between underdetermined theories have caused deep disagreements between supporters and critics of String Theory [5]. In particular, the idea that a theory could be “conﬁrmed” purely by theoretical arguments [6] is a claim met with strong resistance. There is good reason for that. Bayesian methods can be obtained from logical considerations. And they have way to change probabilities from theoretical considerations. In this paper, I will show why inductive methods are a necessity and which conclusions we can draw from them. I will explain why human cognition makes normative methods to choose theories a real necessity [7]. There are too many traps in our brains, we just can’t trust our feelings to pick the best ideas. I will 2

text: Neptune

label: paragraph

text: was

text: predicted

text: because

text: the

text: calculated

text: orbit

text: of

text: Uranus

text: did

text: not

text: match

text: its

text: observed

text: behavior.

text: Dirac

text: predicted

text: anti-particles

text: as

text: a

text: second

text: solution

text: to

text: his

text: equations

text: before

text: they

text: were

text: observed.

text: Physicists

text: love

text: those

text: stories.

text: We

text: all

text: know

text: them

text: well,

text: as

text: great

text: examples

text: of

text: the

text: power

text: of

text: our

text: calculations.

text: Cautionary

text: tales,

text: such

text: as

text: the

text: belief

text: that

text: light

text: needed

text: an

text: aether

text: to

text: propagate,

text: are

text: often

text: told

text: as

text: steps

text: to

text: a

text: better

text: theory

text: and

text: the

text: lesson

text: in

text: caution,

text: lost.

text: And

text: it

text: is

text: not

text: in

text: the

text: precision

text: of

text: our

text: theories

text: that

text: physicists

text: can

text: claim

text: success.

text: In

text: most

text: cases,

text: our

text: theories

text: have

text: provided

text: us

text: with

text: predictions

text: that,

text: given

text: the

text: ac-

text: curacy

text: of

text: our

text: experiments,

text: were

text: also

text: very

text: discriminating.

text: Newtonian

text: Mechanics

text: and

text: General

text: relativity

text: provide

text: almost

text: identical

text: predictions

text: for

text: many

text: systems.

text: But

text: there

text: are

text: plenty

text: of

text: cases

text: where

text: the

text: expected

text: behavior

text: is

text: very

text: diﬀerent.

text: And

text: we

text: can

text: make

text: experiments

text: that

text: show

text: General

text: Relativity

text: is

text: a

text: better

text: theory.

text: This

text: can

text: be

text: said

text: with

text: certainty

text: or,

text: more

text: exactly,

text: we

text: can

text: be

text: so

text: close

text: to

text: certainty

text: that

text: the

text: diﬀerence

text: is

text: pure

text: technicality.

text: Chances

text: smaller

text: than

text: in

text: can

text: be

text: called

text: zero,

text: without

text: much

text: abuse

text: of

text: language,

text: after

text: all.

text: We

text: have

text: had

text: success

text: both

text: in

text: obtaining

text: very

text: precise

text: theories,

text: as

text: well

text: as

text: very

text: discriminating

text: ones.

text: Under

text: these

text: circumstances,

text: epistemological

text: considerations

text: were

text: not

text: necessary.

text: Everything

text: works

text: ﬁne

text: if

text: you

text: know

text: epistemology

text: well.

text: But

text: things

text: also

text: work

text: ﬁne

text: if

text: you

text: know

text: nothing

text: about

text: epistemological

text: problems.

text: We

text: have

text: been

text: living

text: in

text: a

text: epistemically

text: safe

text: world.

text: But

text: that

text: is

text: not

text: always

text: the

text: case.

text: As

text: we

text: look

text: at

text: the

text: status

text: of

text: our

text: current

text: theories

text: and

text: how

text: the

text: community

text: works

text: with

text: them,

text: it

text: is

text: not

text: hard

text: to

text: notice

text: that

text: there

text: is

text: something

text: odd

text: going

text: on.

text: We

text: have

text: never-ending

text: discussions

text: about

text: the

text: status

text: of

text: string

text: theory.

text: Those

text: discussion

text: even

text: include

text: the

text: question

text: of

text: whether

text: it

text: is

text: an

text: acceptable

text: theory

text: We

text: currently

text: can

text: not

text: make

text: observations

text: that

text: would

text: provide

text: evidence

text: that

text: could

text: allow

text: us

text: to

text: choose

text: between

text: string

text: theory

text: and

text: the

text: standard

text: model.

text: Based

text: on

text: that,

text: some

text: physicists

text: claim

text: string

text: theory

text: is

text: not

text: scientiﬁc.

text: And

text: science,

text: we

text: were

text: taught,

text: must

text: be

text: testable.

text: And

text: it

text: must

text: provide

text: one

text: single

text: answer.

text: Or

text: must

text: it

text: not?

text: Inductive

text: reasoning,

text: such

text: as

text: Bayesian

text: methods,

text: can

text: help

text: us

text: answer

text: that

text: ques-

text: tion

text: Indeed,

text: recent

text: eﬀorts

text: to

text: bring

text: some

text: modern,

text: non

text: dated

text: epistemology

text: to

text: the

text: question

text: have

text: point

text: out

text: that

text: the

text: underdetermination

text: problem

text: is

text: real

text: and

text: something

text: we

text: might

text: have

text: to

text: learn

text: to

text: accept.

text: Underdetermination

text: happens

text: when

text: the

text: predictions

text: of

text: two

text: diﬀerent

text: models

text: are

text: identical.

text: As

text: a

text: consequence,

text: the

text: probability

text: ratio

text: between

text: the

text: two

text: models

text: can

text: not

text: be

text: altered

text: by

text: observing

text: any

text: data.

text: And

text: the

text: initial,

text: often

text: subjective

text: prior

text: guesses

text: remain.

text: Under

text: these

text: circumstances,

text: attempts

text: to

text: determine

text: how

text: to

text: choose

text: between

text: underdetermined

text: theories

text: have

text: caused

text: deep

text: disagreements

text: between

text: supporters

text: and

text: critics

text: of

text: String

text: Theory

text: In

text: particular,

text: the

text: idea

text: that

text: a

text: theory

text: could

text: be

text: “conﬁrmed”

text: purely

text: by

text: theoretical

text: arguments

text: is

text: a

text: claim

text: met

text: with

text: strong

text: resistance.

text: There

text: is

text: good

text: reason

text: for

text: that.

text: Bayesian

text: methods

text: can

text: be

text: obtained

text: from

text: logical

text: considerations.

text: And

text: they

text: have

text: way

text: to

text: change

text: probabilities

text: from

text: theoretical

text: considerations.

text: In

text: this

text: paper,

text: I

text: will

text: show

text: why

text: inductive

text: methods

text: are

text: a

text: necessity

text: and

text: which

text: conclusions

text: we

text: can

text: draw

text: from

text: them.

text: I

text: will

text: explain

text: why

text: human

text: cognition

text: makes

text: normative

text: methods

text: to

text: choose

text: theories

text: a

text: real

text: necessity

text: There

text: are

text: too

text: many

text: traps

text: in

text: our

text: brains,

text: we

text: just

text: can’t

text: trust

text: our

text: feelings

text: to

text: pick

text: the

text: best

text: ideas.

text: I

text: will


text: A question that needed an answer was if only data can provide conﬁrmation. String theory seemed to get its support from theoretical arguments, instead of data. I have shown Solomonoﬀ induction can help us understand better these questions. Dawid’s arguments [6] may sound just plainly wrong from a purely Bayesian point of view. But by understanding how they stand in terms of Solomonoﬀ induction, we were able to see their real weaknesses and strengths. Questions about the simplicity and power of a theory actually have an eﬀect on the weight these theories have. In Bayesian terms, the characteristics of a theory can inﬂuence its prior probability. More powerful theories that can be described as smaller algorithms from where everything can be computed correspond to larger Solomonoﬀ weights. They have larger priors. It makes sense to consider String Theory somehow validated by its theoretical characteristics. This should have happened before we start collecting any data. As theories appear after data is collected, we can only correct our analysis later. Finally, the idea that a strategy that provided us veriﬁable and good theories in the past might do the same again poses an interesting question. Our models of normative induction have nothing to say about this problem, as they assume all theories are known before any data is collected. As we are limited, it makes sense to use this kind of heuristic to evaluate which theoretical paths are more likely to produce good theories. While this heuristic might not make string theory more probable, it does suggest it is worth understanding it better. Acknowledgments The author would like to thank the Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP) for partial support to this research under grant 2014/00551-0. References [1] Eugene Wigner. The unreasonable eﬀectiveness of mathematics in the nat- ural sciences. Communications in Pure and Applied Mathematics, 13(1), 1960. [2] George Ellis and Joe Silk. Scientiﬁc method: Defend the integrity of physics. 516:321–323, 2014. [3] E.T. Jaynes. Probability Theory: The Logic of Science. Cambridge, Cam- bridge University Press, 2003. [4] Richard Dawid. String Theory and the Scientiﬁc Method. Cambridge Uni- versity Press, 2013. [5] Davide Castelvecchi. Feuding physicists turn to philosophy for help. 528:446–447, 2015. [6] Richard Dawid. Underdetermination and theory succession from the per- spective of string theory. 73(3):298–322, 2006. 12

text: A

text: question

text: that

text: needed

text: an

text: answer

text: was

text: if

text: only

text: data

text: can

text: provide

text: conﬁrmation.

text: String

text: theory

text: seemed

text: to

text: get

text: its

text: support

text: from

text: theoretical

text: arguments,

text: instead

text: of

text: data.

text: I

text: have

text: shown

text: Solomonoﬀ

text: induction

text: can

text: help

text: us

text: understand

text: better

text: these

text: questions.

text: Dawid’s

text: arguments

text: may

text: sound

text: just

text: plainly

text: wrong

text: from

text: a

text: purely

text: Bayesian

text: point

text: of

text: view.

text: But

text: by

text: understanding

text: how

text: they

text: stand

text: in

text: terms

text: of

text: Solomonoﬀ

text: induction,

text: we

text: were

text: able

text: to

text: see

text: their

text: real

text: weaknesses

text: and

text: strengths.

text: Questions

text: about

text: the

text: simplicity

text: and

text: power

text: of

text: a

text: theory

text: actually

text: have

text: an

text: eﬀect

text: on

text: the

text: weight

text: these

text: theories

text: have.

text: In

text: Bayesian

text: terms,

text: the

text: characteristics

text: of

text: a

text: theory

text: can

text: inﬂuence

text: its

text: prior

text: probability.

text: More

text: powerful

text: theories

text: that

text: can

text: be

text: described

text: as

text: smaller

text: algorithms

text: from

text: where

text: everything

text: can

text: be

text: computed

text: correspond

text: to

text: larger

text: Solomonoﬀ

text: weights.

text: They

text: have

text: larger

text: priors.

text: It

text: makes

text: sense

text: to

text: consider

text: String

text: Theory

text: somehow

text: validated

text: by

text: its

text: theoretical

text: characteristics.

text: This

text: should

text: have

text: happened

text: before

text: we

text: start

text: collecting

text: any

text: data.

text: As

text: theories

text: appear

text: after

text: data

text: is

text: collected,

text: we

text: can

text: only

text: correct

text: our

text: analysis

text: later.

text: Finally,

text: the

text: idea

text: that

text: a

text: strategy

text: that

text: provided

text: us

text: veriﬁable

text: and

text: good

text: theories

text: in

text: the

text: past

text: might

text: do

text: the

text: same

text: again

text: poses

text: an

text: interesting

text: question.

text: Our

text: models

text: of

text: normative

text: induction

text: have

text: nothing

text: to

text: say

text: about

text: this

text: problem,

text: as

text: they

text: assume

text: all

text: theories

text: are

text: known

text: before

text: any

text: data

text: is

text: collected.

text: As

text: we

text: are

text: limited,

text: it

text: makes

text: sense

text: to

text: use

text: this

text: kind

text: of

text: heuristic

text: to

text: evaluate

text: which

text: theoretical

text: paths

text: are

text: more

text: likely

text: to

text: produce

text: good

text: theories.

text: While

text: this

text: heuristic

text: might

text: not

text: make

text: string

text: theory

text: more

text: probable,

text: it

text: does

text: suggest

text: it

text: is

text: worth

text: understanding

text: it

text: better.

text: Acknowledgments

text: The

text: author

text: would

text: like

text: to

text: thank

text: the

text: Funda¸c˜ao

text: de

text: Amparo

text: `a

text: Pesquisa

text: do

text: Estado

text: de

text: S˜ao

text: Paulo

text: (FAPESP)

text: for

text: partial

text: support

text: to

text: this

text: research

text: under

text: grant

text: References

text: Eugene

text: Wigner.

text: The

text: unreasonable

text: eﬀectiveness

text: of

text: mathematics

text: in

text: the

text: nat-

text: ural

text: sciences.

text: Communications

text: in

text: Pure

text: and

text: Applied

text: Mathematics,

text: George

text: Ellis

text: and

text: Joe

text: Silk.

text: Scientiﬁc

text: method:

text: Defend

text: the

text: integrity

text: of

text: physics.

text: E.T.

text: Jaynes.

text: Probability

text: Theory:

text: The

text: Logic

text: of

text: Science.

text: Cambridge,

text: Cam-

text: bridge

text: University

text: Press,

text: Richard

text: Dawid.

text: String

text: Theory

text: and

text: the

text: Scientiﬁc

text: Method.

text: Cambridge

text: Uni-

text: versity

text: Press,

text: Davide

text: Castelvecchi.

text: Feuding

text: physicists

text: turn

text: to

text: philosophy

text: for

text: help.

text: Richard

text: Dawid.

text: Underdetermination

text: and

text: theory

text: succession

text: from

text: the

text: per-

text: spective

text: of

text: string

text: theory.


text: likely to beneﬁt us if we make the throws ourselves [16]. We believe we have far more control over the world and our own reasoning than we actually do. The examples I just presented form a very small sample of a huge literature in how our natural reasoning fails. Our reasoning fails and, still, we trust our brains and our own evaluations when we should not. This means that, when we have a belief, we tend to be too conﬁdent about it. And, since argumentation might be more about convincing than getting the best answer, we are expected to defend our beliefs even when evidence is against us. Add to this the fact that logic and mathematics can prove nothing without initial axioms. No belief about the real world can be proven true. Deduction requires previous knowledge and inductive methods, while immensely helpful, do not provide certainty. Combining those facts, we can see that, when we believe, we harm our ability to look for truth (or best answers). We should actually hold no beliefs and, wherever possible, replace our own analysis for mathematical models [17]. That is exactly what we need to undestand the arguments in the String Theory debate better. 3 Bayes and Solomonoﬀ: Moving beyond Pop- per An outdated but still inﬂuential attempt on using logic to the problem of theory choice was the Popperian notion of falseability [18]. Popper wanted to solve the demarcation problem, to ﬁnd a way to divide theories in scientiﬁc and non-scientiﬁc ones. He proposed that scientiﬁc theories must include the possi- bility that they could be proved false by an experiment. Unfortunately, from a deductive point of view, no theory can be proved true or false. If an experiment fails to produce a predicted outcome, we know something went wrong. One possibility is, indeed, that the theory is false. But it could also be a problem in the experiment itself. Or the calculations that made the prediction. But there are other fundamental problems as well. When we make any calculations, we assume not only the theory to be tested. We also have to use many other auxil- iary hypothesis about the world [19, 20]. When Newtonian Mechanics failed to predict the correct orbit of Uranus, the theory was not the one to blame. The actual error was the incomplete model of the Solar System. The initial calcula- tions did not include the existence of Neptune. Current attempts to reconcile the movement of galaxies with their observed mass by adding dark matter terms do the same thing. Dark matter terms are proposed as an auxiliary hypothesis. Without them, the predictions for the behavior of galaxies would just be wrong. Maybe dark matter exists, maybe the theory is wrong. Experiments always leave something underdetermined. As a matter of fact, one could go on adding new hypothesis as needed to rescue basically any model. The new theory might become a convoluted model, but we might ﬁnd ways to allow any theory to be rescued and still ﬁt observation. The Ptolemaic description of the Solar System did something like that. Popper acknowledged that there is no deductive solid way to say a theory is correct. But there is also no way to determine a theory is wrong. We are left with induction for this task and the uncertainties that come with it. Luckily, 4

text: likely

text: to

text: beneﬁt

text: us

text: if

text: we

text: make

text: the

text: throws

text: ourselves

text: We

text: believe

text: we

text: have

text: far

text: more

text: control

text: over

text: the

text: world

text: and

text: our

text: own

text: reasoning

text: than

text: we

text: actually

text: do.

text: The

text: examples

text: I

text: just

text: presented

text: form

text: a

text: very

text: small

text: sample

text: of

text: a

text: huge

text: literature

text: in

text: how

text: our

text: natural

text: reasoning

text: fails.

text: Our

text: reasoning

text: fails

text: and,

text: still,

text: we

text: trust

text: our

text: brains

text: and

text: our

text: own

text: evaluations

text: when

text: we

text: should

text: not.

text: This

text: means

text: that,

text: when

text: we

text: have

text: a

text: belief,

text: we

text: tend

text: to

text: be

text: too

text: conﬁdent

text: about

text: it.

text: And,

text: since

text: argumentation

text: might

text: be

text: more

text: about

text: convincing

text: than

text: getting

text: the

text: best

text: answer,

text: we

text: are

text: expected

text: to

text: defend

text: our

text: beliefs

text: even

text: when

text: evidence

text: is

text: against

text: us.

text: Add

text: to

text: this

text: the

text: fact

text: that

text: logic

text: and

text: mathematics

text: can

text: prove

text: nothing

text: without

text: initial

text: axioms.

text: No

text: belief

text: about

text: the

text: real

text: world

text: can

text: be

text: proven

text: true.

text: Deduction

text: requires

text: previous

text: knowledge

text: and

text: inductive

text: methods,

text: while

text: immensely

text: helpful,

text: do

text: not

text: provide

text: certainty.

text: Combining

text: those

text: facts,

text: we

text: can

text: see

text: that,

text: when

text: we

text: believe,

text: we

text: harm

text: our

text: ability

text: to

text: look

text: for

text: truth

text: (or

text: best

text: answers).

text: We

text: should

text: actually

text: hold

text: no

text: beliefs

text: and,

text: wherever

text: possible,

text: replace

text: our

text: own

text: analysis

text: for

text: mathematical

text: models

text: That

text: is

text: exactly

text: what

text: we

text: need

text: to

text: undestand

text: the

text: arguments

text: in

text: the

text: String

text: Theory

text: debate

text: better.

text: Bayes

text: and

text: Solomonoﬀ:

text: Moving

text: beyond

text: Pop-

text: per

text: An

text: outdated

text: but

text: still

text: inﬂuential

text: attempt

text: on

text: using

text: logic

text: to

text: the

text: problem

text: of

text: theory

text: choice

text: was

text: the

text: Popperian

text: notion

text: of

text: falseability

text: Popper

text: wanted

text: to

text: solve

text: the

text: demarcation

text: problem,

text: to

text: ﬁnd

text: a

text: way

text: to

text: divide

text: theories

text: in

text: scientiﬁc

text: and

text: non-scientiﬁc

text: ones.

text: He

text: proposed

text: that

text: scientiﬁc

text: theories

text: must

text: include

text: the

text: possi-

text: bility

text: that

text: they

text: could

text: be

text: proved

text: false

text: by

text: an

text: experiment.

text: Unfortunately,

text: from

text: a

text: deductive

text: point

text: of

text: view,

text: no

text: theory

text: can

text: be

text: proved

text: true

text: or

text: false.

text: If

text: an

text: experiment

text: fails

text: to

text: produce

text: a

text: predicted

text: outcome,

text: we

text: know

text: something

text: went

text: wrong.

text: One

text: possibility

text: is,

text: indeed,

text: that

text: the

text: theory

text: is

text: false.

text: But

text: it

text: could

text: also

text: be

text: a

text: problem

text: in

text: the

text: experiment

text: itself.

text: Or

text: the

text: calculations

text: that

text: made

text: the

text: prediction.

text: But

text: there

text: are

text: other

text: fundamental

text: problems

text: as

text: well.

text: When

text: we

text: make

text: any

text: calculations,

text: we

text: assume

text: not

text: only

text: the

text: theory

text: to

text: be

text: tested.

text: We

text: also

text: have

text: to

text: use

text: many

text: other

text: auxil-

text: iary

text: hypothesis

text: about

text: the

text: world

text: When

text: Newtonian

text: Mechanics

text: failed

text: to

text: predict

text: the

text: correct

text: orbit

text: of

text: Uranus,

text: the

text: theory

text: was

text: not

text: the

text: one

text: to

text: blame.

text: The

text: actual

text: error

text: was

text: the

text: incomplete

text: model

text: of

text: the

text: Solar

text: System.

text: The

text: initial

text: calcula-

text: tions

text: did

text: not

text: include

text: the

text: existence

text: of

text: Neptune.

text: Current

text: attempts

text: to

text: reconcile

text: the

text: movement

text: of

text: galaxies

text: with

text: their

text: observed

text: mass

text: by

text: adding

text: dark

text: matter

text: terms

text: do

text: the

text: same

text: thing.

text: Dark

text: matter

text: terms

text: are

text: proposed

text: as

text: an

text: auxiliary

text: hypothesis.

text: Without

text: them,

text: the

text: predictions

text: for

text: the

text: behavior

text: of

text: galaxies

text: would

text: just

text: be

text: wrong.

text: Maybe

text: dark

text: matter

text: exists,

text: maybe

text: the

text: theory

text: is

text: wrong.

text: Experiments

text: always

text: leave

text: something

text: underdetermined.

text: As

text: a

text: matter

text: of

text: fact,

text: one

text: could

text: go

text: on

text: adding

text: new

text: hypothesis

text: as

text: needed

text: to

text: rescue

text: basically

text: any

text: model.

text: The

text: new

text: theory

text: might

text: become

text: a

text: convoluted

text: model,

text: but

text: we

text: might

text: ﬁnd

text: ways

text: to

text: allow

text: any

text: theory

text: to

text: be

text: rescued

text: and

text: still

text: ﬁt

text: observation.

text: The

text: Ptolemaic

text: description

text: of

text: the

text: Solar

text: System

text: did

text: something

text: like

text: that.

text: Popper

text: acknowledged

text: that

text: there

text: is

text: no

text: deductive

text: solid

text: way

text: to

text: say

text: a

text: theory

text: is

text: correct.

text: But

text: there

text: is

text: also

text: no

text: way

text: to

text: determine

text: a

text: theory

text: is

text: wrong.

text: We

text: are

text: left

text: with

text: induction

text: for

text: this

text: task

text: and

text: the

text: uncertainties

text: that

text: come

text: with

text: it.

text: Luckily,


text: a defense with our desire to win arguments rather than seek for the truth. But it is worth asking if there might not be something hidden in those arguments. Taken at face value, the concept that anything other than new data can alter our evaluation is normatively wrong. While true, this statement is about the idealized version of the reasoning process. Limited humans might neglected to evaluate some relevant aspect of the problem. If later, we discover we have missed something, it might make sense to correct our estimates to account for the neglected facts. Indeed, some of the theoretical considerations proposed by Dawid [6] can be understood as such a correction, while at least one of the arguments is wrong. The theoretical consideration that there is no other choice is false. First of all, there might be unconceived alternatives [33] we have yet to ﬁnd. And we already have other proposals for doing quantum gravity. Loop quantum gravity is one example[32]. More than that, if one looks only at the string of all data X we have today, quantum ﬁeld theory and general relativity seem to describe it all well, despite their incompatibilities. An algorithm where one of the two theories is used depending on the problem with a sensible rule to choose is, indeed, compatible with current X. While we might feel that is inelegant and wrong, we should know better than to use any feelings we might have. Normative rules exist exactly because our feelings often lead us astray. Of course, if there was only one algorithm that generated X and we had access to every possible algorithm, we could say we had no choice. Realistically, if we knew of only one algorithm that generated X, we could still say that is all we have for now. But we are not there. The second theoretical argument used to defend String Theory is internal coherence. This argument goes like this. While String Theory was planned as a way to circumvent problems with inﬁnities in quantum ﬁeld theories, it has achieved much more. Gravity can be obtained as a consequence. String Theory can also help us understand better the concept of supersymmetry as well as black holes entropies. Indeed, as string theory seems capable of explaining more and more about the structure and laws of the Universe, it does start to sound more like a solid theory. But our normative rules don’t include the possibility of changing probabilities in this case. It is here that Solomonoﬀ induction can provide a good illustration about what is going on. Matching the string X with a given algorithm is a simple yes or no veriﬁcation. The output either matches and the predictions of the algorithm are recorded with the proper weight, or the output does not match and the algorithm is discarded. The process of obtaining new data means adding to the string X. When we do that, more algorithms stop matching and are discarded, increasing the importance of those that remain. No considerations about the characteristics of the algorithms are relevant at this point. Not when more data is obtained. But that is not true about the whole process. Indeed, one central part of the process is to assign weights to diﬀerent algorithms. The longer they are, the less weight they have. One short algorithm may have a much larger weight or, in Bayesian terms, a larger prior probability. When we are debating theories about 9

text: a

text: defense

text: with

text: our

text: desire

text: to

text: win

text: arguments

text: rather

text: than

text: seek

text: for

text: the

text: truth.

text: But

text: it

text: is

text: worth

text: asking

text: if

text: there

text: might

text: not

text: be

text: something

text: hidden

text: in

text: those

text: arguments.

text: Taken

text: at

text: face

text: value,

text: the

text: concept

text: that

text: anything

text: other

text: than

text: new

text: data

text: can

text: alter

text: our

text: evaluation

text: is

text: normatively

text: wrong.

text: While

text: true,

text: this

text: statement

text: is

text: about

text: the

text: idealized

text: version

text: of

text: the

text: reasoning

text: process.

text: Limited

text: humans

text: might

text: neglected

text: to

text: evaluate

text: some

text: relevant

text: aspect

text: of

text: the

text: problem.

text: If

text: later,

text: we

text: discover

text: we

text: have

text: missed

text: something,

text: it

text: might

text: make

text: sense

text: to

text: correct

text: our

text: estimates

text: to

text: account

text: for

text: the

text: neglected

text: facts.

text: Indeed,

text: some

text: of

text: the

text: theoretical

text: considerations

text: proposed

text: by

text: Dawid

text: can

text: be

text: understood

text: as

text: such

text: a

text: correction,

text: while

text: at

text: least

text: one

text: of

text: the

text: arguments

text: is

text: wrong.

text: The

text: theoretical

text: consideration

text: that

text: there

text: is

text: no

text: other

text: choice

text: is

text: false.

text: First

text: of

text: all,

text: there

text: might

text: be

text: unconceived

text: alternatives

text: we

text: have

text: yet

text: to

text: ﬁnd.

text: And

text: we

text: already

text: have

text: other

text: proposals

text: for

text: doing

text: quantum

text: gravity.

text: Loop

text: quantum

text: gravity

text: is

text: one

text: More

text: than

text: that,

text: if

text: one

text: looks

text: only

text: at

text: the

text: string

text: of

text: all

text: data

text: X

text: we

text: have

text: today,

text: quantum

text: ﬁeld

text: theory

text: and

text: general

text: relativity

text: seem

text: to

text: describe

text: it

text: all

text: well,

text: despite

text: their

text: incompatibilities.

text: An

text: algorithm

text: where

text: one

text: of

text: the

text: two

text: theories

text: is

text: used

text: depending

text: on

text: the

text: problem

text: with

text: a

text: sensible

text: rule

text: to

text: choose

text: is,

text: indeed,

text: compatible

text: with

text: current

text: X.

text: While

text: we

text: might

text: feel

text: that

text: is

text: inelegant

text: and

text: wrong,

text: we

text: should

text: know

text: better

text: than

text: to

text: use

text: any

text: feelings

text: we

text: might

text: have.

text: Normative

text: rules

text: exist

text: exactly

text: because

text: our

text: feelings

text: often

text: lead

text: us

text: astray.

text: Of

text: course,

text: if

text: there

text: was

text: only

text: one

text: algorithm

text: that

text: generated

text: X

text: and

text: we

text: had

text: access

text: to

text: every

text: possible

text: algorithm,

text: we

text: could

text: say

text: we

text: had

text: no

text: choice.

text: Realistically,

text: if

text: we

text: knew

text: of

text: only

text: one

text: algorithm

text: that

text: generated

text: X,

text: we

text: could

text: still

text: say

text: that

text: is

text: all

text: we

text: have

text: for

text: now.

text: But

text: we

text: are

text: not

text: there.

text: The

text: second

text: theoretical

text: argument

text: used

text: to

text: defend

text: String

text: Theory

text: is

text: internal

text: coherence.

text: This

text: argument

text: goes

text: like

text: this.

text: While

text: String

text: Theory

text: was

text: planned

text: as

text: a

text: way

text: to

text: circumvent

text: problems

text: with

text: inﬁnities

text: in

text: quantum

text: ﬁeld

text: theories,

text: it

text: has

text: achieved

text: much

text: more.

text: Gravity

text: can

text: be

text: obtained

text: as

text: a

text: consequence.

text: String

text: Theory

text: can

text: also

text: help

text: us

text: understand

text: better

text: the

text: concept

text: of

text: supersymmetry

text: as

text: well

text: as

text: black

text: holes

text: entropies.

text: Indeed,

text: as

text: string

text: theory

text: seems

text: capable

text: of

text: explaining

text: more

text: and

text: more

text: about

text: the

text: structure

text: and

text: laws

text: of

text: the

text: Universe,

text: it

text: does

text: start

text: to

text: sound

text: more

text: like

text: a

text: solid

text: theory.

text: But

text: our

text: normative

text: rules

text: don’t

text: include

text: the

text: possibility

text: of

text: changing

text: probabilities

text: in

text: this

text: case.

text: It

text: is

text: here

text: that

text: Solomonoﬀ

text: induction

text: can

text: provide

text: a

text: good

text: illustration

text: about

text: what

text: is

text: going

text: on.

text: Matching

text: the

text: string

text: X

text: with

text: a

text: given

text: algorithm

text: is

text: a

text: simple

text: yes

text: or

text: no

text: veriﬁcation.

text: The

text: output

text: either

text: matches

text: and

text: the

text: predictions

text: of

text: the

text: algorithm

text: are

text: recorded

text: with

text: the

text: proper

text: weight,

text: or

text: the

text: output

text: does

text: not

text: match

text: and

text: the

text: algorithm

text: is

text: discarded.

text: The

text: process

text: of

text: obtaining

text: new

text: data

text: means

text: adding

text: to

text: the

text: string

text: X.

text: When

text: we

text: do

text: that,

text: more

text: algorithms

text: stop

text: matching

text: and

text: are

text: discarded,

text: increasing

text: the

text: importance

text: of

text: those

text: that

text: remain.

text: No

text: considerations

text: about

text: the

text: characteristics

text: of

text: the

text: algorithms

text: are

text: relevant

text: at

text: this

text: point.

text: Not

text: when

text: more

text: data

text: is

text: obtained.

text: But

text: that

text: is

text: not

text: true

text: about

text: the

text: whole

text: process.

text: Indeed,

text: one

text: central

text: part

text: of

text: the

text: process

text: is

text: to

text: assign

text: weights

text: to

text: diﬀerent

text: algorithms.

text: The

text: longer

text: they

text: are,

text: the

text: less

text: weight

text: they

text: have.

text: One

text: short

text: algorithm

text: may

text: have

text: a

text: much

text: larger

text: weight

text: or,

text: in

text: Bayesian

text: terms,

text: a

text: larger

text: prior

text: probability.

text: When

text: we

text: are

text: debating

text: theories

text: about


text: 4th order ﬁve-stage Runge-Kutta scheme of Carpenter and Kennedy [38], and the time-step is chosen based on the global estimate C dt = min (cid:107) (cid:107) (cid:107) CF(cid:107)L (cid:107) − (cid:107) , (12) k supx∈Ω C(x) 2 CN Jf L∞(∂Dk) J 1 L∞(Dk) where C = O(N2) is the order-dependent constant in the surface polynomial trace inequality [14] and N C is a tunable global CFL constant. This estimate is derived by bounding the eigenvalues of the spatial CFL DG discretization matrix appearing in the semi-discrete system of ODEs. We note that the usual factor of h arises through the term (cid:13) (cid:13) 1(cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) (cid:13)(cid:107) −(cid:13)(cid:107) = O(h) Jf ∞ J 1 ∞ L (∂Dk) L (Dk) − − − due to the fact that J 1 ∞ = O(h d) and Jf ∞ = O(hd 1) in d dimensions. L (Dk) L (Dk) Finally, in all following experiments, we use τ = τ = 1 and C = 1 unless speciﬁed otherwise. We v σ CFL have arbitrarily chosen these parameters for simplicity, though a more nuanced choice of penalty parameters and CFL constant may improve numerical and computational performance for certain problems. We note that because dt is derived through an upper bound on the spectral radius of the discretization matrix, this estimate of the timestep is rather conservative, and we have observed that it is possible to take C > 1 CFL without losing stability under our choice of timestepping scheme. 6.1 Spectra and choice of penalty parameter We ﬁrst verify the energy stability of the WADG method for arbitrary heterogeneous media. We use a stiﬀness matrix constructed using similarity transforms, such that at every quadrature point, C(x) = ≤ ≤ UDUT , where D is a diagonal matrix with random positive entries d D d and U is a random min ii max unitary matrix. Let A denote the matrix induced by the global semi-discrete DG formulation, such that h the time evolution of the solution v, σ is governed by ∂Q = A Q, ∂t h with Q denotes a vector of degrees of freedom for (v, σ). Figure 1 shows computed eigenvalues of A for − h [d , d ] = [1/10, 1] and [10 5, 1], τ = τ = 0 and τ = τ = 1 under discretization parameters N = 4 min max v σ v σ − and h = 1/4. In both cases, the largest real part of any eigenvalue is O(10 14), verifying the energy stability of the WADG discretization for arbitrary media. For practical simulations, the choice of τ , τ remains to be speciﬁed. Taking τ , τ > 0 results in v σ v σ damping of under-resolved spurious components of the solution [21]; however, a naive selection of these penalty parameters can result in an overly restrictive time-step restriction for stability. We wish to choose (cid:107) (cid:107) τ , τ as large as possible without increasing the value of A when using a central ﬂux (i.e. τ = τ = 0). v σ (cid:107) (cid:107) h v σ For example, in Figure 1, we observe that the value of A for a central ﬂux is roughly half as large as the (cid:107) (cid:107) h (cid:107) (cid:107) value of A when taking τ = τ = 1. We note that the growth in A when τ = τ = 1 is due to the h v σ h v σ large negative real part of the extremal eigenvalues of A , which mirrors observations in [21] that a subset −∞ h of the eigenvalues of A approach as the penalty parameter increases. h Initial numerical experiments suggest that varying the penalty parameters spatially and scaling τ and τ v σ independently of each other can oﬀset the artiﬁcial stiﬀness induced by a naive choice of penalty parameters. For example, material coeﬃcients can be taken into account by scaling the penalty parameters such that (cid:112) ﬂux terms are dimensionally consistent. One example of such a scali(cid:112)ng is (cid:107){{ }}(cid:107) {{ }} 1 τ = γ sup C(x) ρ(x) , τ = γ sup (cid:107){{ }}(cid:107) {{ }}, v v x∈f σ σ x∈f C(x) ρ(x) where the supremum is taken locally over each face and γ , γ are dimensionless constants. We note that the v σ optimal choice of scaling is outside of the scope of this current paper, and will be explored in future work. 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: order

text: ﬁve-stage

text: Runge-Kutta

text: scheme

text: of

text: Carpenter

text: and

text: Kennedy

text: and

text: the

text: time-step

text: is

text: chosen

text: based

text: on

text: the

text: global

text: estimate

text: C

label: equation

text: dt

text: min

text: k

text: supx∈Ω

text: C(x)

text: CN

text: Jf

text: L∞(∂Dk)

text: J

text: L∞(Dk)

text: where

text: C

text: is

text: the

text: order-dependent

text: constant

text: in

text: the

text: surface

text: polynomial

text: trace

text: inequality

text: and

text: N

text: C

text: is

text: a

text: tunable

text: global

text: CFL

text: constant.

text: This

text: estimate

text: is

text: derived

text: by

text: bounding

text: the

text: eigenvalues

text: of

text: the

text: spatial

text: CFL

text: DG

text: discretization

text: matrix

text: appearing

text: in

text: the

text: semi-discrete

text: system

text: of

text: ODEs.

text: We

text: note

text: that

text: the

text: usual

text: factor

text: of

text: h

text: arises

text: through

text: the

text: term

text: O(h)

text: Jf

text: J

text: L

text: (∂Dk)

text: L

text: (Dk)

text: due

text: to

text: the

text: fact

text: that

text: J

text: O(h

text: d)

text: and

text: Jf

text: O(hd

text: in

text: d

text: dimensions.

text: L

text: (Dk)

text: L

text: (Dk)

text: Finally,

text: in

text: all

text: following

text: experiments,

text: we

text: use

text: τ

text: τ

text: and

text: C

text: unless

text: speciﬁed

text: otherwise.

text: We

text: v

text: σ

text: CFL

text: have

text: arbitrarily

text: chosen

text: these

text: parameters

text: for

text: simplicity,

text: though

text: a

text: more

text: nuanced

text: choice

text: of

text: penalty

text: parameters

text: and

text: CFL

text: constant

text: may

text: improve

text: numerical

text: and

text: computational

text: performance

text: for

text: certain

text: problems.

text: We

text: note

text: that

text: because

text: dt

text: is

text: derived

text: through

text: an

text: upper

text: bound

text: on

text: the

text: spectral

text: radius

text: of

text: the

text: discretization

text: matrix,

text: this

text: estimate

text: of

text: the

text: timestep

text: is

text: rather

text: conservative,

text: and

text: we

text: have

text: observed

text: that

text: it

text: is

text: possible

text: to

text: take

text: C

text: CFL

text: without

text: losing

text: stability

text: under

text: our

text: choice

text: of

text: timestepping

text: scheme.

text: Spectra

text: and

text: choice

text: of

text: penalty

text: parameter

text: We

text: ﬁrst

text: verify

text: the

text: energy

text: stability

text: of

text: the

text: WADG

text: method

text: for

text: arbitrary

text: heterogeneous

text: media.

text: We

text: use

text: a

text: stiﬀness

text: matrix

text: constructed

text: using

text: similarity

text: transforms,

text: such

text: that

text: at

text: every

text: quadrature

text: point,

text: C(x)

text: UDUT

text: where

text: D

text: is

text: a

text: diagonal

text: matrix

text: with

text: random

text: positive

text: entries

text: d

text: D

text: d

text: and

text: U

text: is

text: a

text: random

text: min

text: ii

text: max

text: unitary

text: matrix.

text: Let

text: A

text: denote

text: the

text: matrix

text: induced

text: by

text: the

text: global

text: semi-discrete

text: DG

text: formulation,

text: such

text: that

text: h

text: the

text: time

text: evolution

text: of

text: the

text: solution

text: v,

text: σ

text: is

text: governed

text: by

text: ∂Q

text: A

text: Q,

text: ∂t

text: h

text: with

text: Q

text: denotes

text: a

text: vector

text: of

text: degrees

text: of

text: freedom

text: for

text: (v,

text: σ).

text: Figure

text: shows

text: computed

text: eigenvalues

text: of

text: A

text: for

text: h

text: [d

text: d

text: and

text: τ

text: τ

text: and

text: τ

text: τ

text: under

text: discretization

text: parameters

text: N

text: min

text: max

text: v

text: σ

text: v

text: σ

text: and

text: h

text: In

text: both

text: cases,

text: the

text: largest

text: real

text: part

text: of

text: any

text: eigenvalue

text: is

text: verifying

text: the

text: energy

text: stability

text: of

text: the

text: WADG

text: discretization

text: for

text: arbitrary

text: media.

text: For

text: practical

text: simulations,

text: the

text: choice

text: of

text: τ

text: τ

text: remains

text: to

text: be

text: speciﬁed.

text: Taking

text: τ

text: τ

text: results

text: in

text: v

text: σ

text: v

text: σ

text: damping

text: of

text: under-resolved

text: spurious

text: components

text: of

text: the

text: solution

text: however,

text: a

text: naive

text: selection

text: of

text: these

text: penalty

text: parameters

text: can

text: result

text: in

text: an

text: overly

text: restrictive

text: time-step

text: restriction

text: for

text: stability.

text: We

text: wish

text: to

text: choose

text: τ

text: τ

text: as

text: large

text: as

text: possible

text: without

text: increasing

text: the

text: value

text: of

text: A

text: when

text: using

text: a

text: central

text: ﬂux

text: (i.e.

text: τ

text: τ

text: v

text: σ

text: h

text: v

text: σ

text: For

text: example,

text: in

text: Figure

text: we

text: observe

text: that

text: the

text: value

text: of

text: A

text: for

text: a

text: central

text: ﬂux

text: is

text: roughly

text: half

text: as

text: large

text: as

text: the

text: h

text: value

text: of

text: A

text: when

text: taking

text: τ

text: τ

text: We

text: note

text: that

text: the

text: growth

text: in

text: A

text: when

text: τ

text: τ

text: is

text: due

text: to

text: the

text: h

text: v

text: σ

text: h

text: v

text: σ

text: large

text: negative

text: real

text: part

text: of

text: the

text: extremal

text: eigenvalues

text: of

text: A

text: which

text: mirrors

text: observations

text: in

text: that

text: a

text: subset

text: h

text: of

text: the

text: eigenvalues

text: of

text: A

text: approach

text: as

text: the

text: penalty

text: parameter

text: increases.

text: h

text: Initial

text: numerical

text: experiments

text: suggest

text: that

text: varying

text: the

text: penalty

text: parameters

text: spatially

text: and

text: scaling

text: τ

text: and

text: τ

text: v

text: σ

text: independently

text: of

text: each

text: other

text: can

text: oﬀset

text: the

text: artiﬁcial

text: stiﬀness

text: induced

text: by

text: a

text: naive

text: choice

text: of

text: penalty

text: parameters.

text: For

text: example,

text: material

text: coeﬃcients

text: can

text: be

text: taken

text: into

text: account

text: by

text: scaling

text: the

text: penalty

text: parameters

text: such

text: that

text: ﬂux

text: terms

text: are

text: dimensionally

text: consistent.

text: One

text: example

text: of

text: such

text: a

text: is

text: τ

text: γ

text: sup

text: C(x)

text: ρ(x)

text: τ

text: γ

text: sup

text: v

text: v

text: x∈f

text: σ

text: σ

text: x∈f

text: C(x)

text: ρ(x)

text: where

text: the

text: supremum

text: is

text: taken

text: locally

text: over

text: each

text: face

text: and

text: γ

text: γ

text: are

text: dimensionless

text: constants.

text: We

text: note

text: that

text: the

text: v

text: σ

text: optimal

text: choice

text: of

text: scaling

text: is

text: outside

text: of

text: the

text: scope

text: of

text: this

text: current

text: paper,

text: and

text: will

text: be

text: explored

text: in

text: future

text: work.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (a) τv = τσ = 0 (b) τv = τσ = 1 Figure 1: Spectra for N = 4 and h = 1/4, using randomly chosen C(x) with eigenvalues between [d , 1] min at each quadrature point. For both τ = τ = 0 and τ = τ = 1, the largest real part of the spectra is − v σ v σ O(10 14). 6.2 Analytic solutions Next, we study the accuracy and convergence of weight-adjusted DG method for several analytical solutions in linear elasticity. In all cases, the solution is expressed in terms of the displacement vector (cid:0) (cid:1) u(x, t) = (u , . . . , u ). Initial conditions for velocity and stress are computed through 1 d ∂u 1 ∇ ∇ v(x, t) = , σ = C u + uT . ∂t 2 (cid:16) (cid:17) Unless otherwise stated, we report relative L2 erro(cid:80)rs for all components of the solution U = (v, σ) (cid:16) (cid:17) (cid:107)U(cid:107)− U(cid:107) h(cid:107)L2(Ω) = mi=(cid:80)1 (cid:107)Ui − Ui,h(cid:107)2L2(Ω) 1/2 . U (cid:107) (cid:107) 1/2 L2(Ω) m U 2 i=1 i L2(Ω) 6.2.1 Harmonic oscillation of a square We ﬁrst examine convergence on a unit square domain with λ = µ = ρ = 1. The components of the displacement vector are given by u (x, y, t) = cos(ωπt) cos(πx) sin(πy) 1 − u (x, y, t) = cos(ωπt) sin(πx) cos(πy), √ 2 where ω = 2µ. Zero traction boundary conditions are imposed. Figure 2 shows L2 errors computed at time T = 5, using uniform triangular meshes constructed by bisecting a uniform mesh of quadrilaterals along the diagonal. For N = 1, . . . , 5, O(hN+1) rates of convergence are observed when using the penalty ﬂux with τ = τ = v σ 1. When using a central ﬂux (with τ = τ = 0), we observe a so-called “even-odd” pattern [23, 16], where v σ the convergence rate is O(hN) for N odd and between O(hN+1/2) and O(hN+1) for N even. This behavior improves upon the theoretical estimate derived in [9], which is O(hN) for a suﬃciently small time-step size. For quasi-uniform meshes, the optimal rate of convergence of spatial L2 errors under uniform mesh reﬁnement is O(hN+1) [23], which is greater than the O(hN+1/2) rate of convergence which can be proven 20 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: τv

text: τσ

text: (b)

text: τv

text: τσ

text: Figure

text: Spectra

label: caption

text: for

text: N

text: and

text: h

text: using

text: randomly

text: chosen

text: C(x)

text: with

text: eigenvalues

text: between

text: [d

text: min

text: at

text: each

text: quadrature

text: point.

text: For

text: both

text: τ

text: τ

text: and

text: τ

text: τ

text: the

text: largest

text: real

text: part

text: of

text: the

text: spectra

text: is

text: v

text: σ

text: v

text: σ

label: section

text: Analytic

text: solutions

text: Next,

text: we

text: study

text: the

text: accuracy

text: and

text: convergence

text: of

text: weight-adjusted

text: DG

text: method

text: for

text: several

text: analytical

text: solutions

text: in

text: linear

text: elasticity.

text: In

text: all

text: cases,

text: the

text: solution

text: is

text: expressed

text: in

text: terms

text: of

text: the

text: displacement

text: vector

text: u(x,

text: t)

text: (u

text: u

text: Initial

text: conditions

text: for

text: velocity

text: and

text: stress

text: are

text: computed

text: through

text: d

text: ∂u

text: v(x,

text: t)

text: σ

text: C

text: u

text: uT

text: ∂t

text: Unless

text: otherwise

text: stated,

text: we

text: report

text: relative

text: for

text: all

text: components

text: of

text: the

text: solution

text: U

text: (v,

text: σ)

text: U

text: m

text: U

text: i

text: Harmonic

text: oscillation

text: of

text: a

text: square

text: We

text: ﬁrst

text: examine

text: convergence

text: on

text: a

text: unit

text: square

text: domain

text: with

text: λ

text: µ

text: ρ

text: The

text: components

text: of

text: the

text: displacement

text: vector

text: are

text: given

text: by

text: u

text: (x,

text: y,

text: t)

text: cos(ωπt)

text: cos(πx)

text: sin(πy)

text: u

text: (x,

text: y,

text: t)

text: cos(ωπt)

text: sin(πx)

text: cos(πy),

text: where

text: ω

text: Zero

text: traction

text: boundary

text: conditions

text: are

text: imposed.

text: Figure

text: shows

text: errors

text: computed

text: at

text: time

text: T

text: using

text: uniform

text: triangular

text: meshes

text: constructed

text: by

text: bisecting

text: a

text: uniform

text: mesh

text: of

text: quadrilaterals

text: along

text: the

text: diagonal.

text: For

text: N

text: rates

text: of

text: convergence

text: are

text: observed

text: when

text: using

text: the

text: penalty

text: ﬂux

text: with

text: τ

text: τ

text: v

text: σ

text: When

text: using

text: a

text: central

text: ﬂux

text: (with

text: τ

text: τ

text: we

text: observe

text: a

text: so-called

text: “even-odd”

text: pattern

text: where

text: v

text: σ

text: the

text: convergence

text: rate

text: is

text: O(hN)

text: for

text: N

text: odd

text: and

text: between

text: and

text: for

text: N

text: even.

text: This

text: behavior

text: improves

text: upon

text: the

text: theoretical

text: estimate

text: derived

text: in

text: which

text: is

text: O(hN)

text: for

text: a

text: suﬃciently

text: small

text: time-step

text: size.

text: For

text: quasi-uniform

text: meshes,

text: the

text: optimal

text: rate

text: of

text: convergence

text: of

text: spatial

text: errors

text: under

text: uniform

text: mesh

text: reﬁnement

text: is

text: which

text: is

text: greater

text: than

text: the

text: rate

text: of

text: convergence

text: which

text: can

text: be

text: proven

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 100 1 100 1.04 1 1 2.30 − − 10 3 2.65 1 10 3 1 3.00 3.05 1 − 1 − 10 6 10 6 4.02 4.63 1 1 N = 1 N = 1 5.01 5.02 N = 2 N = 2 10−9 N = 3 10−9 1 N = 3 N = 4 5.89 N = 4 N = 5 N = 5 − − 10 12 − 10 12 − 10 1 100 10 1 100 Mesh size h Mesh size h (a) τv = τσ = 0 (central ﬂux) (b) τv = τσ = 1 (penalty ﬂux) Figure 2: Convergence of L2 errors for harmonic oscillation. for dissipative DG discretizations on general meshes [39] (though optimal rates of convergence are often observed in numerical experiments). We note that for N = 4 and N = 5, we observe results for both ﬂuxes which are better than the 4th order accuracy of our time-stepping scheme. This is most likely due to the benign nature of the solution and the choice of timestep (12), which scales as O(h/N2). For N = 4, 5, the results of Figure 2 suggest that the resulting timestep is small enough such that temporal errors of O(dt4) are small relative to spatial discretization errors of O(hN+1). 6.2.2 Rayleigh and Lamb waves Next, we examine the convergence of WADG for Rayleigh and Lamb waves, both of which test the imposition of traction-free boundary conditions. Rayleigh waves are elastic surface wave(cid:18)s which decay exponentially(cid:19)away from the surface. These waves are given by the displacement vector (cid:112) √ u (x, y, t) = (cid:18)e−ωx 1−(cid:19)ξ2 −cos(ω((cid:32)y + crt)) (cid:33) (cid:113)1 ξ2 sin(ω(y + c t)) (cid:113) r + ξ22 − 1 e−ωx 1−2ξµ2+µλ sin(ω(cyo+s(ωc (ty))+/ cr1t)−) ξ2µ , (cid:115) √ r 2µ+λ (cid:18) (cid:19) (cid:112) where ω is the wavespeed, c is the Rayleigh phase velocity c = ξ µ, and ξ satisﬁes r r − − ξ2µ − ξ2 − 2 1 ξ2 1 1 = 0. 2µ + λ 2 In our computations, we use ρ = µ = λ = 1, ξ = 0.949554083888034, and ω = 2π [20]. We solve on × the domain [0, 2] [0, 1] using a sequence of uniform triangular meshes, and enforce traction-free boundary conditions at x = 0 and exact Dirichlet boundary conditions at x = 2. Periodic boundary conditions are applied at y = 0 and y = 1. Lamb waves are supported by elastic waveguides with traction-free (free surface) boundary conditions at the top and bottom of the domain. The displacement of these waves is given by − − − u (x, y, t) = ( kB cos(py) qB cos(qy)) sin(kx ωt) 1 1 2 − − u (x, y, t) = ( pB sin(py) + kB sin(qy)) cos(kx ωt) 2 1 2 21 srorre 2L srorre 2L ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: N

text: N

text: N

text: N

text: N

text: N

text: N

text: N

text: N

text: Mesh

text: size

text: h

text: Mesh

text: size

text: h

text: (a)

text: τv

text: τσ

text: (central

text: ﬂux)

text: (b)

text: τv

text: τσ

text: (penalty

text: ﬂux)

text: Figure

text: Convergence

text: of

text: errors

text: for

text: harmonic

text: oscillation.

text: for

text: dissipative

text: DG

text: discretizations

text: on

text: general

text: meshes

text: (though

text: optimal

text: rates

text: of

text: convergence

text: are

text: often

text: observed

text: in

text: numerical

text: experiments).

text: We

text: note

text: that

text: for

text: N

text: and

text: N

text: we

text: observe

text: results

text: for

text: both

text: ﬂuxes

text: which

text: are

text: better

text: than

text: the

text: order

text: accuracy

text: of

text: our

text: time-stepping

text: scheme.

text: This

text: is

text: most

text: likely

text: due

text: to

text: the

text: benign

text: nature

text: of

text: the

text: solution

text: and

text: the

text: choice

text: of

text: timestep

text: which

text: scales

text: as

text: For

text: N

text: the

text: results

text: of

text: Figure

text: suggest

text: that

text: the

text: resulting

text: timestep

text: is

text: small

text: enough

text: such

text: that

text: temporal

text: errors

text: of

text: are

text: small

text: relative

text: to

text: spatial

text: discretization

text: errors

text: of

text: Rayleigh

text: and

text: Lamb

text: waves

text: Next,

text: we

text: examine

text: the

text: convergence

text: of

text: WADG

text: for

text: Rayleigh

text: and

text: Lamb

text: waves,

text: both

text: of

text: which

text: test

text: the

text: imposition

text: of

text: traction-free

text: boundary

text: conditions.

text: Rayleigh

text: waves

text: are

text: elastic

text: surface

text: which

text: decay

text: from

text: the

text: surface.

text: These

text: waves

text: are

text: given

text: by

text: the

text: displacement

text: vector

text: u

text: (x,

text: y,

text: t)

text: crt))

text: ξ2

text: sin(ω(y

text: c

text: t))

text: r

text: e−ωx

text: 1−2ξµ2+µλ

text: sin(ω(cyo+s(ωc

text: (ty))+/

text: ξ2µ

text: r

text: 2µ+λ

text: where

text: ω

text: is

text: the

text: wavespeed,

text: c

text: is

text: the

text: Rayleigh

text: phase

text: velocity

text: c

text: ξ

text: µ,

text: and

text: ξ

text: satisﬁes

text: r

text: r

text: ξ2µ

text: ξ2

text: ξ2

text: 2µ

text: λ

text: In

text: our

text: computations,

text: we

text: use

text: ρ

text: µ

text: λ

text: ξ

text: and

text: ω

text: 2π

text: We

text: solve

text: on

text: the

text: domain

text: using

text: a

text: sequence

text: of

text: uniform

text: triangular

text: meshes,

text: and

text: enforce

text: traction-free

text: boundary

text: conditions

text: at

text: x

text: and

text: exact

text: Dirichlet

text: boundary

text: conditions

text: at

text: x

text: Periodic

text: boundary

text: conditions

text: are

text: applied

text: at

text: y

text: and

text: y

text: Lamb

text: waves

text: are

text: supported

text: by

text: elastic

text: waveguides

text: with

text: traction-free

text: (free

text: surface)

text: boundary

text: conditions

text: at

text: the

text: top

text: and

text: bottom

text: of

text: the

text: domain.

text: The

text: displacement

text: of

text: these

text: waves

text: is

text: given

text: by

text: u

text: (x,

text: y,

text: t)

text: kB

text: cos(py)

text: qB

text: cos(qy))

text: sin(kx

text: ωt)

text: u

text: (x,

text: y,

text: t)

text: pB

text: sin(py)

text: kB

text: sin(qy))

text: cos(kx

text: ωt)

text: srorre

text: srorre

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6.2.4 Stoneley waves A Stoneley wave is supported along the interface between two solids [43]. Like Rayleigh waves, Stoneley waves decay exponentially away from the interfa(cid:40)ce, and test the eﬀectiveness of numerical ﬂuxes across interfaces. We follow [20, 42] and use discontinuous media deﬁned by (10, 3, 3) y > 0 (ρ, λ, µ) = . (1, 1, 1) y < 0. (cid:40) (cid:0)(cid:0) (cid:1) (cid:1) The displacement vector for a Stoneley wave is then given by (cid:0)(cid:0) (cid:1) (cid:1) − − − u1(x, y, t) = (cid:40)RRee (cid:0)(cid:0)(cid:0)(cid:0)i−kkBb11epBk1be1−pykb+1pkyb+1siBk2Be2ek−b(cid:1)(cid:1)1ksby1syei(ekiy(kx(cid:1)ω(cid:1)−t)ωt), , yy >< 00 − − u2(x, y, t) = RRee ikkbB2p3Bek3be2kpby2py +kbi2ksBB44eekkbb22ssyy eeii((kkxx−ωωtt)) ,, yy >< 00 , (cid:115) (cid:115) where c is the Stoneley wave speed, and st − c2 − c2 k = ω/c , b = 1 st , b = 1 st , j = 1, 2. st jp (2µ + λ )/ρ js (µ )/ρ j j j j j The Stoneley wave speed c can be determined based material parameters and interface conditions, and the st amplitudes B , B , B , B are determined from c up to scaling by a constant. For the parameters used 1 2 3 4 st − in this study, we take c = 0.546981324213884, B = i0.2952173626624, B = 0.6798795208473, B = st − 1 2 3 i0.5220044931212, and B = 0.9339639688697 [20]. We assume k = 1, which gives ω = c . 4 − × − st We solve on the domain [ 1, 1] [ 5, 5], and enforce Dirichlet boundary conditions at all boundaries using the exact solution. Figure 5 shows L2 errors for two uniform meshes of triangles constructed by bisecting a × quadrilateral mesh of K 5K elements. Figure 5b shows errors at time T = 5 when K is even and 1D 1D 1D the mesh is ﬁtted to the interface at y = 0, while Figure 5d shows errors when K is odd and the interface 1D cuts through element interiors. When the mesh is ﬁtted to the interface, computed convergence rates using penalty ﬂuxes match the theoretical O(hN+1/2) rate. When using central ﬂuxes, we observe O(hN) rates of convergence. This matches the behavior observed when using central ﬂuxes for the Rayleigh wave problem, instead of the odd-even pattern of convergence observed when using central ﬂuxes for the harmonic oscillation and Lamb wave solutions. When the mesh is not ﬁtted to the interface exactly, we compute the application of the weight-adjusted mass matrix inverse using a quadrature rule from Xiao and Gimbutas [31] which is exact for degree 2N + 1 polynomials. Since the values of ρ, µ, and λ are positive at all quadrature points, the method is energy stable. However, since the exact solution is discontinuous, the error in elements cut by the interface is O(1), resulting in L2 errors which converge at rate O(h1/2) for both penalty and central ﬂuxes. We have also computed errors on a sequence of unﬁtted unstructured uniform meshes, as well as on a sequence of uniform meshes with randomly perturbed vertex positions. In both cases, similar O(h1/2) rates of convergence were observed. We note that, when using piecewise constant approximations of µ and λ, we observe the same O(h1/2) convergence rate, though errors are roughly twice as large in magnitude. 6.2.5 Convergence to manufactured and reference solutions To check the accuracy of the method for problems with smoothly varying heterogeneous media, we follow [16] and consider a manufactured solution. We assume isotropic media, and incorporate variations into the stiﬀness matrix C by taking λ such that λ(x, y) = λ + λ˜(x, y), 0 24 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Stoneley

text: waves

text: A

text: Stoneley

text: wave

text: is

text: supported

text: along

text: the

text: interface

text: between

text: two

text: solids

text: Like

text: Rayleigh

text: waves,

text: Stoneley

text: waves

text: decay

text: exponentially

text: away

text: from

text: the

text: and

text: test

text: the

text: eﬀectiveness

text: of

text: numerical

text: ﬂuxes

text: across

text: interfaces.

text: We

text: follow

text: and

text: use

text: discontinuous

text: media

text: deﬁned

text: by

text: y

text: (ρ,

text: λ,

text: µ)

text: y

text: The

text: displacement

text: vector

text: for

text: a

text: Stoneley

text: wave

text: is

text: then

text: given

text: by

text: y,

text: t)

text: yy

text: y,

text: t)

text: RRee

text: eeii((kkxx−ωωtt))

text: yy

text: where

text: c

text: is

text: the

text: Stoneley

text: wave

text: speed,

text: and

text: st

text: k

text: ω/c

text: b

text: st

text: b

text: st

text: j

text: st

text: jp

text: λ

text: )/ρ

text: js

text: (µ

text: )/ρ

text: j

text: j

text: j

text: j

text: j

text: The

text: Stoneley

text: wave

text: speed

text: c

text: can

text: be

text: determined

text: based

text: material

text: parameters

text: and

text: interface

text: conditions,

text: and

text: the

text: st

text: amplitudes

text: B

text: B

text: B

text: B

text: are

text: determined

text: from

text: c

text: up

text: to

text: scaling

text: by

text: a

text: constant.

text: For

text: the

text: parameters

text: used

text: st

text: in

text: this

text: study,

text: we

text: take

text: c

text: B

text: B

text: B

text: st

text: and

text: B

text: We

text: assume

text: k

text: which

text: gives

text: ω

text: c

text: st

text: We

text: solve

text: on

text: the

text: domain

text: and

text: enforce

text: Dirichlet

text: boundary

text: conditions

text: at

text: all

text: boundaries

text: using

text: the

text: exact

text: solution.

text: Figure

text: shows

text: errors

text: for

text: two

text: uniform

text: meshes

text: of

text: triangles

text: constructed

text: by

text: bisecting

text: a

text: quadrilateral

text: mesh

text: of

text: K

text: elements.

text: Figure

text: shows

text: errors

text: at

text: time

text: T

text: when

text: K

text: is

text: even

text: and

text: the

text: mesh

text: is

text: ﬁtted

text: to

text: the

text: interface

text: at

text: y

text: while

text: Figure

text: shows

text: errors

text: when

text: K

text: is

text: odd

text: and

text: the

text: interface

text: cuts

text: through

text: element

text: interiors.

text: When

text: the

text: mesh

text: is

text: ﬁtted

text: to

text: the

text: interface,

text: computed

text: convergence

text: rates

text: using

text: penalty

text: ﬂuxes

text: match

text: the

text: theoretical

text: rate.

text: When

text: using

text: central

text: ﬂuxes,

text: we

text: observe

text: O(hN)

text: rates

text: of

text: convergence.

text: This

text: matches

text: the

text: behavior

text: observed

text: when

text: using

text: central

text: ﬂuxes

text: for

text: the

text: Rayleigh

text: wave

text: problem,

text: instead

text: of

text: the

text: odd-even

text: pattern

text: of

text: convergence

text: observed

text: when

text: using

text: central

text: ﬂuxes

text: for

text: the

text: harmonic

text: oscillation

text: and

text: Lamb

text: wave

text: solutions.

text: When

text: the

text: mesh

text: is

text: not

text: ﬁtted

text: to

text: the

text: interface

text: exactly,

text: we

text: compute

text: the

text: application

text: of

text: the

text: weight-adjusted

text: mass

text: matrix

text: inverse

text: using

text: a

text: quadrature

text: rule

text: from

text: Xiao

text: and

text: Gimbutas

text: which

text: is

text: exact

text: for

text: degree

text: polynomials.

text: Since

text: the

text: values

text: of

text: ρ,

text: µ,

text: and

text: λ

text: are

text: positive

text: at

text: all

text: quadrature

text: points,

text: the

text: method

text: is

text: energy

text: stable.

text: However,

text: since

text: the

text: exact

text: solution

text: is

text: discontinuous,

text: the

text: error

text: in

text: elements

text: cut

text: by

text: the

text: interface

text: is

text: resulting

text: in

text: errors

text: which

text: converge

text: at

text: rate

text: for

text: both

text: penalty

text: and

text: central

text: ﬂuxes.

text: We

text: have

text: also

text: computed

text: errors

text: on

text: a

text: sequence

text: of

text: unﬁtted

text: unstructured

text: uniform

text: meshes,

text: as

text: well

text: as

text: on

text: a

text: sequence

text: of

text: uniform

text: meshes

text: with

text: randomly

text: perturbed

text: vertex

text: positions.

text: In

text: both

text: cases,

text: similar

text: rates

text: of

text: convergence

text: were

text: observed.

text: We

text: note

text: that,

text: when

text: using

text: piecewise

text: constant

text: approximations

text: of

text: µ

text: and

text: λ,

text: we

text: observe

text: the

text: same

text: convergence

text: rate,

text: though

text: errors

text: are

text: roughly

text: twice

text: as

text: large

text: in

text: magnitude.

text: Convergence

text: to

text: manufactured

text: and

text: reference

text: solutions

text: To

text: check

text: the

text: accuracy

text: of

text: the

text: method

text: for

text: problems

text: with

text: smoothly

text: varying

text: heterogeneous

text: media,

text: we

text: follow

text: and

text: consider

text: a

text: manufactured

text: solution.

text: We

text: assume

text: isotropic

text: media,

text: and

text: incorporate

text: variations

text: into

text: the

text: stiﬀness

text: matrix

text: C

text: by

text: taking

text: λ

text: such

text: that

text: λ(x,

text: y)

text: λ

text: λ˜(x,

text: y),

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (a) Computational mesh (b) Piecewise constant coeﬃcients (c) Smooth coeﬃcients Figure 12: Mesh and xz, xy slices of v at T = .5. The order of approximation is taken to be N = 5. 1 where w(x) = .5 cos(3πx) cos(3πy) cos(3πz). Forcing is applied to the x-component of velocity through a (cid:0) (cid:1) smoothed point source and Ricker wavelet − − − − − (cid:107) − (cid:107) f(x, t) = 1 2(πf (t t ))2 e (πf0(t t0))2e (a x x0 )2 0 0 where x = (0, 0, .1)T , a = 100, f = 10, and t = 1/f . Figure 12 shows the x-velocity of the computed 0 0 0 0 solution at T = .5, and Figure 12a shows the unstructured mesh of 222824 tetrahedral elements of degree N = 5 used to compute both solutions. In order to capture the discontinuity in material parameters, the elements of this mesh are made to conform to the z = 0 plane. The mesh resolution and degree are chosen to resolve the spatial variation of the smoothed point source present in the forcing function. For piecewise constant coeﬃcients (using the average of each coeﬃcient over an element), spurious reﬂections are observed in the solution. When smoothly varying coeﬃcients are resolved within an element using WADG, these spurious reﬂections disappear. These computations are performed on an Nvidia GTX 980 GPU, following the implementation of GPU- accelerated DG methods outlined in [11]. This approach breaks the computational work for each time-step into volume and surface kernels (for the evaluation of the DG formulation) and an update kernel (for the application of a time integration method). In this implementation, we apply the weight-adjusted mass matrix inverse within the update kernel as well. Strategies for volume and surface kernels follow [11], while computational approaches for WADG are outlined in [2]. Non-invasive GPU-accelerated implementations of WADG are described in [2], where kernels for the acoustic wave equation in isotropic media are re-used. We re-write high order DG methods based on explicit inversion of weighted mass matrices (as done in [16, 17]) into a similar non-invasive form with equivalent storage. The semi-discrete form of standard DG yields a system of ODEs over each element dU M = A U, w dt h where A U denotes the evaluation of the DG right hand side for some local vector U. Multiplying by an h un-weighted mass matrix on both sides gives − dU − M 1M = M 1A U, w dt h (cid:0) (cid:1) The right hand side is the same as the right hand side for the case when the weighting function is w(x) = 1, and can re-use DG kernels for isotropic wave propagation. The inﬂuence of the spatially varying coeﬃcient is incorporated by inverting the weighted projection matrix P = M−1M −1 = M−1M and applying it w w w 31 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: (a)

text: Computational

text: mesh

text: (b)

text: Piecewise

text: constant

text: coeﬃcients

text: (c)

text: Smooth

text: coeﬃcients

text: Figure

text: Mesh

text: and

text: xz,

text: xy

text: slices

text: of

text: v

text: at

text: T

text: The

text: order

text: of

text: approximation

text: is

text: taken

text: to

text: be

text: N

text: where

text: w(x)

text: Forcing

text: is

text: applied

text: to

text: the

text: x-component

text: of

text: velocity

text: through

text: a

text: smoothed

text: point

text: source

text: and

text: Ricker

text: wavelet

text: f(x,

text: t)

text: (t

text: t

text: e

text: (a

text: x

text: where

text: x

text: a

text: f

text: and

text: t

text: Figure

text: shows

text: the

text: x-velocity

text: of

text: the

text: computed

text: solution

text: at

text: T

text: and

text: Figure

text: shows

text: the

text: unstructured

text: mesh

text: of

text: tetrahedral

text: elements

text: of

text: degree

text: N

text: used

text: to

text: compute

text: both

text: solutions.

text: In

text: order

text: to

text: capture

text: the

text: discontinuity

text: in

text: material

text: parameters,

text: the

text: elements

text: of

text: this

text: mesh

text: are

text: made

text: to

text: conform

text: to

text: the

text: z

text: plane.

text: The

text: mesh

text: resolution

text: and

text: degree

text: are

text: chosen

text: to

text: resolve

text: the

text: spatial

text: variation

text: of

text: the

text: smoothed

text: point

text: source

text: present

text: in

text: the

text: forcing

text: function.

text: For

text: piecewise

text: constant

text: coeﬃcients

text: (using

text: the

text: average

text: of

text: each

text: coeﬃcient

text: over

text: an

text: element),

text: spurious

text: reﬂections

text: are

text: observed

text: in

text: the

text: solution.

text: When

text: smoothly

text: varying

text: coeﬃcients

text: are

text: resolved

text: within

text: an

text: element

text: using

text: WADG,

text: these

text: spurious

text: reﬂections

text: disappear.

text: These

text: computations

text: are

text: performed

text: on

text: an

text: Nvidia

text: GTX

text: GPU,

text: following

text: the

text: implementation

text: of

text: GPU-

text: accelerated

text: DG

text: methods

text: outlined

text: in

text: This

text: approach

text: breaks

text: the

text: computational

text: work

text: for

text: each

text: time-step

text: into

text: volume

text: and

text: surface

text: kernels

text: (for

text: the

text: evaluation

text: of

text: the

text: DG

text: formulation)

text: and

text: an

text: update

text: kernel

text: (for

text: the

text: application

text: of

text: a

text: time

text: integration

text: method).

text: In

text: this

text: implementation,

text: we

text: apply

text: the

text: weight-adjusted

text: mass

text: matrix

text: inverse

text: within

text: the

text: update

text: kernel

text: as

text: well.

text: Strategies

text: for

text: volume

text: and

text: surface

text: kernels

text: follow

text: while

text: computational

text: approaches

text: for

text: WADG

text: are

text: outlined

text: in

text: Non-invasive

text: GPU-accelerated

text: implementations

text: of

text: WADG

text: are

text: described

text: in

text: where

text: kernels

text: for

text: the

text: acoustic

text: wave

text: equation

text: in

text: isotropic

text: media

text: are

text: re-used.

text: We

text: re-write

text: high

text: order

text: DG

text: methods

text: based

text: on

text: explicit

text: inversion

text: of

text: weighted

text: mass

text: matrices

text: (as

text: done

text: in

text: into

text: a

text: similar

text: non-invasive

text: form

text: with

text: equivalent

text: storage.

text: The

text: semi-discrete

text: form

text: of

text: standard

text: DG

text: yields

text: a

text: system

text: of

text: ODEs

text: over

text: each

text: element

text: dU

text: M

text: A

text: U,

text: w

text: dt

text: h

text: where

text: A

text: U

text: denotes

text: the

text: evaluation

text: of

text: the

text: DG

text: right

text: hand

text: side

text: for

text: some

text: local

text: vector

text: U.

text: Multiplying

text: by

text: an

text: h

text: un-weighted

text: mass

text: matrix

text: on

text: both

text: sides

text: gives

text: dU

text: M

text: M

text: U,

text: w

text: dt

text: h

text: The

text: right

text: hand

text: side

text: is

text: the

text: same

text: as

text: the

text: right

text: hand

text: side

text: for

text: the

text: case

text: when

text: the

text: weighting

text: function

text: is

text: w(x)

text: and

text: can

text: re-use

text: DG

text: kernels

text: for

text: isotropic

text: wave

text: propagation.

text: The

text: inﬂuence

text: of

text: the

text: spatially

text: varying

text: coeﬃcient

text: is

text: incorporated

text: by

text: inverting

text: the

text: weighted

text: projection

text: matrix

text: P

text: −1

text: and

text: applying

text: it

text: w

text: w

text: w

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##