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text: to directly compare scalar signals. Moreover, it is shown [15] Schmidhuber, J., 2015. Deep learning in neural networks: that for the case of multidimensional signals, the effects of An overview. Neural networks, 61, pp.85-117. nuisance groups (such as those due to changes in pose, per- [16] Lee H, Pham P, Largman Y, Ng AY. Unsupervised feature spective, etc.) which act “externally” (i.e., at each instant of learning for audio classiﬁcation using convolutional deep be- time) can be eliminated either a priori by using invariants, lief networks. In Advances in neural information processing or simultaneously with the the temporal reparameterization. systems 2009 (pp. 1096-1104). [17] Mahony, R., Hamel, T., Morin, P., Malis, E., “Nonlinear Acknowledgements The author thanks Ms. Yixin Gao, Complementary Filters on the Special Linear Group,” Inter- Mr. Qianli Ma, Mr. Sipu Ruan, and Profs. Raman Arora national Journal of Control, 85(10):1557–1573, 2012. and Mauro Maggioni for useful discussions leading to this [18] Hartley, R., Zisserman, A., Multiple View Geometry in Com- work. This work was performed under Ofﬁce of Naval puter Vision, Cambridge University Press, 2003. Research Award N00014-17-1-2142 (GRANT12203025) [19] Gao, Y., “Unsupervised Pattern and Structure Discovery in under code 311. Surgical Data,” PhD Disseration (in progress), April, 2017. References [1] Chirikjian, G.S., Harmonic Analysis for Engineers and Ap- plied Scientists, Dover, Mineola, NY 2016. [2] Chirikjian, G.S., Stochastic Models, Information Theory, and Lie Groups, Vol 2., Birkha¨user, Boston, 2012. [3] Makadia, A., Daniilidis, K.,“Spherical Correlation of Vi- sual Representations for 3D Model Retrieval,” International Journal of Computer Vision, 89(2-3):193–210, 2010. [4] Kamien, M.I., Schwartz, N.L., Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, 2nd ed., North-Holland, New York, 1991. (Dover edition, 2012). [5] Ratcliffe J., Foundations of Hyperbolic Manifolds, Springer, New York, 2006. [6] Cheeger, J., Ebin, D.G., Comparison Theorems in Rieman- nian Geometry, North-Holland Publishing Company, Ams- terdam, 1975. [7] Benedetti, R., Petronio, C., Lectures on Hyperbolic Geome- try, Springer, New York, 2012. [8] Troutman, J.L., Variational Calculus and Optimal Control: Optimization with Elementary Convexity, Springer, New York, 2012. [9] Eberlein P., Geometry of Nonpositively Curved Manifolds, University of Chicago Press, 1996. [10] Berndt, D.J. Clifford, J., 1994, July. Using dynamic time warping to ﬁnd patterns in time series. In KDD workshop (Vol. 10, No. 16, pp. 359-370). [11] Salvador S, Chan P., “Toward Accurate Dynamic Time Warping in Linear Time and Space,” Intelligent Data Analy- sis, 11(5): 561–580, 2007. [12] Rabiner, L., Rosenberg, A. and Levinson, S., 1978. Con- siderations in dynamic time warping algorithms for discrete word recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(6), pp.575-582. [13] Soatto S. Actionable information in vision. InComputer Vi- sion, 2009 IEEE 12th International Conference on 2009 Sep 29 (pp. 2138-2145). IEEE. [14] LeCun, Y., Bengio, Y. and Hinton, G., 2015. Deep learning. Nature, 521(7553), pp.436-444. 9

text: to

label: paragraph

text: directly

text: compare

text: scalar

text: signals.

text: Moreover,

text: it

text: is

text: shown

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text: Schmidhuber,

text: J.,

text: Deep

text: learning

text: in

text: neural

text: networks:

text: that

text: for

text: the

text: case

text: of

text: multidimensional

text: signals,

text: the

text: effects

text: of

text: An

text: overview.

text: Neural

text: networks,

text: nuisance

text: groups

text: (such

text: as

text: those

text: due

text: to

text: changes

text: in

text: pose,

text: per-

text: Lee

text: H,

text: Pham

text: P,

text: Largman

text: Y,

text: Ng

text: AY.

text: Unsupervised

text: feature

text: spective,

text: etc.)

text: which

text: act

text: “externally”

text: (i.e.,

text: at

text: each

text: instant

text: of

text: learning

text: for

text: audio

text: classiﬁcation

text: using

text: convolutional

text: deep

text: be-

text: time)

text: can

text: be

text: eliminated

text: either

text: a

text: priori

text: by

text: using

text: invariants,

text: lief

text: networks.

text: In

text: Advances

text: in

text: neural

text: information

text: processing

text: or

text: simultaneously

text: with

text: the

text: the

text: temporal

text: reparameterization.

text: systems

text: (pp.

text: Mahony,

text: R.,

text: Hamel,

text: T.,

text: Morin,

text: P.,

text: Malis,

text: E.,

text: “Nonlinear

text: Acknowledgements

text: The

text: author

text: thanks

text: Ms.

text: Yixin

text: Gao,

text: Complementary

text: Filters

text: on

text: the

text: Special

text: Linear

text: Group,”

text: Inter-

text: Mr.

text: Qianli

text: Ma,

text: Mr.

text: Sipu

text: Ruan,

text: and

text: Profs.

text: Raman

text: Arora

text: national

text: Journal

text: of

text: Control,

text: and

text: Mauro

text: Maggioni

text: for

text: useful

text: discussions

text: leading

text: to

text: this

text: Hartley,

text: R.,

text: Zisserman,

text: A.,

text: Multiple

text: View

text: Geometry

text: in

text: Com-

text: work.

text: This

text: work

text: was

text: performed

text: under

text: Ofﬁce

text: of

text: Naval

text: puter

text: Vision,

text: Cambridge

text: University

text: Press,

text: Research

text: Award

text: Gao,

text: Y.,

text: “Unsupervised

text: Pattern

text: and

text: Structure

text: Discovery

text: in

text: under

text: code

text: Surgical

text: Data,”

text: PhD

text: Disseration

text: (in

text: progress),

text: April,

text: References

text: Chirikjian,

text: G.S.,

text: Harmonic

text: Analysis

text: for

text: Engineers

text: and

text: Ap-

text: plied

text: Scientists,

text: Dover,

text: Mineola,

text: NY

text: Chirikjian,

text: G.S.,

text: Stochastic

text: Models,

text: Information

text: Theory,

text: and

text: Lie

text: Groups,

text: Vol

text: Birkha¨user,

text: Boston,

text: Makadia,

text: A.,

text: Daniilidis,

text: K.,“Spherical

text: Correlation

text: of

text: Vi-

text: sual

text: Representations

text: for

text: Model

text: Retrieval,”

text: International

text: Journal

text: of

text: Computer

text: Vision,

text: Kamien,

text: M.I.,

text: Schwartz,

text: N.L.,

text: Dynamic

text: Optimization:

text: The

text: Calculus

text: of

text: Variations

text: and

text: Optimal

text: Control

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text: Economics

text: and

text: Management,

text: ed.,

text: North-Holland,

text: New

text: York,

text: (Dover

text: edition,

text: Ratcliffe

text: J.,

text: Foundations

text: of

text: Hyperbolic

text: Manifolds,

text: Springer,

text: New

text: York,

text: Cheeger,

text: J.,

text: Ebin,

text: D.G.,

text: Comparison

text: Theorems

text: in

text: Rieman-

text: nian

text: Geometry,

text: North-Holland

text: Publishing

text: Company,

text: Ams-

text: terdam,

text: Benedetti,

text: R.,

text: Petronio,

text: C.,

text: Lectures

text: on

text: Hyperbolic

text: Geome-

text: try,

text: Springer,

text: New

text: York,

text: Troutman,

text: J.L.,

text: Variational

text: Calculus

text: and

text: Optimal

text: Control:

text: Optimization

text: with

text: Elementary

text: Convexity,

text: Springer,

text: New

text: York,

text: Eberlein

text: P.,

text: Geometry

text: of

text: Nonpositively

text: Curved

text: Manifolds,

text: University

text: of

text: Chicago

text: Press,

text: Berndt,

text: D.J.

text: Clifford,

text: J.,

text: July.

text: Using

text: dynamic

text: time

text: warping

text: to

text: ﬁnd

text: patterns

text: in

text: time

text: series.

text: In

text: KDD

text: workshop

text: (Vol.

text: No.

text: pp.

text: Salvador

text: S,

text: Chan

text: P.,

text: “Toward

text: Accurate

text: Dynamic

text: Time

text: Warping

text: in

text: Linear

text: Time

text: and

text: Space,”

text: Intelligent

text: Data

text: Analy-

text: sis,

text: Rabiner,

text: L.,

text: Rosenberg,

text: A.

text: and

text: Levinson,

text: S.,

text: Con-

text: siderations

text: in

text: dynamic

text: time

text: warping

text: algorithms

text: for

text: discrete

text: word

text: recognition.

text: IEEE

text: Transactions

text: on

text: Acoustics,

text: Speech,

text: and

text: Signal

text: Processing,

text: Soatto

text: S.

text: Actionable

text: information

text: in

text: vision.

text: InComputer

text: Vi-

text: sion,

text: IEEE

text: International

text: Conference

text: on

text: Sep

text: (pp.

text: IEEE.

text: LeCun,

text: Y.,

text: Bengio,

text: Y.

text: and

text: Hinton,

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text: Deep

text: learning.

text: Nature,


text: 14 S. Colombi and C. Alard (cid:48) (cid:48) − − j , with the additional constraints i j = 0 if i i = 1 or j j = 1 to max m m min max min max gain even more time. § Again, similarly as stated at the end of , the above algorithm is not guaranteed to provide a smooth interpolation if the error on the remapping associated to the metric elements becomes larger than some threshold. However, this should have little consequence on the algorithm accuracy and the gain of speed in high dimensional phase- space is considerable with such a procedure. Indeed, if E becomes too large, the map loss of accuracy on the characteristics themselves is much more dramatic than small discontinuities introduced on their interpolation. 2.3.3. Estimates of errors on the Lagrangian position and choice of the margin parameters α and β In the two subsections above, we deﬁned two parameters α and β but did not provide any numerical value. In principle these parameters could be user-deﬁned, but we chose to ﬁx them: √ (cid:32) (cid:33) 3 (cid:39) α = 0.43, (2.41) 4 √ 1 − 3 (cid:39) β = 1 0.067. (2.42) 2 2 The idea behind this choice is the following. Firstly, according to Figure 5, since we do − not want to consider any diagonal element of metric if i i = 1 or j j = 1, min max min max we must have for full consistency, √ (cid:54) − 3 (cid:39) β 1 0.13. (2.43) 2 Because there is some error on the reconstructed value of Q, we take half this value, (cid:32) (cid:33) assuming that √ (cid:54) ≡ 1 − 3 (cid:39) E /∆x , E /∆v E 1 0.067. (2.44) x m y m max 2 2 − Hence α = 1/2 E is given by equation (2.41) to take into account this upper max bound for the errors. With this setting, the algorithm is fully consistent only if the uncertainty on the remapped trajectories remains small enough, but even if E /∆x or (cid:112) x m E /∆v are larger than E , it still works. Simply function Q (P ) is not guaranteed v m max r to be completely smooth anymore. Note that in our algorithm, a global error E = (E /∆x )2 + (E /∆v )2 is map x m v m estimated during the percolation process by comparing the value of Q proposed by a neighbouring element of metric to the value of Q proposed by the element of metric under consideration using the Lagrangian distance d deﬁned in equation (2.32). Such an error can in principle be used to force global remapping if E exceeds some threshold, map for instance E to preserve full smoothness of function Q (P ). While this is an option max § r in our code, we do not use it for the tests performed in . Indeed, we found diﬃcult to ﬁnd some reliable criterion on E and preferred to keep the number n of time steps map s between full resampling ﬁxed and see how the results depend on the choice of n , being s aware of the fact that this is potentially suboptimal. Note also that E and E could be x v ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: Colombi

text: and

text: C.

text: Alard

text: j

text: with

text: the

text: additional

text: constraints

text: i

text: j

text: if

text: i

text: i

text: or

text: j

text: j

text: to

text: max

text: m

text: m

text: min

text: max

text: min

text: max

text: gain

text: even

text: more

text: time.

text: Again,

text: similarly

text: as

text: stated

text: at

text: the

text: end

text: of

text: the

text: above

text: algorithm

text: is

text: not

text: guaranteed

text: to

text: provide

text: a

text: smooth

text: interpolation

text: if

text: the

text: error

text: on

text: the

text: remapping

text: associated

text: to

text: the

text: metric

text: elements

text: becomes

text: larger

text: than

text: some

text: threshold.

text: However,

text: this

text: should

text: have

text: little

text: consequence

text: on

text: the

text: algorithm

text: accuracy

text: and

text: the

text: gain

text: of

text: speed

text: in

text: high

text: dimensional

text: phase-

text: space

text: is

text: considerable

text: with

text: such

text: a

text: procedure.

text: Indeed,

text: if

text: E

text: becomes

text: too

text: large,

text: the

text: map

text: loss

text: of

text: accuracy

text: on

text: the

text: characteristics

text: themselves

text: is

text: much

text: more

text: dramatic

text: than

text: small

text: discontinuities

text: introduced

text: on

text: their

text: interpolation.

text: Estimates

text: of

text: errors

text: on

text: the

text: Lagrangian

text: position

text: and

text: choice

text: of

text: the

text: margin

text: parameters

text: α

text: and

text: β

text: In

text: the

text: two

text: subsections

text: above,

text: we

text: deﬁned

text: two

text: parameters

text: α

text: and

text: β

text: but

text: did

text: not

text: provide

text: any

text: numerical

text: value.

text: In

text: principle

text: these

text: parameters

text: could

text: be

text: user-deﬁned,

text: but

text: we

text: chose

text: to

text: ﬁx

text: them:

text: α

text: β

text: The

text: idea

text: behind

text: this

text: choice

text: is

text: the

text: following.

text: Firstly,

text: according

text: to

text: Figure

text: since

text: we

text: do

text: not

text: want

text: to

text: consider

text: any

text: diagonal

text: element

text: of

text: metric

text: if

text: i

text: i

text: or

text: j

text: j

text: min

text: max

text: min

text: max

text: we

text: must

text: have

text: for

text: full

text: consistency,

text: β

text: Because

text: there

text: is

text: some

text: error

text: on

text: the

text: reconstructed

text: value

text: of

text: Q,

text: we

text: take

text: half

text: this

text: value,

text: assuming

text: that

text: E

text: /∆x

text: E

text: /∆v

text: E

text: x

text: m

text: y

text: m

text: max

text: Hence

text: α

text: E

text: is

text: given

text: by

text: equation

text: to

text: take

text: into

text: account

text: this

text: upper

text: max

text: bound

text: for

text: the

text: errors.

text: With

text: this

text: setting,

text: the

text: algorithm

text: is

text: fully

text: consistent

text: only

text: if

text: the

text: uncertainty

text: on

text: the

text: remapped

text: trajectories

text: remains

text: small

text: enough,

text: but

text: even

text: if

text: E

text: /∆x

text: or

text: x

text: m

text: E

text: /∆v

text: are

text: larger

text: than

text: E

text: it

text: still

text: works.

text: Simply

text: function

text: Q

text: (P

text: is

text: not

text: guaranteed

text: v

text: m

text: max

text: r

text: to

text: be

text: completely

text: smooth

text: anymore.

text: Note

text: that

text: in

text: our

text: algorithm,

text: a

text: global

text: error

text: E

text: (E

text: /∆x

text: (E

text: /∆v

text: is

text: map

text: x

text: m

text: v

text: m

text: estimated

text: during

text: the

text: percolation

text: process

text: by

text: comparing

text: the

text: value

text: of

text: Q

text: proposed

text: by

text: a

text: neighbouring

text: element

text: of

text: metric

text: to

text: the

text: value

text: of

text: Q

text: proposed

text: by

text: the

text: element

text: of

text: metric

text: under

text: consideration

text: using

text: the

text: Lagrangian

text: distance

text: d

text: deﬁned

text: in

text: equation

text: Such

text: an

text: error

text: can

text: in

text: principle

text: be

text: used

text: to

text: force

text: global

text: remapping

text: if

text: E

text: exceeds

text: some

text: threshold,

text: map

text: for

text: instance

text: E

text: to

text: preserve

text: full

text: smoothness

text: of

text: function

text: Q

text: (P

text: While

text: this

text: is

text: an

text: option

text: max

text: r

text: in

text: our

text: code,

text: we

text: do

text: not

text: use

text: it

text: for

text: the

text: tests

text: performed

text: in

text: Indeed,

text: we

text: found

text: diﬃcult

text: to

text: ﬁnd

text: some

text: reliable

text: criterion

text: on

text: E

text: and

text: preferred

text: to

text: keep

text: the

text: number

text: n

text: of

text: time

text: steps

text: map

text: s

text: between

text: full

text: resampling

text: ﬁxed

text: and

text: see

text: how

text: the

text: results

text: depend

text: on

text: the

text: choice

text: of

text: n

text: being

text: s

text: aware

text: of

text: the

text: fact

text: that

text: this

text: is

text: potentially

text: suboptimal.

text: Note

text: also

text: that

text: E

text: and

text: E

text: could

text: be

text: x

text: v

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A “metric” semi-Lagrangian Vlasov-Poisson solver 15 (im-‐1,jm+1) (im,jm+1) Δxm (im+1,jm+1) Δv m/2 h Δx -‐ h x m x 1 2 1 (im-‐1,jm) 3 (im4, jm) 3 hv (im+1,jm) 1 2 1 (i -‐1,j -‐1) (i ,j -‐1) (i +1,j -‐1) m m m m m m Figure 5. Scheme determining the number of neighbouring elements of metric to consider for the interpolation (2.36) used to reconstruct the Lagrangian coordinate Q of a test particle P in the region of inﬂuence of metric element (i , j ). This ﬁgure also justiﬁes equation (2.43). Each m m element of metric has an ellipsoid region of inﬂuence (ocher ellipse) for computing the weights in equation (2.36). Outside this area, the element of metric does not contribute. Therefore, in the √ √ absence of uncertainty on the reconstruction of the Lagrangian coordinate, one can safely deﬁne − − 4 kinds of regions, as shown on the ﬁgure, with h = (1 3/2)∆x and h = (1 3/2)∆v , − x m v − m steaming from the ellipse equations (1 h /∆x )2 + 1/4 = 1 and 1/4 + (1 h /∆v )2 = 1. x− m − v m In region 4, only the cross composed of (i 1, j ), (i + 1, j ), (i , j 1), (i , j + 1) and m m m m m m m m (i , j ) contributes to the interpolation (2.36). In regions 2 and 3, only 4 elements of metric m m − contribute, namely (i 1, j ), (i + 1, j ), (i , j + 1) and (i , j ) for the upper area 2. In m m m m m m m m regions labelled by 1, four elements of metric contribute, e.g. (i , j ), (i + 1, j ), (i , j + 1), m m m m m m (i + 1, j + 1) in the upper right quadrant. In the presence of uncertainty, we need to change m m the extensions of these regions to take it into account, as discussed further in the main text. estimated as the residuals from the third order description of the local metric elements, but this is very costly in highly dimensional phase-space. 2.4. Equations of motion In 2D-dimensional phase-space, the equations characterising the evolution of an ele- ment of metric with time are given by the following expressions with some obvious new notations: ∂x = v, (2.45) ∂t ∂v = a, (2.46) ∂t ∂Ti,j = T(cid:88) , i (cid:54) D, (2.47) ∂t i+D,j D ∂T ∂a i+D,j = i T , (2.48) ∂t ∂x r,j r r=1 ∂Hi,j,k = H(cid:88) , i (cid:54) D(cid:88), (2.49) ∂t i+D,j,k ∂H D ∂a D ∂2a i+D,j,k = i H + i T T , (2.50) ∂t ∂x r,j,k ∂x ∂x r,j m,k r r m r=1 r,m=1 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: “metric”

text: semi-Lagrangian

text: Vlasov-Poisson

text: solver

font: default

text: Δxm

text: Δv

text: h

text: Δx

text: h

text: x

text: m

text: x

text: jm)

text: hv

text: (i

text: (i

text: ,j

text: (i

text: m

text: m

text: m

text: m

text: m

text: m

text: Figure

text: Scheme

label: caption

text: determining

text: the

text: number

text: of

text: neighbouring

text: elements

text: of

text: metric

text: to

text: consider

text: for

text: the

text: interpolation

text: used

text: to

text: reconstruct

text: the

text: Lagrangian

text: coordinate

text: Q

text: of

text: a

text: test

text: particle

text: P

text: in

text: the

text: region

text: of

text: inﬂuence

text: of

text: metric

text: element

text: (i

text: j

text: This

text: ﬁgure

text: also

text: justiﬁes

text: equation

text: Each

text: m

text: m

text: element

text: of

text: metric

text: has

text: an

text: ellipsoid

text: region

text: of

text: inﬂuence

text: (ocher

text: ellipse)

text: for

text: computing

text: the

text: weights

text: in

text: equation

text: Outside

text: this

text: area,

text: the

text: element

text: of

text: metric

text: does

text: not

text: contribute.

text: Therefore,

text: in

text: the

text: absence

text: of

text: uncertainty

text: on

text: the

text: reconstruction

text: of

text: the

text: Lagrangian

text: coordinate,

text: one

text: can

text: safely

text: deﬁne

text: kinds

text: of

text: regions,

text: as

text: shown

text: on

text: the

text: ﬁgure,

text: with

text: h

text: and

text: h

text: x

text: m

text: v

text: m

text: steaming

text: from

text: the

text: ellipse

text: equations

text: h

text: /∆x

text: and

text: h

text: /∆v

text: x−

text: m

text: v

text: m

text: In

text: region

text: only

text: the

text: cross

text: composed

text: of

text: (i

text: j

text: (i

text: j

text: (i

text: j

text: (i

text: j

text: and

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: (i

text: j

text: contributes

text: to

text: the

text: interpolation

text: In

text: regions

text: and

text: only

text: elements

text: of

text: metric

text: m

text: m

text: contribute,

text: namely

text: (i

text: j

text: (i

text: j

text: (i

text: j

text: and

text: (i

text: j

text: for

text: the

text: upper

text: area

text: In

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: regions

text: labelled

text: by

text: four

text: elements

text: of

text: metric

text: contribute,

text: e.g.

text: (i

text: j

text: (i

text: j

text: (i

text: j

text: m

text: m

text: m

text: m

text: m

text: m

text: (i

text: j

text: in

text: the

text: upper

text: right

text: quadrant.

text: In

text: the

text: presence

text: of

text: uncertainty,

text: we

text: need

text: to

text: change

text: m

text: m

text: the

text: extensions

text: of

text: these

text: regions

text: to

text: take

text: it

text: into

text: account,

text: as

text: discussed

text: further

text: in

text: the

text: main

text: text.

text: estimated

text: as

text: the

text: residuals

text: from

text: the

text: third

text: order

text: description

text: of

text: the

text: local

text: metric

text: elements,

text: but

text: this

text: is

text: very

text: costly

text: in

text: highly

text: dimensional

text: phase-space.

text: Equations

text: of

text: motion

text: In

text: phase-space,

text: the

text: equations

text: characterising

text: the

text: evolution

text: of

text: an

text: ele-

text: ment

text: of

text: metric

text: with

text: time

text: are

text: given

text: by

text: the

text: following

text: expressions

text: with

text: some

text: obvious

text: new

text: notations:

text: ∂x

text: v,

text: ∂t

text: ∂v

text: a,

text: ∂t

text: ∂Ti,j

text: i

text: D,

text: ∂t

text: i+D,j

text: D

text: ∂T

text: ∂a

text: i+D,j

text: i

text: T

text: ∂t

text: ∂x

text: r,j

text: r

text: ∂Hi,j,k

text: i

text: ∂t

text: i+D,j,k

text: ∂H

text: D

text: ∂a

text: D

text: i+D,j,k

text: i

text: H

text: i

text: T

text: T

text: ∂t

text: ∂x

text: r,j,k

text: ∂x

text: ∂x

text: r,j

text: m,k

text: r

text: r

text: m

text: ##LTFigure##

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: ##LTLine##


text: 16 S. Colombi and C. Alard where a is the acceleration calculated by solving Poisson equation. Note thus that we need to have access at all times to the gradient and the Hessian of the gravitational force. In these equations we have dropped the “m” subscript for simplicity. The actual time step implementation uses a splitting algorithm with a drift phase during half a time step where only spatial positions are modiﬁed, followed with a kick phase where only velocities are modiﬁed during a full time step using the acceleration computed after the drift, followed again by a drift during half a time step, where spatial positions are modiﬁed using the new velocities. The ﬁrst drift step or predictor is thus written 1 x(t + ∆t/2) = x(t) + v(t)∆t, (2.51) 2 1 (cid:54) T (t + ∆t/2) = T (t) + T (t)∆t, i D, (2.52) i,j i,j 2 i+D,j 1 (cid:54) H (t + ∆t/2) = H (t) + H ∆t, i D, i,j,k i,j,k 2 i+D,j,k (2.53) after which Poisson equation is solved to compute the acceleration a(t + ∆t/2), its gradient and its Hessian. To compute the acceleration, we ﬁrst need to estimate the phase-space distribution function f(x, v, t + ∆t/2) on the Eulerian grid. To map f back to initial conditions (or previous stage after full remapping), we use the elements of metric as transformed by the predictor step. The kick phase reads (cid:88) v(t + ∆t) = v(t) + a(t + ∆t/2)∆t, (2.54) D ∂a T (t + ∆t) = T (t) + i T (t + ∆t/2) ∆t, i+D,j i,j ∂x r,j r r=1 (cid:88) (2.55) D ∂a Hi+D,j,k(t + ∆t) = Hi+(cid:88)D,j,k(t) + ∂xi Hr,j,k(t + ∆t/2) ∆t r r=1 D ∂2a + i T (t + ∆t/2) T (t + ∆t/2) ∆t. (2.56) ∂x ∂x r,j m,k r m r,m=1 Finally the second drift phase, or corrector, is expressed as follows: 1 x(t + ∆t) = x(t + ∆t/2) + v(t + ∆t)∆t, (2.57) 2 1 (cid:54) T (t + ∆t) = T (t + ∆t/2) + T (t + ∆t)∆t, i D, (2.58) i,j i,j 2 i+D,j 1 (cid:54) H (t + ∆t) = H (t + ∆t/2) + H (t + ∆t)∆t, i D. (2.59) i,j,k i,j,k 2 i+D,j,k With our predictor-corrector scheme, it is possible to adopt a slowly variable time step. In this case, we use the following dynamical constraint (see, e.g., Alard and Colombi 2005; Colombi & Touma 2014), ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: Colombi

text: and

text: C.

text: Alard

text: where

text: a

text: is

text: the

text: acceleration

text: calculated

text: by

text: solving

text: Poisson

text: equation.

text: Note

text: thus

text: that

text: we

text: need

text: to

text: have

text: access

text: at

text: all

text: times

text: to

text: the

text: gradient

text: and

text: the

text: Hessian

text: of

text: the

text: gravitational

text: force.

text: In

text: these

text: equations

text: we

text: have

text: dropped

text: the

text: “m”

text: subscript

text: for

text: simplicity.

text: The

text: actual

text: time

text: step

text: implementation

text: uses

text: a

text: splitting

text: algorithm

text: with

text: a

text: drift

text: phase

text: during

text: half

text: a

text: time

text: step

text: where

text: only

text: spatial

text: positions

text: are

text: modiﬁed,

text: followed

text: with

text: a

text: kick

text: phase

text: where

text: only

text: velocities

text: are

text: modiﬁed

text: during

text: a

text: full

text: time

text: step

text: using

text: the

text: acceleration

text: computed

text: after

text: the

text: drift,

text: followed

text: again

text: by

text: a

text: drift

text: during

text: half

text: a

text: time

text: step,

text: where

text: spatial

text: positions

text: are

text: modiﬁed

text: using

text: the

text: new

text: velocities.

text: The

text: ﬁrst

text: drift

text: step

text: or

text: predictor

text: is

text: thus

text: written

text: x(t

text: x(t)

text: v(t)∆t,

text: T

text: (t

text: T

text: (t)

text: T

text: (t)∆t,

text: i

text: D,

text: i,j

text: i,j

text: i+D,j

text: H

text: (t

text: H

text: (t)

text: H

text: ∆t,

text: i

text: D,

text: i,j,k

text: i,j,k

text: i+D,j,k

text: after

text: which

text: Poisson

text: equation

text: is

text: solved

text: to

text: compute

text: the

text: acceleration

text: a(t

text: its

text: gradient

text: and

text: its

text: Hessian.

text: To

text: compute

text: the

text: acceleration,

text: we

text: ﬁrst

text: need

text: to

text: estimate

text: the

text: phase-space

text: distribution

text: function

text: f(x,

text: v,

text: t

text: on

text: the

text: Eulerian

text: grid.

text: To

text: map

text: f

text: back

text: to

text: initial

text: conditions

text: (or

text: previous

text: stage

text: after

text: full

text: remapping),

text: we

text: use

text: the

text: elements

text: of

text: metric

text: as

text: transformed

text: by

text: the

text: predictor

text: step.

text: The

text: kick

text: phase

text: reads

text: v(t

text: ∆t)

text: v(t)

text: a(t

text: D

text: ∂a

text: T

text: (t

text: ∆t)

text: T

text: (t)

text: i

text: T

text: (t

text: ∆t,

text: i+D,j

text: i,j

text: ∂x

text: r,j

text: r

text: D

text: ∂a

text: Hi+D,j,k(t

text: ∆t)

text: ∂xi

text: Hr,j,k(t

text: ∆t

text: r

text: D

text: i

text: T

text: (t

text: T

text: (t

text: ∆t.

text: ∂x

text: ∂x

text: r,j

text: m,k

text: r

text: m

text: Finally

text: the

text: second

text: drift

text: phase,

text: or

text: corrector,

text: is

text: expressed

text: as

text: follows:

text: x(t

text: ∆t)

text: x(t

text: v(t

text: ∆t)∆t,

text: T

text: (t

text: ∆t)

text: T

text: (t

text: T

text: (t

text: ∆t)∆t,

text: i

text: D,

text: i,j

text: i,j

text: i+D,j

text: H

text: (t

text: ∆t)

text: H

text: (t

text: H

text: (t

text: ∆t)∆t,

text: i

text: D.

text: i,j,k

text: i,j,k

text: i+D,j,k

text: With

text: our

text: predictor-corrector

text: scheme,

text: it

text: is

text: possible

text: to

text: adopt

text: a

text: slowly

text: variable

text: time

text: step.

text: In

text: this

text: case,

text: we

text: use

text: the

text: following

text: dynamical

text: constraint

text: (see,

text: e.g.,

text: Alard

text: and

text: Colombi

text: Colombi

text: Touma

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 S. Colombi and C. Alard Designation ∆ ∆ n t m s CPU I 0.002 0.02 25 1.0 II 0.002 0.04 25 1.0 III 0.002 0.01 25 1.2 IV 0.002 0.005 25 1.4 V 0.002 0.02 100 1.1 VI 0.002 0.02 50 1.1 VII 0.002 0.02 10 1.2 VIII 0.002 0.02 5 1.5 IX 0.002 0.02 1 3.2 X 0.001 0.01 25 4.0 XI 0.004 0.04 25 0.4 XII 0.0005 – – 8.8 XIII 0.00707 – – 4.5 XIV 0.001 – – 2.3 XV 0.00141 – – 1.2 XVI 0.002 – – 0.7 Table 1. Parameters of the simulations performed in this work, namely the grid cell size, ∆ = ∆x = ∆y, the initial separation between each metric element, ∆ = ∆x = ∆y , the m m m number of sub-cycles, n , between resamplings and the approximate total CPU time expressed s in units of the time spent for simulation I. The calculation of t was performed for the CPU simulations with initial conditions corresponding to the random set of stationary clouds as § described in , but the results do not change drastically for other initial conditions. A horizontal line separates the runs performed with Vlamet (I–XI) from those performed with the standard splitting algorithm (XII–XVI). Additionally, the time step ∆t was chosen constant, given by ∆t = 0.01, 0.01 and 0.0025 for the stationary, Gaussian and random set of halos, respectively. The two last choices come from the constraints on the time step obtained in the waterbag runs by Colombi & Touma (2014). (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 1 1 − 2 − 3 − 1 1 2 a(x) = w a(i 1) + w2 a(i) + + w a(i + 1) (2.65) 2 2 4 2 2 (cid:98) − (cid:101) i = (x x )/∆x (2.66) mid − − w = (x x )/∆x i (2.67) mid and likewise for its gradient and its second derivative. Note that we also tested linear interpolation, which leads in practice to nearly the same results as TSC for all the numerical tests we discuss in next section. Clearly, the way we estimate the force and its successive derivatives is probably too naive but actually works pretty well for all the experiments we did so we did not bother to try to improve this part of our algorithm. However, this should be investigated in a future work. 3. Numerical experiments To test the method, we performed an ensemble of simulations of which the parameters § are listed in Table 1. In , we compare the results obtained with Vlamet to the standard splitting algorithm of Cheng and Knorr (1976) with cubic B-spline interpolation as well as the extremely accurate entropy conserving waterbag simulations of Colombi & Touma ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: Colombi

text: and

text: C.

text: Alard

text: Designation

label: table

text: n

text: t

text: m

text: s

text: CPU

text: I

text: II

text: III

text: IV

text: V

text: VI

text: VII

text: VIII

text: IX

text: X

text: XI

text: XII

text: XIII

text: XIV

text: XV

text: XVI

text: Table

text: Parameters

text: of

text: the

text: simulations

text: performed

text: in

text: this

text: work,

text: namely

text: the

text: grid

text: cell

text: size,

text: ∆x

text: ∆y,

text: the

text: initial

text: separation

text: between

text: each

text: metric

text: element,

text: ∆x

text: ∆y

text: the

text: m

text: m

text: m

text: number

text: of

text: sub-cycles,

text: n

text: between

text: resamplings

text: and

text: the

text: approximate

text: total

text: CPU

text: time

text: expressed

text: s

text: in

text: units

text: of

text: the

text: time

text: spent

text: for

text: simulation

text: I.

text: The

text: calculation

text: of

text: t

text: was

text: performed

text: for

text: the

text: CPU

text: simulations

text: with

text: initial

text: conditions

text: corresponding

text: to

text: the

text: random

text: set

text: of

text: stationary

text: clouds

text: as

text: described

text: in

text: but

text: the

text: results

text: do

text: not

text: change

text: drastically

text: for

text: other

text: initial

text: conditions.

text: A

text: horizontal

text: line

text: separates

text: the

text: runs

text: performed

text: with

text: Vlamet

text: (I–XI)

text: from

text: those

text: performed

text: with

text: the

text: standard

text: splitting

text: algorithm

text: (XII–XVI).

text: Additionally,

text: the

text: time

text: step

text: ∆t

text: was

text: chosen

text: constant,

text: given

text: by

text: ∆t

text: and

text: for

text: the

text: stationary,

text: Gaussian

text: and

text: random

text: set

text: of

text: halos,

text: respectively.

text: The

text: two

text: last

text: choices

text: come

text: from

text: the

text: constraints

text: on

text: the

text: time

text: step

text: obtained

text: in

text: the

text: waterbag

text: runs

text: by

text: Colombi

text: Touma

text: a(x)

text: w

text: a(i

text: a(i)

text: w

text: a(i

text: i

text: (x

text: x

text: )/∆x

text: mid

text: w

text: (x

text: x

text: )/∆x

text: i

text: mid

text: and

text: likewise

text: for

text: its

text: gradient

text: and

text: its

text: second

text: derivative.

text: Note

text: that

text: we

text: also

text: tested

text: linear

text: interpolation,

text: which

text: leads

text: in

text: practice

text: to

text: nearly

text: the

text: same

text: results

text: as

text: TSC

text: for

text: all

text: the

text: numerical

text: tests

text: we

text: discuss

text: in

text: next

text: section.

text: Clearly,

text: the

text: way

text: we

text: estimate

text: the

text: force

text: and

text: its

text: successive

text: derivatives

text: is

text: probably

text: too

text: naive

text: but

text: actually

text: works

text: pretty

text: well

text: for

text: all

text: the

text: experiments

text: we

text: did

text: so

text: we

text: did

text: not

text: bother

text: to

text: try

text: to

text: improve

text: this

text: part

text: of

text: our

text: algorithm.

text: However,

text: this

text: should

text: be

text: investigated

text: in

text: a

text: future

text: work.

text: Numerical

text: experiments

text: To

text: test

text: the

text: method,

text: we

text: performed

text: an

text: ensemble

text: of

text: simulations

text: of

text: which

text: the

text: parameters

text: are

text: listed

text: in

text: Table

text: In

text: we

text: compare

text: the

text: results

text: obtained

text: with

text: Vlamet

text: to

text: the

text: standard

text: splitting

text: algorithm

text: of

text: Cheng

text: and

text: Knorr

text: with

text: cubic

text: B-spline

text: interpolation

text: as

text: well

text: as

text: the

text: extremely

text: accurate

text: entropy

text: conserving

text: waterbag

text: simulations

text: of

text: Colombi

text: Touma

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text: 22 S. Colombi and C. Alard Figure 7. Deviation from stationarity for simulations with ∆ = 0.002. Left panel shows the diﬀerence δf between the phase-space density measured at t = 100 and the initial one for simulation I. Darker regions correspond to larger values of δf, which ranges in the interval ∈ − δf [ 0.0014, 0.0014]. On right panel, the value of δf(x, 0) is plotted for simulations I, II, III, IV (i.e. varying inter-element of metric distance ∆ while keeping spatial resolution ∆ = 0.002 m ﬁxed) and XVI. Note that the purple and blue curves nearly superpose on each other. In practice, to compare the output of Vlamet to the waterbag result of CT14, we take R = 1, ρ = 4, σ = 0.2 and η = 0.02, which makes the total mass of the system G G G approximately equal to unity for a Gaussian truncated at 5 sigmas. The measured maximum projected density of this system, is, after relaxation, ρ = 2.95, so simulating max this system up to t = 100 corresponds to following it during approximately 39 dynamical times according to equation (3.6). Figure 8 shows, at various times, the phase-space distribution function obtained from run I and the waterbag run Gaussian84 of CT14. The agreement between both simulations seems visually close to perfect. A more careful examination of the details of the phase-space distribution function shows small deviations which can be reduced by augmenting the resolution of the Vlamet run. This is illustrated by Fig. 9, which displays on a small interval, the projected density for runs I, X, XI (diﬀerent resolutions ∆ but ﬁxed ns = 25 and ﬁxed ratio ∆m/∆ = 10) and Gaussian84. What is particularly interesting in this ﬁgure are the turquoise curves corresponding to runs performed with the splitting algorithm, namely XII, XIV and XVI. We notice a very good mach between Vlamet and the splitting algorithm but only if this latter is run at twice the resolution of Vlamet! This ﬁgure already demonstrates the signiﬁcantly weaker level of diﬀusion of Vlamet compared to the splitting algorithm. • The ensemble of stationary clouds corresponds to a set of halos following the stationary solution (3.3), created exactly as in CT14. In particular, their proﬁle follows equation (3.3) with ρ = 6 and their velocity dispersion, σ , ranges in the interval S S [0.005, 0.1]. The components are added on the top of each other in phase-space, to obtain the desired initial distribution function f (x, v), which is apodized as in equation (3.4) ini with η = 0.05. The interesting fact about this system is that it was found experimentally by CT14 to be chaotic. In ﬁgure 10, the phase-space distribution function is shown at various times for simulation I and run Random of CT14. At t = 25, the agreement between both simulations is impressive, despite the fact that the phase-space distribution function is sampled with only three waterbags in the Random run of CT14. At t = 80, the system presents remarkable structures: some halos have been destroyed by tidal eﬀects and a complex ﬁlamentary structure has grown. At this time, the Vlamet and the waterbag runs diverge from each other, but remain still in agreement at the macroscopic level. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: Colombi

text: and

text: C.

text: Alard

text: Figure

text: Deviation

text: from

text: stationarity

text: for

text: simulations

text: with

text: Left

text: panel

text: shows

text: the

text: diﬀerence

text: δf

text: between

text: the

text: phase-space

text: density

text: measured

text: at

text: t

text: and

text: the

text: initial

text: one

text: for

text: simulation

text: I.

text: Darker

text: regions

text: correspond

text: to

text: larger

text: values

text: of

text: δf,

text: which

text: ranges

text: in

text: the

text: interval

text: δf

text: On

text: right

text: panel,

text: the

text: value

text: of

text: δf(x,

text: is

text: plotted

text: for

text: simulations

text: I,

text: II,

text: III,

text: IV

text: (i.e.

text: varying

text: inter-element

text: of

text: metric

text: distance

text: while

text: keeping

text: spatial

text: resolution

text: m

text: ﬁxed)

text: and

text: XVI.

text: Note

text: that

text: the

text: purple

text: and

text: blue

text: curves

text: nearly

text: superpose

text: on

text: each

text: other.

text: In

text: practice,

text: to

text: compare

text: the

text: output

text: of

text: Vlamet

text: to

text: the

text: waterbag

text: result

text: of

text: we

text: take

text: R

text: ρ

text: σ

text: and

text: η

text: which

text: makes

text: the

text: total

text: mass

text: of

text: the

text: system

text: G

text: G

text: G

text: approximately

text: equal

text: to

text: unity

text: for

text: a

text: Gaussian

text: truncated

text: at

text: sigmas.

text: The

text: measured

text: maximum

text: projected

text: density

text: of

text: this

text: system,

text: is,

text: after

text: relaxation,

text: ρ

text: so

text: simulating

text: max

text: this

text: system

text: up

text: to

text: t

text: corresponds

text: to

text: following

text: it

text: during

text: approximately

text: dynamical

text: times

text: according

text: to

text: equation

text: Figure

text: shows,

text: at

text: various

text: times,

text: the

text: phase-space

text: distribution

text: function

text: obtained

text: from

text: run

text: I

text: and

text: the

text: waterbag

text: run

text: of

text: The

text: agreement

text: between

text: both

text: simulations

text: seems

text: visually

text: close

text: to

text: perfect.

text: A

text: more

text: careful

text: examination

text: of

text: the

text: details

text: of

text: the

text: phase-space

text: distribution

text: function

text: shows

text: small

text: deviations

text: which

text: can

text: be

text: reduced

text: by

text: augmenting

text: the

text: resolution

text: of

text: the

text: Vlamet

text: run.

text: This

text: is

text: illustrated

text: by

text: Fig.

text: which

text: displays

text: on

text: a

text: small

text: interval,

text: the

text: projected

text: density

text: for

text: runs

text: I,

text: X,

text: XI

text: (diﬀerent

text: resolutions

text: but

text: ﬁxed

text: ns

text: and

text: ﬁxed

text: ratio

text: ∆m/∆

text: and

text: What

text: is

text: particularly

text: interesting

text: in

text: this

text: ﬁgure

text: are

text: the

text: turquoise

text: curves

text: corresponding

text: to

text: runs

text: performed

text: with

text: the

text: splitting

text: algorithm,

text: namely

text: XII,

text: XIV

text: and

text: XVI.

text: We

text: notice

text: a

text: very

text: good

text: mach

text: between

text: Vlamet

text: and

text: the

text: splitting

text: algorithm

text: but

text: only

text: if

text: this

text: latter

text: is

text: run

text: at

text: twice

text: the

text: resolution

text: of

text: Vlamet!

text: This

text: ﬁgure

text: already

text: demonstrates

text: the

text: signiﬁcantly

text: weaker

text: level

text: of

text: diﬀusion

text: of

text: Vlamet

text: compared

text: to

text: the

text: splitting

text: algorithm.

text: The

text: ensemble

text: of

text: stationary

text: clouds

text: corresponds

text: to

text: a

text: set

text: of

text: halos

text: following

text: the

text: stationary

text: solution

text: created

text: exactly

text: as

text: in

text: In

text: particular,

text: their

text: proﬁle

text: follows

text: equation

text: with

text: ρ

text: and

text: their

text: velocity

text: dispersion,

text: σ

text: ranges

text: in

text: the

text: interval

text: S

text: S

text: The

text: components

text: are

text: added

text: on

text: the

text: top

text: of

text: each

text: other

text: in

text: phase-space,

text: to

text: obtain

text: the

text: desired

text: initial

text: distribution

text: function

text: f

text: (x,

text: v),

text: which

text: is

text: apodized

text: as

text: in

text: equation

text: ini

text: with

text: η

text: The

text: interesting

text: fact

text: about

text: this

text: system

text: is

text: that

text: it

text: was

text: found

text: experimentally

text: by

text: to

text: be

text: chaotic.

text: In

text: ﬁgure

text: the

text: phase-space

text: distribution

text: function

text: is

text: shown

text: at

text: various

text: times

text: for

text: simulation

text: I

text: and

text: run

text: Random

text: of

text: At

text: t

text: the

text: agreement

text: between

text: both

text: simulations

text: is

text: impressive,

text: despite

text: the

text: fact

text: that

text: the

text: phase-space

text: distribution

text: function

text: is

text: sampled

text: with

text: only

text: three

text: waterbags

text: in

text: the

text: Random

text: run

text: of

text: At

text: t

text: the

text: system

text: presents

text: remarkable

text: structures:

text: some

text: halos

text: have

text: been

text: destroyed

text: by

text: tidal

text: eﬀects

text: and

text: a

text: complex

text: ﬁlamentary

text: structure

text: has

text: grown.

text: At

text: this

text: time,

text: the

text: Vlamet

text: and

text: the

text: waterbag

text: runs

text: diverge

text: from

text: each

text: other,

text: but

text: remain

text: still

text: in

text: agreement

text: at

text: the

text: macroscopic

text: level.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A “metric” semi-Lagrangian Vlasov-Poisson solver 23 Figure 8. Phase-space density in the simulations with Gaussian initial conditions. The simulation I with the metric method is compared to a waterbag run of Colombi & Touma (2014). In the later case, the phase-space distribution function is represented with 84 waterbags. ##LTFigure##

text: A

text: “metric”

text: semi-Lagrangian

text: Vlasov-Poisson

text: solver

text: Figure

text: Phase-space

text: density

text: in

text: the

text: simulations

text: with

text: Gaussian

text: initial

text: conditions.

text: The

text: simulation

text: I

text: with

text: the

text: metric

text: method

text: is

text: compared

text: to

text: a

text: waterbag

text: run

text: of

text: Colombi

text: Touma

text: In

text: the

text: later

text: case,

text: the

text: phase-space

text: distribution

text: function

text: is

text: represented

text: with

text: waterbags.

text: ##LTFigure##

label: figure


text: A “metric” semi-Lagrangian Vlasov-Poisson solver 25 Figure 10. Phase-space density in the simulations with random initial conditions. The simulation I with the metric method is compared to a waterbag run of Colombi & Touma (2014). In the later case, the phase-space distribution function is represented with only with three waterbag levels. ##LTFigure##

text: A

text: “metric”

text: semi-Lagrangian

text: Vlasov-Poisson

text: solver

text: Figure

text: Phase-space

text: density

text: in

text: the

text: simulations

text: with

text: random

text: initial

text: conditions.

text: The

text: simulation

text: I

text: with

text: the

text: metric

text: method

text: is

text: compared

text: to

text: a

text: waterbag

text: run

text: of

text: Colombi

text: Touma

text: In

text: the

text: later

text: case,

text: the

text: phase-space

text: distribution

text: function

text: is

text: represented

text: with

text: only

text: with

text: three

text: waterbag

text: levels.

text: ##LTFigure##


text: 26 S. Colombi and C. Alard where a(x, t) is the gravitational force and M the total mass. These integral are total computed directly as sums on the grid used to sample the phase-space distribution (cid:90) function and the force. In addition, we check for conservation of the following Casimir − | | S = f(x, v) log f(x, v) dxdv, (3.11) which is numerically estimated the same way as total energy. This quantity would reduce to Gibbs entropy if positivity of the phase-space distribution function was pre- served. While we know it is not rigorously correct, we shall still designate S by entropy. We also tested for total mass conservation, as well as L1 and L2 norms, but do not show the results here, to avoid superﬂuous content. Indeed, energy and entropy were found in practice to be suﬃciently representative of the system when it comes to test the accuracy of Vlamet. Figures 11, 12 and 13 show E and S in the simulations we performed for the stationary, the Gaussian and the random set of halos, respectively. These ﬁgures allow us to test thoroughly the performances of Vlamet and to compare it to the splitting algorithm. They are supplemented with Figure 14, which displays the corresponding maximum mismatch between the Lagrangian positions predicates from neighbouring metric elements, as § deﬁned in . The most striking result when examining Figs. 12 and 13 is the superior behaviour of Vlamet compared to the standard splitting method with respect to entropy, in agreement with the preliminary conclusions driven from the analysis of Fig. 9. Indeed, to obtain comparable entropy conservation in Vlamet and the splitting algorithm, it is required to have a higher spatial resolution in the latter. Of course, the results depend on the choice of ns and of the ratio ∆xm/∆. For instance, Vlamet runs with ns = 25 and ∆xm/∆ = 10 perform as well as the splitting algorithm with twice the spatial resolution. But if n is s increased to 100, it is required to increase the resolution by a factor as large as 2.8 (middle right panel of Fig. 13) or 4 (middle right panel of Fig. 12) in the splitting algorithm to perform as well as Vlamet. The price to pay might be energy conservation, which is in general not as good in Vlamet as in the splitting algorithm, that preserves energy at a ∼ − level better than 10 5 regardless of initial conditions and spatial resolution. However, energy conservation violation in Vlamet can be kept under control with the appropriate combination of parameter n and ratio ∆ /∆, as discussed further below. On the same s m ground, we notice, when examining Fig. 11, that the splitting algorithm seems to preserve better the thermal equilibrium solution than Vlamet, as already noticed on right panel of Fig. 7, unless n or/and ∆ /∆ are chosen small enough. However, one has to be aware s m that the smoothness of the thermal equilibrium solution provides a very good ground for the splitting algorithm. Further investigation of Figs. 11, 12, 13 and 14 allows us to study more in details how Vlamet behaves with the various control parameters: • Increasing overall resolution, i.e. decreasing ∆ = ∆x = ∆y and ∆ = ∆x = ∆y m m m by some factor improves both energy conservation and entropy conservation, as can be seen from last line of panels of all the ﬁgures, as expected. Note that adopting ∆x = ∆y and ∆x = ∆y as we did in this work is not necessarily optimal, but is a natural choice m m in the code units if total mass of the system is of the order of unity. • Changing the resampling frequency parameter n aﬀects both entropy and energy, s but to a much larger extent the former than the latter. In particular, increasing n up s to 100 clearly improves entropy at an acceptable cost for energy conservation, of which ∼ × − the violation remains below the 5 10 4 level, as can be seen from the second line of

text: S.

text: Colombi

text: and

text: C.

text: Alard

text: where

text: a(x,

text: t)

text: is

text: the

text: gravitational

text: force

text: and

text: M

text: the

text: total

text: mass.

text: These

text: integral

text: are

text: total

text: computed

text: directly

text: as

text: sums

text: on

text: the

text: grid

text: used

text: to

text: sample

text: the

text: phase-space

text: distribution

label: equation

text: function

text: and

text: the

text: force.

text: In

text: addition,

text: we

text: check

text: for

text: conservation

text: of

text: the

text: following

text: Casimir

text: S

text: f(x,

text: v)

text: log

text: f(x,

text: v)

text: dxdv,

text: which

text: is

text: numerically

text: estimated

text: the

text: same

text: way

text: as

text: total

text: energy.

text: This

text: quantity

text: would

text: reduce

text: to

text: Gibbs

text: entropy

text: if

text: positivity

text: of

text: the

text: phase-space

text: distribution

text: function

text: was

text: pre-

text: served.

text: While

text: we

text: know

text: it

text: is

text: not

text: rigorously

text: correct,

text: we

text: shall

text: still

text: designate

text: S

text: by

text: entropy.

text: We

text: also

text: tested

text: for

text: total

text: mass

text: conservation,

text: as

text: well

text: as

text: and

text: norms,

text: but

text: do

text: not

text: show

text: the

text: results

text: here,

text: to

text: avoid

text: superﬂuous

text: content.

text: Indeed,

text: energy

text: and

text: entropy

text: were

text: found

text: in

text: practice

text: to

text: be

text: suﬃciently

text: representative

text: of

text: the

text: system

text: when

text: it

text: comes

text: to

text: test

text: the

text: accuracy

text: of

text: Vlamet.

text: Figures

text: and

text: show

text: E

text: and

text: S

text: in

text: the

text: simulations

text: we

text: performed

text: for

text: the

text: stationary,

text: the

text: Gaussian

text: and

text: the

text: random

text: set

text: of

text: halos,

text: respectively.

text: These

text: ﬁgures

text: allow

text: us

text: to

text: test

text: thoroughly

text: the

text: performances

text: of

text: Vlamet

text: and

text: to

text: compare

text: it

text: to

text: the

text: splitting

text: algorithm.

text: They

text: are

text: supplemented

text: with

text: Figure

text: which

text: displays

text: the

text: corresponding

text: maximum

text: mismatch

text: between

text: the

text: Lagrangian

text: positions

text: predicates

text: from

text: neighbouring

text: metric

text: elements,

text: as

text: deﬁned

text: in

text: The

text: most

text: striking

text: result

text: when

text: examining

text: Figs.

text: and

text: is

text: the

text: superior

text: behaviour

text: of

text: Vlamet

text: compared

text: to

text: the

text: standard

text: splitting

text: method

text: with

text: respect

text: to

text: entropy,

text: in

text: agreement

text: with

text: the

text: preliminary

text: conclusions

text: driven

text: from

text: the

text: analysis

text: of

text: Fig.

text: Indeed,

text: to

text: obtain

text: comparable

text: entropy

text: conservation

text: in

text: Vlamet

text: and

text: the

text: splitting

text: algorithm,

text: it

text: is

text: required

text: to

text: have

text: a

text: higher

text: spatial

text: resolution

text: in

text: the

text: latter.

text: Of

text: course,

text: the

text: results

text: depend

text: on

text: the

text: choice

text: of

text: ns

text: and

text: of

text: the

text: ratio

text: ∆xm/∆.

text: For

text: instance,

text: Vlamet

text: runs

text: with

text: ns

text: and

text: ∆xm/∆

text: perform

text: as

text: well

text: as

text: the

text: splitting

text: algorithm

text: with

text: twice

text: the

text: spatial

text: resolution.

text: But

text: if

text: n

text: is

text: s

text: increased

text: to

text: it

text: is

text: required

text: to

text: increase

text: the

text: resolution

text: by

text: a

text: factor

text: as

text: large

text: as

text: (middle

text: right

text: panel

text: of

text: Fig.

text: or

text: (middle

text: right

text: panel

text: of

text: Fig.

text: in

text: the

text: splitting

text: algorithm

text: to

text: perform

text: as

text: well

text: as

text: Vlamet.

text: The

text: price

text: to

text: pay

text: might

text: be

text: energy

text: conservation,

text: which

text: is

text: in

text: general

text: not

text: as

text: good

text: in

text: Vlamet

text: as

text: in

text: the

text: splitting

text: algorithm,

text: that

text: preserves

text: energy

text: at

text: a

text: level

text: better

text: than

text: regardless

text: of

text: initial

text: conditions

text: and

text: spatial

text: resolution.

text: However,

text: energy

text: conservation

text: violation

text: in

text: Vlamet

text: can

text: be

text: kept

text: under

text: control

text: with

text: the

text: appropriate

text: combination

text: of

text: parameter

text: n

text: and

text: ratio

text: as

text: discussed

text: further

text: below.

text: On

text: the

text: same

text: s

text: m

text: ground,

text: we

text: notice,

text: when

text: examining

text: Fig.

text: that

text: the

text: splitting

text: algorithm

text: seems

text: to

text: preserve

text: better

text: the

text: thermal

text: equilibrium

text: solution

text: than

text: Vlamet,

text: as

text: already

text: noticed

text: on

text: right

text: panel

text: of

text: Fig.

text: unless

text: n

text: or/and

text: are

text: chosen

text: small

text: enough.

text: However,

text: one

text: has

text: to

text: be

text: aware

text: s

text: m

text: that

text: the

text: smoothness

text: of

text: the

text: thermal

text: equilibrium

text: solution

text: provides

text: a

text: very

text: good

text: ground

text: for

text: the

text: splitting

text: algorithm.

text: Further

text: investigation

text: of

text: Figs.

text: and

text: allows

text: us

text: to

text: study

text: more

text: in

text: details

text: how

text: Vlamet

text: behaves

text: with

text: the

text: various

text: control

text: parameters:

text: Increasing

text: overall

text: resolution,

text: i.e.

text: decreasing

text: ∆x

text: ∆y

text: and

text: ∆x

text: ∆y

text: m

text: m

text: m

text: by

text: some

text: factor

text: improves

text: both

text: energy

text: conservation

text: and

text: entropy

text: conservation,

text: as

text: can

text: be

text: seen

text: from

text: last

text: line

text: of

text: panels

text: of

text: all

text: the

text: ﬁgures,

text: as

text: expected.

text: Note

text: that

text: adopting

text: ∆x

text: ∆y

text: and

text: ∆x

text: ∆y

text: as

text: we

text: did

text: in

text: this

text: work

text: is

text: not

text: necessarily

text: optimal,

text: but

text: is

text: a

text: natural

text: choice

text: m

text: m

text: in

text: the

text: code

text: units

text: if

text: total

text: mass

text: of

text: the

text: system

text: is

text: of

text: the

text: order

text: of

text: unity.

text: Changing

text: the

text: resampling

text: frequency

text: parameter

text: n

text: aﬀects

text: both

text: entropy

text: and

text: energy,

text: s

text: but

text: to

text: a

text: much

text: larger

text: extent

text: the

text: former

text: than

text: the

text: latter.

text: In

text: particular,

text: increasing

text: n

text: up

text: s

text: to

text: clearly

text: improves

text: entropy

text: at

text: an

text: acceptable

text: cost

text: for

text: energy

text: conservation,

text: of

text: which

text: the

text: violation

text: remains

text: below

text: the

text: level,

text: as

text: can

text: be

text: seen

text: from

text: the

text: second

text: line

text: of


text: 30 S. Colombi and C. Alard Figure 14. The maximum reconstructed displacement mismatch between neighbouring § elements of metric, Emap, as deﬁned in , as a function of time for the Vlamet runs considered in the energy and entropy plots shown in previous ﬁgures. E is expressed in units of spatial map resolution, ∆. increase the spatial resolution by a factor 2 to 4 in the splitting algorithm runs, and this factor might turn to become even larger after additional optimisation: some work is indeed still needed to understand how to use Vlamet in the best way, while keeping errors, in particular deviations from energy conservation, under control. Indeed, despite the numerous runs we did, we did not span all the possible range of control parameters of Vlamet. In particular, we did not explore thoroughly the case when the separation ∼ ∼ between elements of metric approaches the spatial resolution, e.g 2.5 < ∆ /∆ < 5, m which should allow for even more seldom resamplings than what we achieved. The gain in eﬀective spatial resolution of Vlamet compared to the splitting algorithm compensates largely for the cost of the code and makes our method potentially extremely competitive in 4 and 6 dimensional phase-space. Our current implementation treats iso- lated self-gravitating systems in 2 dimensional phase-space and uses parallel acceleration on shared memory architecture with the OpenMP library. The next step, quite relevant to galactic dynamics consists in generalising the code to 4 and 6 dimensional phase- space. This extension will require the introduction of adaptive mesh reﬁnement with e.g. a k-d tree structure to focus on regions of phase-space of interest as well as parallel programming on distributed memory architecture. This should not present, in principle, § any major diﬃculty, as discussed in . Note also that generalisation to plasmas is obvious and could represent a way to improve the accuracy of current codes. Still, our implementation remains quite naive and some more work is needed for full ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: Colombi

text: and

text: C.

text: Alard

text: Figure

text: The

text: maximum

text: reconstructed

text: displacement

text: mismatch

text: between

text: neighbouring

text: elements

text: of

text: metric,

text: Emap,

text: as

text: deﬁned

text: in

text: as

text: a

text: function

text: of

text: time

text: for

text: the

text: Vlamet

text: runs

text: considered

text: in

text: the

text: energy

text: and

text: entropy

text: plots

text: shown

text: in

text: previous

text: ﬁgures.

text: E

text: is

text: expressed

text: in

text: units

text: of

text: spatial

text: map

text: resolution,

text: increase

text: the

text: spatial

text: resolution

text: by

text: a

text: factor

text: to

text: in

text: the

text: splitting

text: algorithm

text: runs,

text: and

text: this

text: factor

text: might

text: turn

text: to

text: become

text: even

text: larger

text: after

text: additional

text: optimisation:

text: some

text: work

text: is

text: indeed

text: still

text: needed

text: to

text: understand

text: how

text: to

text: use

text: Vlamet

text: in

text: the

text: best

text: way,

text: while

text: keeping

text: errors,

text: in

text: particular

text: deviations

text: from

text: energy

text: conservation,

text: under

text: control.

text: Indeed,

text: despite

text: the

text: numerous

text: runs

text: we

text: did,

text: we

text: did

text: not

text: span

text: all

text: the

text: possible

text: range

text: of

text: control

text: parameters

text: of

text: Vlamet.

text: In

text: particular,

text: we

text: did

text: not

text: explore

text: thoroughly

text: the

text: case

text: when

text: the

text: separation

text: between

text: elements

text: of

text: metric

text: approaches

text: the

text: spatial

text: resolution,

text: e.g

text: m

text: which

text: should

text: allow

text: for

text: even

text: more

text: seldom

text: resamplings

text: than

text: what

text: we

text: achieved.

text: The

text: gain

text: in

text: eﬀective

text: spatial

text: resolution

text: of

text: Vlamet

text: compared

text: to

text: the

text: splitting

text: algorithm

text: compensates

text: largely

text: for

text: the

text: cost

text: of

text: the

text: code

text: and

text: makes

text: our

text: method

text: potentially

text: extremely

text: competitive

text: in

text: and

text: dimensional

text: phase-space.

text: Our

text: current

text: implementation

text: treats

text: iso-

text: lated

text: self-gravitating

text: systems

text: in

text: dimensional

text: phase-space

text: and

text: uses

text: parallel

text: acceleration

text: on

text: shared

text: memory

text: architecture

text: with

text: the

text: OpenMP

text: library.

text: The

text: next

text: step,

text: quite

text: relevant

text: to

text: galactic

text: dynamics

text: consists

text: in

text: generalising

text: the

text: code

text: to

text: and

text: dimensional

text: phase-

text: space.

text: This

text: extension

text: will

text: require

text: the

text: introduction

text: of

text: adaptive

text: mesh

text: reﬁnement

text: with

text: e.g.

text: a

text: k-d

text: tree

text: structure

text: to

text: focus

text: on

text: regions

text: of

text: phase-space

text: of

text: interest

text: as

text: well

text: as

text: parallel

text: programming

text: on

text: distributed

text: memory

text: architecture.

text: This

text: should

text: not

text: present,

text: in

text: principle,

text: any

text: major

text: diﬃculty,

text: as

text: discussed

text: in

text: Note

text: also

text: that

text: generalisation

text: to

text: plasmas

text: is

text: obvious

text: and

text: could

text: represent

text: a

text: way

text: to

text: improve

text: the

text: accuracy

text: of

text: current

text: codes.

text: Still,

text: our

text: implementation

text: remains

text: quite

text: naive

text: and

text: some

text: more

text: work

text: is

text: needed

text: for

text: full

text: ##LTFigure##

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: ##LTLine##


text: A “metric” semi-Lagrangian Vlasov-Poisson solver 9 (i ,j +1) n n (i ,j +1) (i +1,j +1) c c c c Q=(q ,q ) x v (i ,j ) (i +1,j ) c c c c (i -‐1,j ) (i ,j ) (i +1,j ) n n n n n n (i ,j -‐1) n n Figure 2. Procedure employed to select the points of the grid used to perform second order interpolation of the phase-space distribution function at position Q in 2D-dimensional phase-space. One ﬁrst ﬁnds the nearest grid point G to Q, G = (i , j ) for D = 1, and select n n n n all the segments of the grid composed of 3 grid points and centered on G : one obtains the n red cross for D = 1. Then one selects, for each plane passing through a pair of such segments the four elements of the grid so that the square composed of these grid elements contains the projection of Q: one of them, (i + 1, j + 1) on the ﬁgure, provides one additional point for the c c interpolation. plane such that the square composed of this grid site plus 3 contained in the 2 segments contains the projection of Q and add it to our list of points deﬁning the second order hypersurface. There are 6 and 15 such points in 4 and 6 dimensions, respectively, and we thus end up using this procedure with respectively 15 and 28 points as needed to interpolate our hypersurface, with straightforward generalisation of equation (2.18) to perform the interpolation. 2.2.2. Standard third order spline The third order interpolation we use employs standard cubic B-splines as very clearly described in e.g. Crouseilles et al. (2009), so we do not provide the details here, except the edge conditions, which are ∂f | | f(q , q ) = 0, = 0, q = q + n ∆x, (2.24) x v ∂q x x,mid x x ∂f | | f(q , q ) = 0, = 0, q = q + n ∆v, (2.25) x v ∂q v v,mid v v and this can be trivially generalised to 4- or 6-dimensional phase-space. 2.3. Tracing back the characteristics in Lagrangian space: percolation algorithm To determine the value of f(P , t), we have to trace the trajectory of a test particle coinciding with P back to its Lagrangian, initial position Q (the position at time of remapping, in general). To do this, we need to ﬁnd the closest element of metric to this test particle and use equation (2.7) to compute Q. However, the closest element of metric, from the dynamical point of view, has to be found in Lagrangian space, which complicates the procedure. Furthermore, two close test particles might end up associated ##LTFigure## ##LTLine## ##LTLine##

text: A

text: “metric”

text: semi-Lagrangian

text: Vlasov-Poisson

text: solver

text: (i

text: ,j

text: n

text: n

text: (i

text: ,j

text: (i

text: c

text: c

text: c

text: c

text: Q=(q

text: ,q

text: x

text: v

text: (i

text: ,j

text: (i

text: c

text: c

text: c

text: c

text: (i

text: (i

text: ,j

text: (i

text: n

text: n

text: n

text: n

text: n

text: n

text: (i

text: ,j

text: n

text: n

text: Figure

text: Procedure

text: employed

text: to

text: select

text: the

text: points

text: of

text: the

text: grid

text: used

text: to

text: perform

text: second

text: order

text: interpolation

text: of

text: the

text: phase-space

text: distribution

text: function

text: at

text: position

text: Q

text: in

text: phase-space.

text: One

text: ﬁrst

text: ﬁnds

text: the

text: nearest

text: grid

text: point

text: G

text: to

text: Q,

text: G

text: (i

text: j

text: for

text: D

text: and

text: select

text: n

text: n

text: n

text: n

text: all

text: the

text: segments

text: of

text: the

text: grid

text: composed

text: of

text: grid

text: points

text: and

text: centered

text: on

text: G

text: one

text: obtains

text: the

text: n

text: red

text: cross

text: for

text: D

text: Then

text: one

text: selects,

text: for

text: each

text: plane

text: passing

text: through

text: a

text: pair

text: of

text: such

text: segments

text: the

text: four

text: elements

text: of

text: the

text: grid

text: so

text: that

text: the

text: square

text: composed

text: of

text: these

text: grid

text: elements

text: contains

text: the

text: projection

text: of

text: Q:

text: one

text: of

text: them,

text: (i

text: j

text: on

text: the

text: ﬁgure,

text: provides

text: one

text: additional

text: point

text: for

text: the

text: c

text: c

text: interpolation.

text: plane

text: such

text: that

text: the

text: square

text: composed

text: of

text: this

text: grid

text: site

text: plus

text: contained

text: in

text: the

text: segments

text: contains

text: the

text: projection

text: of

text: Q

text: and

text: add

text: it

text: to

text: our

text: list

text: of

text: points

text: deﬁning

text: the

text: second

text: order

text: hypersurface.

text: There

text: are

text: and

text: such

text: points

text: in

text: and

text: dimensions,

text: respectively,

text: and

text: we

text: thus

text: end

text: up

text: using

text: this

text: procedure

text: with

text: respectively

text: and

text: points

text: as

text: needed

text: to

text: interpolate

text: our

text: hypersurface,

text: with

text: straightforward

text: generalisation

text: of

text: equation

text: to

text: perform

text: the

text: interpolation.

text: Standard

text: third

text: order

text: spline

text: The

text: third

text: order

text: interpolation

text: we

text: use

text: employs

text: standard

text: cubic

text: B-splines

text: as

text: very

text: clearly

text: described

text: in

text: e.g.

text: Crouseilles

text: et

text: al.

text: so

text: we

text: do

text: not

text: provide

text: the

text: details

text: here,

text: except

text: the

text: edge

text: conditions,

text: which

text: are

text: ∂f

text: f(q

text: q

text: q

text: q

text: n

text: ∆x,

text: x

text: v

text: ∂q

text: x

text: x,mid

text: x

text: x

text: ∂f

text: f(q

text: q

text: q

text: q

text: n

text: ∆v,

text: x

text: v

text: ∂q

text: v

text: v,mid

text: v

text: v

text: and

text: this

text: can

text: be

text: trivially

text: generalised

text: to

text: or

text: phase-space.

text: Tracing

text: back

text: the

text: characteristics

text: in

text: Lagrangian

text: space:

text: percolation

text: algorithm

text: To

text: determine

text: the

text: value

text: of

text: f(P

text: t),

text: we

text: have

text: to

text: trace

text: the

text: trajectory

text: of

text: a

text: test

text: particle

text: coinciding

text: with

text: P

text: back

text: to

text: its

text: Lagrangian,

text: initial

text: position

text: Q

text: (the

text: position

text: at

text: time

text: of

text: remapping,

text: in

text: general).

text: To

text: do

text: this,

text: we

text: need

text: to

text: ﬁnd

text: the

text: closest

text: element

text: of

text: metric

text: to

text: this

text: test

text: particle

text: and

text: use

text: equation

text: to

text: compute

text: Q.

text: However,

text: the

text: closest

text: element

text: of

text: metric,

text: from

text: the

text: dynamical

text: point

text: of

text: view,

text: has

text: to

text: be

text: found

text: in

text: Lagrangian

text: space,

text: which

text: complicates

text: the

text: procedure.

text: Furthermore,

text: two

text: close

text: test

text: particles

text: might

text: end

text: up

text: associated

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: ACFI-T17-11 Prepared for submission to JCAP Generation of gravitational waves from symmetry restoration during inﬂation Cody Goolsby-Cole and Lorenzo Sorbo Amherst Center for Fundamental Interactions, Department of Physics, University of Mas- sachusetts, Amherst, MA 01003 Abstract. We discuss the possibility of a feature in the spectrum of inﬂationary gravitational waves sourced by a scalar ﬁeld χ whose vacuum ﬂuctuations are ampliﬁed by a rapidly time dependent mass. Unlike previous work which has focused on the case in which the mass of the ﬁeld χ vanishes only for an instant before becoming massive again, we study a system where the scalar ﬁeld becomes and remains massless through the end of inﬂation as the consequence of the restoration of a shift symmetry. After applying appropriate constraints to our parameters, we ﬁnd, for future CMB experiments, a small contribution to the tensor- ∼ − to-scalar ratio which can be at most of the order r 10 5. At smaller scales probed by gravitational interferometers, on the other hand, the energy density in the gravitational ∼ − waves produced this way might be above the projected sensitivity of LISA, Ω h2 10 13, GW in a narrow region of parameter space. If there is more than one χ species, then these amplitudes are enhanced by a factor equal to the number of those species.

text: Prepared

text: for

text: submission

text: to

text: JCAP

text: Generation

label: title

text: of

text: gravitational

text: waves

text: from

text: symmetry

text: restoration

text: during

text: inﬂation

text: Cody

label: author

text: Goolsby-Cole

text: and

text: Lorenzo

text: Sorbo

text: Amherst

text: Center

text: for

text: Fundamental

text: Interactions,

text: Department

text: of

text: Physics,

text: University

text: of

text: Mas-

text: sachusetts,

text: Amherst,

text: MA

text: Abstract.

text: We

label: abstract

text: discuss

text: the

text: possibility

text: of

text: a

text: feature

text: in

text: the

text: spectrum

text: of

text: inﬂationary

text: gravitational

text: waves

text: sourced

text: by

text: a

text: scalar

text: ﬁeld

text: χ

text: whose

text: vacuum

text: ﬂuctuations

text: are

text: ampliﬁed

text: by

text: a

text: rapidly

text: time

text: dependent

text: mass.

text: Unlike

text: previous

text: work

text: which

text: has

text: focused

text: on

text: the

text: case

text: in

text: which

text: the

text: mass

text: of

text: the

text: ﬁeld

text: χ

text: vanishes

text: only

text: for

text: an

text: instant

text: before

text: becoming

text: massive

text: again,

text: we

text: study

text: a

text: system

text: where

text: the

text: scalar

text: ﬁeld

text: becomes

text: and

text: remains

text: massless

text: through

text: the

text: end

text: of

text: inﬂation

text: as

text: the

text: consequence

text: of

text: the

text: restoration

text: of

text: a

text: shift

text: symmetry.

text: After

text: applying

text: appropriate

text: constraints

text: to

text: our

text: parameters,

text: we

text: ﬁnd,

text: for

text: future

text: CMB

text: experiments,

text: a

text: small

text: contribution

text: to

text: the

text: tensor-

text: to-scalar

text: ratio

text: which

text: can

text: be

text: at

text: most

text: of

text: the

text: order

text: r

text: At

text: smaller

text: scales

text: probed

text: by

text: gravitational

text: interferometers,

text: on

text: the

text: other

text: hand,

text: the

text: energy

text: density

text: in

text: the

text: gravitational

text: waves

text: produced

text: this

text: way

text: might

text: be

text: above

text: the

text: projected

text: sensitivity

text: of

text: LISA,

text: Ω

text: GW

text: in

text: a

text: narrow

text: region

text: of

text: parameter

text: space.

text: If

text: there

text: is

text: more

text: than

text: one

text: χ

text: species,

text: then

text: these

text: amplitudes

text: are

text: enhanced

text: by

text: a

text: factor

text: equal

text: to

text: the

text: number

text: of

text: those

text: species.


text: transparent way of performing such a renormalization. On a Minkowski background, in fact, the frequency of the ﬁeld χˆ (which is just its momentum) evolves adiabatically after τ∗, so that we can fully use the formalism of Bogolyubov coeﬃcients. This means that we † would decompose the ﬁeld in terms of new creation/annihilation operators ˆb( ), where the ˆb k k operator multiplies the positive frequency component of the mode functions of χˆ for times τ > τ∗, when the frequency of the modes of χˆ is adiabatically evolving. Then we would † normal order the χˆ(k , τ ) χˆ(k , τ ) operator in terms of the ˆb( ) ladder operators, and not 1 1† 2 2 k in terms of the original aˆ( ) ones, that were used to quantize χˆ for τ < τ∗. This means that k observers born after τ = τ∗ would renormalize away the vacuum ﬂuctuations of the mode † functions deﬁned for τ > τ∗ by normal ordering the operators ˆb( ), that correspond to their † k † ( ) ( ) notion of a particle. Writing the relationship between the b and the a operators as k k ∗ − † ˆb (τ) = α(k, τ) aˆ + β ( k, τ) aˆ− , (3.25) k k k we would obtain (cid:20)(cid:18) (cid:19) (cid:112) (cid:82) (cid:104)χˆ(p, τ(cid:48))χˆ(q, τ(cid:48)(cid:48))(cid:105) = (cid:18) δ((cid:82)3)(p(cid:48) + q(cid:82)) (cid:48)(cid:48) ei ττ(cid:48)(cid:48)(cid:48) ωp β∗(−p, τ(cid:0)(cid:48))β(−p, τ(cid:48)(cid:48)) +(cid:1)(cid:19)h(cid:21).c. (3.26) 2 ω (τ ) ω (τ ) p p (cid:48) (cid:48)(cid:48) − − (cid:48) ∗ (cid:48)(cid:48) (cid:48) ↔ (cid:48)(cid:48) + e i τ ωp i τ ωpα(p, τ )β (p, τ ) + τ τ , h.c. . As we will also discuss in Appendix , the prescription (3.26) above is equivalent to setting (cid:2) (cid:3) (cid:104) (cid:48) (cid:48)(cid:48) (cid:105) (cid:48) (cid:48)(cid:48) − (cid:48) (cid:48)(cid:48) χ(p, τ )χ(q, τ ) = δ(3)(p + q) χ(p, τ ) χ(p, τ ) χ˜(p, τ ) χ˜(p, τ ) , (3.27) where, in the case of particles on a Minkowskian background, χ˜(p, τ) = e√−ipτ corre- → 2p sponds to the mode functions in the absence of particle creation, Λ 0. In its turn, this χ means that the procedure presented above corresponds precisely to that of adiabatic regular- ization, where one subtracts from the UV-divergent propagator its adiabatic part to obtain a ﬁnite result. Let us now go back to the production of quanta of χˆ in de Sitter space. As discussed − above, since we are talking about massless particles in de Sitter space, we have ω2 = k2 2/τ2 (cid:46) − that for k 1/τ is not evolving adiabatically. Therefore the prescription (3.26) cannot be applied to this case. However, the prescription (3.27), that on a Minkowski background is equivalent to (3.26), can be applied to our de Sitter(cid:12)background, once we set (cid:12) (cid:12) χ˜(k, τ) = χ(k, τ) . (3.28) → Λχ 0 Based on the above considerations,(cid:2)we will use (cid:3) (cid:104) (cid:48) (cid:48)(cid:48) (cid:105) (cid:48) (cid:48)(cid:48) − (cid:48) (cid:48)(cid:48) χ(p, τ )χ(q, τ ) = δ(3)(p + q) χ(p, τ ) χ(p, τ ) χ˜(p, τ ) χ˜(p, τ ) , (3.29) – 10 – ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: transparent

text: way

text: of

text: performing

text: such

text: a

text: renormalization.

text: On

text: a

text: Minkowski

text: background,

text: in

text: fact,

text: the

text: frequency

text: of

text: the

text: ﬁeld

text: χˆ

text: (which

text: is

text: just

text: its

text: momentum)

text: evolves

text: adiabatically

text: after

text: τ∗,

text: so

text: that

text: we

text: can

text: fully

text: use

text: the

text: formalism

text: of

text: Bogolyubov

text: coeﬃcients.

text: This

text: means

text: that

text: we

text: would

text: decompose

text: the

text: ﬁeld

text: in

text: terms

text: of

text: new

text: creation/annihilation

text: operators

text: ˆb(

text: where

text: the

text: ˆb

text: k

text: k

text: operator

text: multiplies

text: the

text: positive

text: frequency

text: component

text: of

text: the

text: mode

text: functions

text: of

text: χˆ

text: for

text: times

text: τ

text: τ∗,

text: when

text: the

text: frequency

text: of

text: the

text: modes

text: of

text: χˆ

text: is

text: adiabatically

text: evolving.

text: Then

text: we

text: would

text: normal

text: order

text: the

text: χˆ(k

text: τ

text: χˆ(k

text: τ

text: operator

text: in

text: terms

text: of

text: the

text: ˆb(

text: ladder

text: operators,

text: and

text: not

text: 1†

text: k

text: in

text: terms

text: of

text: the

text: original

text: aˆ(

text: ones,

text: that

text: were

text: used

text: to

text: quantize

text: χˆ

text: for

text: τ

text: τ∗.

text: This

text: means

text: that

text: k

text: observers

text: born

text: after

text: τ

text: τ∗

text: would

text: renormalize

text: away

text: the

text: vacuum

text: ﬂuctuations

text: of

text: the

text: mode

text: functions

text: deﬁned

text: for

text: τ

text: τ∗

text: by

text: normal

text: ordering

text: the

text: operators

text: ˆb(

text: that

text: correspond

text: to

text: their

text: k

text: notion

text: of

text: a

text: particle.

text: Writing

text: the

text: relationship

text: between

text: the

text: b

text: and

text: the

text: a

text: operators

text: as

text: k

text: k

text: ˆb

text: (τ)

text: α(k,

text: τ)

text: aˆ

text: β

text: k,

text: τ)

text: aˆ−

text: k

text: k

text: k

text: we

text: would

text: obtain

text: ei

text: ωp

text: β∗(−p,

text: ω

text: (τ

text: ω

text: (τ

text: p

text: p

text: e

text: i

text: τ

text: ωp

text: i

text: τ

text: ωpα(p,

text: τ

text: )β

text: (p,

text: τ

text: τ

text: τ

text: h.c.

text: As

text: we

text: will

text: also

text: discuss

text: in

text: Appendix

text: the

text: prescription

text: above

text: is

text: equivalent

text: to

text: setting

text: χ(p,

text: τ

text: )χ(q,

text: τ

text: q)

text: χ(p,

text: τ

text: χ(p,

text: τ

text: χ˜(p,

text: τ

text: χ˜(p,

text: τ

text: where,

text: in

text: the

text: case

text: of

text: particles

text: on

text: a

text: Minkowskian

text: background,

text: χ˜(p,

text: τ)

text: e√−ipτ

text: corre-

text: sponds

text: to

text: the

text: mode

text: functions

text: in

text: the

text: absence

text: of

text: particle

text: creation,

text: Λ

text: In

text: its

text: turn,

text: this

text: χ

text: means

text: that

text: the

text: procedure

text: presented

text: above

text: corresponds

text: precisely

text: to

text: that

text: of

text: adiabatic

text: regular-

text: ization,

text: where

text: one

text: subtracts

text: from

text: the

text: UV-divergent

text: propagator

text: its

text: adiabatic

text: part

text: to

text: obtain

text: a

text: ﬁnite

text: result.

text: Let

text: us

text: now

text: go

text: back

text: to

text: the

text: production

text: of

text: quanta

text: of

text: χˆ

text: in

text: de

text: Sitter

text: space.

text: As

text: discussed

text: above,

text: since

text: we

text: are

text: talking

text: about

text: massless

text: particles

text: in

text: de

text: Sitter

text: space,

text: we

text: have

text: ω2

text: that

text: for

text: k

text: is

text: not

text: evolving

text: adiabatically.

text: Therefore

text: the

text: prescription

text: cannot

text: be

text: applied

text: to

text: this

text: case.

text: However,

text: the

text: prescription

text: that

text: on

text: a

text: Minkowski

text: background

text: is

text: equivalent

text: to

text: can

text: be

text: applied

text: to

text: our

text: de

text: once

text: we

text: set

text: χ˜(k,

text: τ)

text: χ(k,

text: τ)

text: Λχ

text: Based

text: on

text: the

text: above

text: will

text: use

text: χ(p,

text: τ

text: )χ(q,

text: τ

text: q)

text: χ(p,

text: τ

text: χ(p,

text: τ

text: χ˜(p,

text: τ

text: χ˜(p,

text: τ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##