text: – 9 – values (see Vasyunin et al. 2008; Albertsson et al. 2013). The same is found for the nebular model with the standard grains but 100 times lower UV-desorption yield of 10−5. Next, we studied how the calculated water D/H ratios are affected if only a limited set of the nuclear spin-state processes is included in our chemical network. For that test we calculated a model in which only the para states of H , H+, and H+ isotopologues were considered, implying more efﬁcient deuterium isotope 2 2 3 exchange. The water D/H ratios calculated with this model coincide with the ratios calculated with our ∼ standard model with ortho/para-species (within the factor of 3 in intrinsic chemical uncertainties). This result is a combination of two extreme situations. First, the initial abundances are calculated using the physical conditions of a dark, dense, 10 K cloud core. At such conditions almost the entire H population 2 exists in the lowest energy para-state, and the isotope exchange reactions involving ortho-species are still out of equilibrium. This leads to similar values of the initial water D/H ratio. Second, in the water de- (cid:46) fractionation zone in the nebula, at r 10 − 30 AU, temperatures are higher than 30 K. These temperatures are high enough to enable rapid gas-phase de-fractionation regardless of the dominant nuclear spin-state form of H , H+, and H+, mainly due to X-ray- and CRP-driven reprocessing of CO by He+, followed by 2 2 3 rapid gas-phase production of H O. This combination of effects leads to the somewhat surprising result that 2 a reduced network produce roughly the same abundances as the complete network. The initial H2 ortho:para ratio in cold dark clouds is unknown but predicted to be low (< 0.1 Pagani et al. 2009). However, later, during the warmer protostellar phase this ratio is likely modiﬁed, such that there are more ortho−H and less para−H molecules. As such the ortho:para ratio can be closer to the 2 2 statistical ortho:para ratio of 3:1 in the initial phase of the protoplanetary disk. We take the statistical value 3:1 as the upper extreme of the H ortho:para ratio and initiate another run of the laminar model with initial 2 abundances resulting from a TMC-1 model with an initial 3:1 ortho:para ratio for H . 2 In Figure 5 we show the D/H ratios of solid-phase HDO between 0.8 and 30 AU for the laminar model using our previous “cold” TMC-1 initial abundance (left panel) and “warm” TMC-1 initial abundance (middle panel), and in the right panel the D/H ratio determined from column densities is shown. In the ∼ ∼ disk midplane there are small but signiﬁcant differences where D/H ratios decrease from 10−2 down to 10−4 −10−3 around 10 AU. This cause the column density D/H ratios in the right panel to drop approximately ∼ (cid:38) an order of magnitude while the minimum and maximum D/H ratios remain unchanged at 1 AU and 30 AU, respectively. Thus our conclusion that carbonaceous asteroids formed at 2-3 AU will inherit Earth ∼ water-like D/H ratios remains valid. The fact that the H O D/H ratio gets lower at intermediate radii ( 10 2 AU) will only help us to achieve a larger zone where Hartley-2-like comets with the Earth water D/H ratio ∼ could form, while still leaving a zone for the Oort comets to form at 10 AU. Finally, we discuss the effects of the underlying chemistry on the D/H ratios throughout the disk.In the midplane, temperatures are low enough to drive a rich surface chemistry, where O atoms are adsorbed on the grain surfaces and become hydrogenated, forming H O, and/or reacting with deuterium atoms to form HDO 2 or D O. There is a small chance for the water molecules to be released into the gas-phase directly through 2 chemisorption, but more likely they remain on the grains and desorb as material is transported into warmer regions, i.e. the inner regions or the atmosphere, where water, and its D/H ratio, will be reprocessed.

text: values

text: (see

text: Vasyunin

text: et

text: al.

text: Albertsson

text: et

text: al.

text: The

text: same

text: is

text: found

text: for

text: the

text: nebular

text: model

text: with

text: the

text: standard

text: grains

text: but

text: times

text: lower

text: UV-desorption

text: yield

text: of

text: Next,

text: we

text: studied

text: how

text: the

text: calculated

text: water

text: D/H

text: ratios

text: are

text: affected

text: if

text: only

text: a

text: limited

text: set

text: of

text: the

text: nuclear

text: spin-state

text: processes

text: is

text: included

text: in

text: our

text: chemical

text: network.

text: For

text: that

text: test

text: we

text: calculated

text: a

text: model

text: in

text: which

text: only

text: the

text: para

text: states

text: of

text: H

text: H+,

text: and

text: H+

text: isotopologues

text: were

text: considered,

text: implying

text: more

text: efﬁcient

text: deuterium

text: isotope

text: exchange.

text: The

text: water

text: D/H

text: ratios

text: calculated

text: with

text: this

text: model

text: coincide

text: with

text: the

text: ratios

text: calculated

text: with

text: our

text: standard

text: model

text: with

text: ortho/para-species

text: (within

text: the

text: factor

text: of

text: in

text: intrinsic

text: chemical

text: uncertainties).

text: This

text: result

text: is

text: a

text: combination

text: of

text: two

text: extreme

text: situations.

text: First,

text: the

text: initial

text: abundances

text: are

text: calculated

text: using

text: the

text: physical

text: conditions

text: of

text: a

text: dark,

text: dense,

text: K

text: cloud

text: core.

text: At

text: such

text: conditions

text: almost

text: the

text: entire

text: H

text: population

text: exists

text: in

text: the

text: lowest

text: energy

text: para-state,

text: and

text: the

text: isotope

text: exchange

text: reactions

text: involving

text: ortho-species

text: are

text: still

text: out

text: of

text: equilibrium.

text: This

text: leads

text: to

text: similar

text: values

text: of

text: the

text: initial

text: water

text: D/H

text: ratio.

text: Second,

text: in

text: the

text: water

text: de-

text: fractionation

text: zone

text: in

text: the

text: nebula,

text: at

text: r

text: AU,

text: temperatures

text: are

text: higher

text: than

text: K.

text: These

text: temperatures

text: are

text: high

text: enough

text: to

text: enable

text: rapid

text: gas-phase

text: de-fractionation

text: regardless

text: of

text: the

text: dominant

text: nuclear

text: spin-state

text: form

text: of

text: H

text: H+,

text: and

text: H+,

text: mainly

text: due

text: to

text: X-ray-

text: and

text: CRP-driven

text: reprocessing

text: of

text: CO

text: by

text: He+,

text: followed

text: by

text: rapid

text: gas-phase

text: production

text: of

text: H

text: O.

text: This

text: combination

text: of

text: effects

text: leads

text: to

text: the

text: somewhat

text: surprising

text: result

text: that

text: a

text: reduced

text: network

text: produce

text: roughly

text: the

text: same

text: abundances

text: as

text: the

text: complete

text: network.

text: The

text: initial

text: ortho:para

text: ratio

text: in

text: cold

text: dark

text: clouds

text: is

text: unknown

text: but

text: predicted

text: to

text: be

text: low

text: Pagani

text: et

text: al.

text: However,

text: later,

text: during

text: the

text: warmer

text: protostellar

text: phase

text: this

text: ratio

text: is

text: likely

text: modiﬁed,

text: such

text: that

text: there

text: are

text: more

text: ortho−H

text: and

text: less

text: para−H

text: molecules.

text: As

text: such

text: the

text: ortho:para

text: ratio

text: can

text: be

text: closer

text: to

text: the

text: statistical

text: ortho:para

text: ratio

text: of

text: in

text: the

text: initial

text: phase

text: of

text: the

text: protoplanetary

text: disk.

text: We

text: take

text: the

text: statistical

text: value

text: as

text: the

text: upper

text: extreme

text: of

text: the

text: H

text: ortho:para

text: ratio

text: and

text: initiate

text: another

text: run

text: of

text: the

text: laminar

text: model

text: with

text: initial

text: abundances

text: resulting

text: from

text: a

text: model

text: with

text: an

text: initial

text: ortho:para

text: ratio

text: for

text: H

text: In

text: Figure

text: we

text: show

text: the

text: D/H

text: ratios

text: of

text: solid-phase

text: HDO

text: between

text: and

text: AU

text: for

text: the

text: laminar

text: model

text: using

text: our

text: previous

text: “cold”

text: initial

text: abundance

text: (left

text: panel)

text: and

text: “warm”

text: initial

text: abundance

text: (middle

text: panel),

text: and

text: in

text: the

text: right

text: panel

text: the

text: D/H

text: ratio

text: determined

text: from

text: column

text: densities

text: is

text: shown.

text: In

text: the

text: disk

text: midplane

text: there

text: are

text: small

text: but

text: signiﬁcant

text: differences

text: where

text: D/H

text: ratios

text: decrease

text: from

text: down

text: to

text: around

text: AU.

text: This

text: cause

text: the

text: column

text: density

text: D/H

text: ratios

text: in

text: the

text: right

text: panel

text: to

text: drop

text: approximately

text: an

text: order

text: of

text: magnitude

text: while

text: the

text: minimum

text: and

text: maximum

text: D/H

text: ratios

text: remain

text: unchanged

text: at

text: AU

text: and

text: AU,

text: respectively.

text: Thus

text: our

text: conclusion

text: that

text: carbonaceous

text: asteroids

text: formed

text: at

text: AU

text: will

text: inherit

text: Earth

text: water-like

text: D/H

text: ratios

text: remains

text: valid.

text: The

text: fact

text: that

text: the

text: H

text: O

text: D/H

text: ratio

text: gets

text: lower

text: at

text: intermediate

text: radii

text: AU)

text: will

text: only

text: help

text: us

text: to

text: achieve

text: a

text: larger

text: zone

text: where

text: comets

text: with

text: the

text: Earth

text: water

text: D/H

text: ratio

text: could

text: form,

text: while

text: still

text: leaving

text: a

text: zone

text: for

text: the

text: Oort

text: comets

text: to

text: form

text: at

text: AU.

text: Finally,

text: we

text: discuss

text: the

text: effects

text: of

text: the

text: underlying

text: chemistry

text: on

text: the

text: D/H

text: ratios

text: throughout

text: the

text: disk.In

text: the

text: midplane,

text: temperatures

text: are

text: low

text: enough

text: to

text: drive

text: a

text: rich

text: surface

text: chemistry,

text: where

text: O

text: atoms

text: are

text: adsorbed

text: on

text: the

text: grain

text: surfaces

text: and

text: become

text: hydrogenated,

text: forming

text: H

text: O,

text: and/or

text: reacting

text: with

text: deuterium

text: atoms

text: to

text: form

text: HDO

text: or

text: D

text: O.

text: There

text: is

text: a

text: small

text: chance

text: for

text: the

text: water

text: molecules

text: to

text: be

text: released

text: into

text: the

text: gas-phase

text: directly

text: through

text: chemisorption,

text: but

text: more

text: likely

text: they

text: remain

text: on

text: the

text: grains

text: and

text: desorb

text: as

text: material

text: is

text: transported

text: into

text: warmer

text: regions,

text: i.e.

text: the

text: inner

text: regions

text: or

text: the

text: atmosphere,

text: where

text: water,

text: and

text: its

text: D/H

text: ratio,

text: will

text: be

text: reprocessed.


text: – 10 – The reprocessing of water occurs in the gas phase, so the relative adsorption and desorption plays an important role. These processes are heavily effected by temperature and density. In the warmer regions, the (cid:46) water abundances steadily increase by 30% during the ﬁrst 10 000 - 100 000 years. As soon as the water reaches the gas-phase, it is reprocessed. After that, the abundances of water ice and water vapor are at quasi steady-state. However, constant evaporation and re-accretion result in cycling of water between solid and gaseous phases. One important fact is that the D/H ratio never drops to the cosmic ratio, even in the warm, inner regions within a few AU of the Sun. In the inner solar nebula there are several important processes at work that sets the D/H ratio. First of all these regions experience strong X-ray or CRP-ionization and water is either ionized or protonated by reacting with various abundant ions, such as H+, HCO+, and their 3 isotopologues: ⇒ × H2O + o−H+3 ⇒ H3O+ + o−H2; 4.5×10−9 cm−3 s−1 β = −0.5 H2O + p−H+3 ⇒ H3O+ + o−H2; 2.3×10−9 cm−3 s−1 β = −0.5 H2O + p−H+3 ⇒H3O+ + p−H2; 2.3×10−9 cm−3 s−1 β = −0.5 H2O + HCO+ H3O+ + CO; 2.1 10−9 cm−3 s−1 β = −0.5 The same set of reactions applies for HDO being protonated into H DO+ (as well as heavier isotopo- 2 logues). The H O+ isotopologues can then reform water through dissociative recombination, but the D/H 3 ratios will be reset, characteristic of the temperature in the environment at which they are reforming: ⇒ × H3O + e− H2O + H; 1.1 10−7 cm−3 s−1 β = −0.5 ⇒ × H2DO + e− H2O + D; 7.3 10−8 cm−3 s−1 β = −0.5 ⇒ × H2DO + e− HDO + H; 7.3 10−8 cm−3 s−1 β = −0.5 The reprocessing of water involves several additional reactions that compete in both lowering and in- creasing the D/H ratio. In the warm inner nebular region the normal H+ fractionation pathway is not active 3 and instead fractionation is driven mostly via the “warm” fractionation route that includes light hydrocar- (cid:46) bons such as CH D+ and C HD+ and is active at T 80 K (e.g., Millar et al. 1989; Roueff et al. 2005; Parise 2 2 ∼ et al. 2009). Consequently, the initially high D/H ratios of light hydrocarbons, 10−3, is not fully equili- ∼ brated with the cosmic D/H ratio of 10−5. Neutral-neutral reactions of these hydrocarbons with atomic or molecular oxygen are able to produce normal or heavy water, e.g.: ⇒ × CH3 + O2 HCO + H2O; 1.66 10−12 cm−3 s−1 (3) Another important channel of the water formation in the gas-phase is through the neutral-neutral reac- tion with OH or OD of H/D or ortho/para−H2/HD:

label: paragraph

text: The

text: reprocessing

text: of

text: water

text: occurs

text: in

text: the

text: gas

text: phase,

text: so

text: the

text: relative

text: adsorption

text: and

text: desorption

text: plays

text: an

text: important

text: role.

text: These

text: processes

text: are

text: heavily

text: effected

text: by

text: temperature

text: and

text: density.

text: In

text: the

text: warmer

text: regions,

text: the

text: water

text: abundances

text: steadily

text: increase

text: by

text: during

text: the

text: ﬁrst

text: years.

text: As

text: soon

text: as

text: the

text: water

text: reaches

text: the

text: gas-phase,

text: it

text: is

text: reprocessed.

text: After

text: that,

text: the

text: abundances

text: of

text: water

text: ice

text: and

text: water

text: vapor

text: are

text: at

text: quasi

text: steady-state.

text: However,

text: constant

text: evaporation

text: and

text: re-accretion

text: result

text: in

text: cycling

text: of

text: water

text: between

text: solid

text: and

text: gaseous

text: phases.

text: One

text: important

text: fact

text: is

text: that

text: the

text: D/H

text: ratio

text: never

text: drops

text: to

text: the

text: cosmic

text: ratio,

text: even

text: in

text: the

text: warm,

text: inner

text: regions

text: within

text: a

text: few

text: AU

text: of

text: the

text: Sun.

text: In

text: the

text: inner

text: solar

text: nebula

text: there

text: are

text: several

text: important

text: processes

text: at

text: work

text: that

text: sets

text: the

text: D/H

text: ratio.

text: First

text: of

text: all

text: these

text: regions

text: experience

text: strong

text: X-ray

text: or

text: CRP-ionization

text: and

text: water

text: is

text: either

text: ionized

text: or

text: protonated

text: by

text: reacting

text: with

text: various

text: abundant

text: ions,

text: such

text: as

text: H+,

text: HCO+,

text: and

text: their

text: isotopologues:

text: β

text: β

text: β

text: HCO+

text: CO;

text: β

text: The

text: same

text: set

text: of

text: reactions

text: applies

text: for

text: HDO

text: being

text: protonated

text: into

text: H

text: DO+

text: (as

text: well

text: as

text: heavier

text: isotopo-

text: logues).

text: The

text: H

text: O+

text: isotopologues

text: can

text: then

text: reform

text: water

text: through

text: dissociative

text: recombination,

text: but

text: the

text: D/H

text: ratios

text: will

text: be

text: reset,

text: characteristic

text: of

text: the

text: temperature

text: in

text: the

text: environment

text: at

text: which

text: they

text: are

text: reforming:

text: e−

text: H;

text: β

text: e−

text: D;

text: β

text: e−

text: HDO

text: H;

text: β

text: The

text: reprocessing

text: of

text: water

text: involves

text: several

text: additional

text: reactions

text: that

text: compete

text: in

text: both

text: lowering

text: and

text: in-

text: creasing

text: the

text: D/H

text: ratio.

text: In

text: the

text: warm

text: inner

text: nebular

text: region

text: the

text: normal

text: H+

text: fractionation

text: pathway

text: is

text: not

text: active

text: and

text: instead

text: fractionation

text: is

text: driven

text: mostly

text: via

text: the

text: “warm”

text: fractionation

text: route

text: that

text: includes

text: light

text: hydrocar-

text: bons

text: such

text: as

text: CH

text: D+

text: and

text: C

text: HD+

text: and

text: is

text: active

text: at

text: T

text: K

text: (e.g.,

text: Millar

text: et

text: al.

text: Roueff

text: et

text: al.

text: Parise

text: et

text: al.

text: Consequently,

text: the

text: initially

text: high

text: D/H

text: ratios

text: of

text: light

text: hydrocarbons,

text: is

text: not

text: fully

text: equili-

text: brated

text: with

text: the

text: cosmic

text: D/H

text: ratio

text: of

text: Neutral-neutral

text: reactions

text: of

text: these

text: hydrocarbons

text: with

text: atomic

text: or

text: molecular

text: oxygen

text: are

text: able

text: to

text: produce

text: normal

text: or

text: heavy

text: water,

text: e.g.:

text: HCO

text: Another

text: important

text: channel

text: of

text: the

text: water

text: formation

text: in

text: the

text: gas-phase

text: is

text: through

text: the

text: neutral-neutral

text: reac-

text: tion

text: with

text: OH

text: or

text: OD

text: of

text: H/D

text: or


text: 16 J.-E. BUSSE, P. GWIAZDA AND A. MARCINIAK-CZOCHRA Applying Lemma 2.4, we obtain(cid:32) (cid:18) (cid:19) ∂∂t uu2((tt,, xx1)) = puu1(cid:18)((tt,, xx2)) 2 1 −(cid:19)1a+(xK1)ρ (cid:33)uu010((xx1))e2(a(x1+1)K−Ma(3x2))t 2 2 2 2 2 1 2 − − a(x ) u (t, x ) 2 1 2 2 1 . 1 + Kρ u (t, x ) 2 2 2 (cid:16) (cid:17) Thus, we can deduce the following bound for u2(t,x1) u2(t,x2) u2(t, x1) ≤ 1 − 1+a(Kx1M)3 −uu0101((xx12))e2(a(x1+1)K−Ma(3x2))t , u (t, x ) 1 a(x ) 2 2 2 where the right hand side tends exponentially to zero, as t tends to inﬁnity. (cid:3) This concludes the proof. Having shown the dynamics of the quotients between solutions at diﬀerent x points, we can prove that the solutions converge to zero outside the set of points with a maximal value of the parameter a(x). of Lemma 2.6. Let x˜ be a point diﬀerent that x¯ and assume that lim →∞ u(t, x˜) > 0. Continuity of a(x) implies that the set of x, such that a(x) > t a(x˜), is an open nonempty set and, therefore, it has a positive measure. Since ∈ − (cid:82) Lemma 2.4 holds for every x, x˜ Ω such that a(x) a(x˜) > 0, we conclude that ∞ u(t, x) tends exponentially to + for every x such that a(x) > a(x˜). This is, (cid:3) however, in contradiction with the uniform boundedness of mass u(t, x)dx. Ω 4. Proof of convergence of the total mass We begin the proof of Theorem 2.7 by showing the following lemma, which allows comparing two dynamical systems. → Lemma 4.1. Let u : R+ (R+)2 be a solution of du = F(u) dt ∈ with a globally stable stationary solution u¯ and let V C1((R+)2) be a Lyapunov function for this equation with compact level sets and the minimum δ achieved at the stationary solution u¯. If u˜ is a solution of du˜ = F(u˜) + f, dt ∈ → → ∞ where f L1(R+), then u˜ u¯ for t . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J.-E.

text: BUSSE,

text: P.

text: GWIAZDA

text: AND

text: A.

text: MARCINIAK-CZOCHRA

text: Applying

text: Lemma

text: we

label: equation

text: ∂∂t

text: a(x

text: u

text: (t,

text: x

text: Kρ

text: u

text: (t,

text: x

text: Thus,

text: we

text: can

text: deduce

text: the

text: following

text: bound

text: for

text: u

text: (t,

text: x

text: a(x

text: where

text: the

text: right

text: hand

text: side

text: tends

text: exponentially

text: to

text: zero,

text: as

text: t

text: tends

text: to

text: inﬁnity.

text: This

text: concludes

text: the

text: proof.

text: Having

text: shown

text: the

text: dynamics

text: of

text: the

text: quotients

text: between

text: solutions

text: at

text: diﬀerent

text: x

text: points,

text: we

text: can

text: prove

text: that

text: the

text: solutions

text: converge

text: to

text: zero

text: outside

text: the

text: set

text: of

text: points

text: with

text: a

text: maximal

text: value

text: of

text: the

text: parameter

text: a(x).

text: of

text: Lemma

text: Let

text: x˜

text: be

text: a

text: point

text: diﬀerent

text: that

text: x¯

text: and

text: assume

text: that

text: lim

text: u(t,

text: x˜)

text: Continuity

text: of

text: a(x)

text: implies

text: that

text: the

text: set

text: of

text: x,

text: such

text: that

text: a(x)

text: t

text: a(x˜),

text: is

text: an

text: open

text: nonempty

text: set

text: and,

text: therefore,

text: it

text: has

text: a

text: positive

text: measure.

text: Since

text: Lemma

text: holds

text: for

text: every

text: x,

text: x˜

text: Ω

text: such

text: that

text: a(x)

text: a(x˜)

text: we

text: conclude

text: that

text: u(t,

text: x)

text: tends

text: exponentially

text: to

text: for

text: every

text: x

text: such

text: that

text: a(x)

text: a(x˜).

text: This

text: is,

text: however,

text: in

text: contradiction

text: with

text: the

text: uniform

text: boundedness

text: of

text: mass

text: u(t,

text: x)dx.

text: Ω

text: Proof

text: of

text: convergence

text: of

text: the

text: total

text: mass

text: We

text: begin

text: the

text: proof

text: of

text: Theorem

text: by

text: showing

text: the

text: following

text: lemma,

text: which

text: allows

text: comparing

text: two

text: dynamical

text: systems.

text: Lemma

text: Let

text: u

text: R+

text: be

text: a

text: solution

text: of

text: du

text: F(u)

text: dt

text: with

text: a

text: globally

text: stable

text: stationary

text: solution

text: u¯

text: and

text: let

text: V

text: be

text: a

text: Lyapunov

text: function

text: for

text: this

text: equation

text: with

text: compact

text: level

text: sets

text: and

text: the

text: minimum

text: δ

text: achieved

text: at

text: the

text: stationary

text: solution

text: u¯.

text: If

text: u˜

text: is

text: a

text: solution

text: of

text: du˜

text: F(u˜)

text: f,

text: dt

text: where

text: f

text: then

text: u˜

text: u¯

text: for

text: t

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 J.-E. BUSSE, P. GWIAZDA AND A. MARCINIAK-CZOCHRA (cid:107)∇ (cid:107) where C = u˜V (u˜(t)) L∞. (cid:90) Integrating the above estimate, we obtain ≤ − t | | − − V (u˜(t)) V (u )e β˜at + f e β˜a(t τ)dτ. a a 0 0 ∈ → → ∞ Since, by deﬁnition, f L1(R+), we conclude that V (u(t)) 0 for t . The a → convergence holds for every a which yields convergence V (u(t)) δ, which is the → (cid:3) minimum of the function V . In turn, it ensures that u(t) u¯. of Theorem 2.7. To apply Lemma 4.1 to our system (6), we consider a ﬁnite di- (cid:18) (cid:19) mensional model obtained by setting a(x) to a constant value a d (cid:18) 2a − (cid:19) v = 1 pv , dt 1 1 + Kv 1 2 d − a − (16) v = 2 1 pv dv , dt 2 1 + Kv 1 2 2 v (0) = v0, 1 1 v (0) = v0. 2 2 A Lyapunov function for this system has been constructed in [Getto et al. 2013]. It takes the form 1 1 (17) V (v , v ) := V (v , v ) + V (v , v ), 1 2 pG(v¯ ) 1 1 2 d 2 1 2 2 where v − − v(cid:90) V (v , v ) := 1 1 ln 1 , 1 1 2 v¯ v¯ 1 1 v − − 1 v2 G(v¯ ) V (v , v ) := 2 1 2 dξ, 2 1 2 v¯ v¯ G(ξ) 2 2 v¯2 (cid:18) (cid:19) (v¯ , v¯ ) is the stationary solution and 1 1 − a ≥ (18) G(v ) := 2 1 for v 0. 2 1 + kv 2 2 Compactness of level sets of V follows from uniform boundedness of model solutions. Direct calculations, presented in [Getto et al. 2013], allow to check that d ≤ (19) V (v (t), v (t)) 0. dt 1 2 We deﬁne (cid:8) (cid:9) ∈ | (20) Σ := (v , v ) R2 v > 0, v > 0 , 1 2 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J.-E.

text: BUSSE,

text: P.

text: GWIAZDA

text: AND

text: A.

text: MARCINIAK-CZOCHRA

text: where

text: C

text: u˜V

text: (u˜(t))

text: L∞.

text: Integrating

text: the

text: above

text: estimate,

text: we

text: obtain

text: t

text: V

text: (u˜(t))

text: V

text: (u

text: )e

text: β˜at

text: f

text: e

text: β˜a(t

text: τ)dτ.

text: a

text: a

text: Since,

text: by

text: deﬁnition,

text: f

text: we

text: conclude

text: that

text: V

text: (u(t))

text: for

text: t

text: The

text: a

text: convergence

text: holds

text: for

text: every

text: a

text: which

text: yields

text: convergence

text: V

text: (u(t))

text: δ,

text: which

text: is

text: the

text: minimum

text: of

text: the

text: function

text: V

text: In

text: turn,

text: it

text: ensures

text: that

text: u(t)

text: u¯.

text: of

text: Theorem

text: To

text: apply

text: Lemma

text: to

text: our

text: system

text: we

text: consider

text: a

text: ﬁnite

text: di-

text: mensional

text: model

text: obtained

text: by

text: setting

text: a(x)

text: to

text: a

text: constant

text: value

text: a

text: d

text: v

text: pv

text: dt

text: Kv

text: d

text: a

text: v

text: pv

text: dv

text: dt

text: Kv

text: v

text: v

text: A

text: Lyapunov

text: function

text: for

text: this

text: system

text: has

text: been

text: constructed

text: in

text: [Getto

text: et

text: al.

text: It

text: takes

text: the

text: form

text: V

text: (v

text: v

text: V

text: (v

text: v

text: V

text: (v

text: v

text: pG(v¯

text: d

text: where

text: v

text: V

text: (v

text: v

text: ln

text: v¯

text: v¯

text: v

text: G(v¯

text: V

text: (v

text: v

text: dξ,

text: v¯

text: v¯

text: G(ξ)

text: (v¯

text: v¯

text: is

text: the

text: stationary

text: solution

text: and

text: a

text: G(v

text: for

text: v

text: kv

text: Compactness

text: of

text: level

text: sets

text: of

text: V

text: follows

text: from

text: uniform

text: boundedness

text: of

text: model

text: solutions.

text: Direct

text: calculations,

text: presented

text: in

text: [Getto

text: et

text: al.

text: allow

text: to

text: check

text: that

text: d

text: V

text: (v

text: (t),

text: v

text: (t))

text: dt

text: We

text: deﬁne

text: Σ

text: (v

text: v

text: v

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: ##LTLine##

text: ##LTLine##


text: CLONAL SELECTION MODEL 21 { } (i) Ω = x¯ : a Convergence to a stationary solution follows from the convergence of mass given by Theorem 2.7. Hence, the solutions converge to measures concen- trated at x¯: −→→∞ t u(t, x) cδ , x¯ where δ is a Dirac measure localised at x¯ and c is a positive constant due x¯ to the positivity of masses. The convergence result can be understood in an appropriate metric on the space of positive Radon measures. We apply here the ﬂat metric ρ , F known also as a bounded Lipschitz distance [Neunzer 1981]. For complete- ness of presentation, we provide in Appendix the deﬁnition and basic prop- erties of this metric. To estimate the distance between a solution u(t, x) and the stationary measure cδ , we use the following inequality for the distance of two measures x¯ µ and µ 1 2 ≤ { } µ µ | − | (23) ρ (µ , µ ) min ρ , ρ W ( 1 , 2 ) + ρ ρ . F 1 2 1 2 1 ρ ρ 1 2 1 2 For the proof of this inequality we refer to [Carillo et al. 2012] and [Jabl(cid:32)on´ski 2013]. Here W (µ1 , µ2 ) denotes Wasserstein distance between two probabilistic 1 ρ1 ρ2 P × measures, for which w(cid:110)e(cid:90)use Lipschitz test(cid:12)functions. Namely, W : (cid:111) (R+) P −→ ∞ (cid:12) 1 (R+) [0, ] in its dual representati(cid:12)on is deﬁned by − ∈ ≤ W (ν , ν ) := sup ψ d(ν ν ) ψ C1(R+), Lip ψ 1 . 1 1 2 1 2 R+ The ﬁrst term on the right hand-side of (23) can be estimated using exponential estimates from Lemma 2.4, which allows concluding about its convergence to zero. Convergence of the second term comparing the two masses is due to Theorem 2.7. Hence, we obtain that ≤ ρ (u(t, x), δ ) k(t), F x¯ where k(t) is monotone, nonnegative and lim k(t) = 0. → ∞ t + L (ii) 1(Ω ) > 0: a If Ω is a set with positive measure, no singularites emerge due to the a uniform boundedness of the total mass. In this case, the solution tends to zero outside Ω and to a positive L1-function on Ω . Following Corollary a a 2.5, we conclude that the exact shape of the limit solution depends on the initial distribution. (cid:3) ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CLONAL

text: SELECTION

text: MODEL

label: list

text: (i)

text: Ω

text: x¯

text: a

text: Convergence

text: to

text: a

text: stationary

text: solution

text: follows

text: from

text: the

text: convergence

text: of

text: mass

text: given

text: by

text: Theorem

text: Hence,

text: the

text: solutions

text: converge

text: to

text: measures

text: concen-

text: trated

text: at

text: x¯:

text: t

text: u(t,

text: x)

text: cδ

text: x¯

text: where

text: δ

text: is

text: a

text: Dirac

text: measure

text: localised

text: at

text: x¯

text: and

text: c

text: is

text: a

text: positive

text: constant

text: due

text: x¯

text: to

text: the

text: positivity

text: of

text: masses.

text: The

text: convergence

text: result

text: can

text: be

text: understood

text: in

text: an

text: appropriate

text: metric

text: on

text: the

text: space

text: of

text: positive

text: Radon

text: measures.

text: We

text: apply

text: here

text: the

text: ﬂat

text: metric

text: ρ

text: F

text: known

text: also

text: as

text: a

text: bounded

text: Lipschitz

text: distance

text: [Neunzer

text: For

text: complete-

text: ness

text: of

text: presentation,

text: we

text: provide

text: in

text: Appendix

text: the

text: deﬁnition

text: and

text: basic

text: prop-

text: erties

text: of

text: this

text: metric.

text: To

text: estimate

text: the

text: distance

text: between

text: a

text: solution

text: u(t,

text: x)

text: and

text: the

text: stationary

text: measure

text: cδ

text: we

text: use

text: the

text: following

text: inequality

text: for

text: the

text: distance

text: of

text: two

text: measures

text: x¯

text: µ

text: and

text: µ

text: µ

text: µ

text: ρ

text: (µ

text: µ

text: min

text: ρ

text: ρ

text: W

text: ρ

text: ρ

text: F

text: ρ

text: ρ

text: For

text: the

text: proof

text: of

text: this

text: inequality

text: we

text: refer

text: to

text: [Carillo

text: et

text: al.

text: and

text: Here

text: W

text: µ2

text: denotes

text: Wasserstein

text: distance

text: between

text: two

text: probabilistic

text: ρ1

text: ρ2

text: P

text: measures,

text: for

text: which

text: Lipschitz

text: Namely,

text: W

text: (R+)

text: P

text: (R+)

text: in

text: its

text: dual

text: is

text: deﬁned

text: by

text: W

text: (ν

text: ν

text: sup

text: ψ

text: d(ν

text: ν

text: ψ

text: Lip

text: ψ

text: R+

text: The

text: ﬁrst

text: term

text: on

text: the

text: right

text: hand-side

text: of

text: can

text: be

text: estimated

text: using

text: exponential

text: estimates

text: from

text: Lemma

text: which

text: allows

text: concluding

text: about

text: its

text: convergence

text: to

text: zero.

text: Convergence

text: of

text: the

text: second

text: term

text: comparing

text: the

text: two

text: masses

text: is

text: due

text: to

text: Theorem

text: Hence,

text: we

text: obtain

text: that

text: ρ

text: (u(t,

text: x),

text: δ

text: k(t),

text: F

text: x¯

text: where

text: k(t)

text: is

text: monotone,

text: nonnegative

text: and

text: lim

text: k(t)

text: t

text: L

text: (ii)

text: a

text: If

text: Ω

text: is

text: a

text: set

text: with

text: positive

text: measure,

text: no

text: singularites

text: emerge

text: due

text: to

text: the

text: a

text: uniform

text: boundedness

text: of

text: the

text: total

text: mass.

text: In

text: this

text: case,

text: the

text: solution

text: tends

text: to

text: zero

text: outside

text: Ω

text: and

text: to

text: a

text: positive

text: on

text: Ω

text: Following

text: Corollary

text: a

text: a

text: we

text: conclude

text: that

text: the

text: exact

text: shape

text: of

text: the

text: limit

text: solution

text: depends

text: on

text: the

text: initial

text: distribution.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 J.-E. BUSSE, P. GWIAZDA AND A. MARCINIAK-CZOCHRA By the deﬁnition of ﬂat metric, it holds | − | ≤ ρ ρ˜ ρ (u λ, u˜ λ). 2 2 F 2 2 Hence, we can infer that d ∂ ≤ 2 dtρF (u1λ, u˜1λ) + ∂tρF (u2λ, u˜2λ) C ρF(cid:26)(u1λ, u˜1λ) (cid:27) + (ρ (u˜ λ, 0) + µ) ρ (u λ, u˜ λ) F 1 F 2 2 ≤ 2 max , ρ (u˜ λ, 0) + d (ρ (u λ, u˜ λ) C F 1 F 1 1 (cid:8) +(cid:9)ρ (u λ, u˜ λ)) . F 2 2 Deﬁning α(t) := max 2 , ρ (u˜ λ, 0) + d , we conclude by applying Gronwall’s C F 1 lemma (cid:0) (cid:82) (cid:1) t ≤ α(s)ds ρF (u1λ, u˜1λ) + ρF (u2λ, u˜2λ) e0 ρF (u1(0)λ, u˜1(0)λ) +ρ (u (0)λ, u˜ (0)λ) F 2 2 (cid:3) 7. Appendix We present here basic results related to the space of positive Radon measures equipped with ﬂat metric ρ , known also as a bounded Lipschitz distance [Neunzer 1981]. F ∈ M M × Deﬁnition 7.1. Let µ, ν +(R+). The distance function ρ : +(R+) M → ∞ F +(R+) [0, ] is deﬁn(cid:110)ed(cid:90)by (cid:12) (cid:111) (cid:12) − ∈ (cid:107) (cid:107) ≤ (29) ρ (µ, ν) := sup ψd(µ ν) ψ C1(R+), ψ ∞ 1 , F W1, R+ where (cid:107) (cid:107) {(cid:107) (cid:107) (cid:107) (cid:107) } ψ ∞ := max ψ ∞, ∂ ψ ∞ . W1, x It metrizes both weak* and narrow topologies on each tight subset of Radon mea- sures with uniformly bounded total variation [Schwartz 1973], [Ambrosio et al. 2000]. Proposition 7.2. Flat metric satisﬁes the following properties: • scale-invariance · · ρ (θ µ, θ ν) = θρ (µ, ν). F F ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J.-E.

text: BUSSE,

text: P.

text: GWIAZDA

text: AND

text: A.

text: MARCINIAK-CZOCHRA

text: By

text: the

text: deﬁnition

text: of

text: ﬂat

text: metric,

text: it

text: holds

text: ρ

text: ρ˜

text: ρ

text: (u

text: λ,

text: u˜

text: λ).

text: F

text: Hence,

text: we

text: can

text: infer

text: that

text: d

text: dtρF

text: ∂tρF

text: C

text: (ρ

text: (u˜

text: λ,

text: µ)

text: ρ

text: (u

text: λ,

text: u˜

text: λ)

text: F

text: F

text: max

text: ρ

text: (u˜

text: λ,

text: d

text: (ρ

text: (u

text: λ,

text: u˜

text: λ)

text: C

text: F

text: F

text: (u

text: λ,

text: u˜

text: λ))

text: F

text: Deﬁning

text: α(t)

text: max

text: ρ

text: (u˜

text: λ,

text: d

text: we

text: conclude

text: by

text: applying

text: Gronwall’s

text: C

text: F

text: lemma

text: t

text: α(s)ds

text: ρF

text: ρF

text: ρF

text: +ρ

text: (u

text: u˜

text: F

text: Appendix

text: We

text: present

text: here

text: basic

text: results

text: related

text: to

text: the

text: space

text: of

text: positive

text: Radon

text: measures

text: equipped

text: with

text: ﬂat

text: metric

text: ρ

text: known

text: also

text: as

text: a

text: bounded

text: Lipschitz

text: distance

text: [Neunzer

text: F

text: M

text: M

text: Deﬁnition

text: Let

text: µ,

text: ν

text: +(R+).

text: The

text: distance

text: function

text: ρ

text: +(R+)

text: M

text: F

text: +(R+)

text: is

text: ρ

text: (µ,

text: ν)

text: sup

text: ψd(µ

text: ν)

text: ψ

text: ψ

text: F

text: R+

text: where

text: ψ

text: max

text: ψ

text: ψ

text: x

text: It

text: metrizes

text: both

text: weak*

text: and

text: narrow

text: topologies

text: on

text: each

text: tight

text: subset

text: of

text: Radon

text: mea-

text: sures

text: with

text: uniformly

text: bounded

text: total

text: variation

text: [Schwartz

text: [Ambrosio

text: et

text: al.

text: Proposition

text: Flat

text: metric

text: satisﬁes

text: the

text: following

text: properties:

text: scale-invariance

text: ρ

text: (θ

text: µ,

text: θ

text: ν)

text: θρ

text: (µ,

text: ν).

text: F

text: F

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 18 Rn 16 1 min Rn k 14 opposite Rn 1 12 opposite min Rn k 10 k) s Ri g(10 8 o l 6 4 2 0 −2 0 50 100 150 200 Number of samples, n Fig. 1. Recursively tracked risks for the 4 hypotheses. In this Monte Carlo simulation, a change from f to f occurs at n = 100, 0 1 (cid:48) ≈ SNR = 3 dB, d = d 1.65. The parameter values were a = c = 1.45 and b = 10000. 0 1 The performance of the proposed sequential two-sided change detection scheme is evaluated next. Simulations were for a ﬁxed change time of n = 500, and parameter values were chosen to satisfy: (1) parameters a and c are chosen to control the probability of incorrect detection according to Theorem 1, and (2) parameter b is chosen to achieve a probability of false alarm of 0.05. Average detection delay is obtained as a function of SNR. Based on Section , it is known that these average detection delays represent the minimum delay cost for that particular SNR given a maximum allowable probability of false alarm of 0.05. The initial state chosen in each simulation realization is f or f with equal probability. Figure 2 shows the performance of the 0 1 sequential two-sided change detector alongside that of CUSUM, the optimal one-sided change detection algorithm assuming the initial state were known. From Figure 2 it is clear that the sequential two-sided detection scheme has delay only slightly greater than that of the optimal one-sided change detector, CUSUM, whose threshold is chosen to give the same allowable probability of false alarm to within 95% statistical conﬁdence. It must be noted that this performance comparison is unfair, being solutions to different problems: the sequential two-sided change detector identiﬁes a change in state given a random initial state, whereas CUSUM identiﬁes a change in state given a known initial state. Rather, the intention of ##LTFigure## ##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##

font: default

text: Rn

text: min

text: Rn

text: k

text: opposite

text: Rn

text: opposite

text: min

text: Rn

text: k

text: k)

text: s

text: Ri

text: o

text: l

text: −2

text: Number

text: of

text: samples,

text: n

text: Fig.

text: Recursively

label: caption

text: tracked

text: risks

text: for

text: the

text: hypotheses.

text: In

text: this

text: Monte

text: Carlo

text: simulation,

text: a

text: change

text: from

text: f

text: to

text: f

text: occurs

text: at

text: n

text: SNR

text: dB,

text: d

text: d

text: The

text: parameter

text: values

text: were

text: a

text: c

text: and

text: b

text: The

text: performance

text: of

text: the

text: proposed

text: sequential

text: two-sided

text: change

text: detection

text: scheme

text: is

text: evaluated

text: next.

text: Simulations

text: were

text: for

text: a

text: ﬁxed

text: change

text: time

text: of

text: n

text: and

text: parameter

text: values

text: were

text: chosen

text: to

text: satisfy:

text: parameters

text: a

text: and

text: c

text: are

text: chosen

text: to

text: control

text: the

text: probability

text: of

text: incorrect

text: detection

text: according

text: to

text: Theorem

text: and

text: parameter

text: b

text: is

text: chosen

text: to

text: achieve

text: a

text: probability

text: of

text: false

text: alarm

text: of

text: Average

text: detection

text: delay

text: is

text: obtained

text: as

text: a

text: function

text: of

text: SNR.

text: Based

text: on

text: Section

text: it

text: is

text: known

text: that

text: these

text: average

text: detection

text: delays

text: represent

text: the

text: minimum

text: delay

text: cost

text: for

text: that

text: particular

text: SNR

text: given

text: a

text: maximum

text: allowable

text: probability

text: of

text: false

text: alarm

text: of

text: The

text: initial

text: state

text: chosen

text: in

text: each

text: simulation

text: realization

text: is

text: f

text: or

text: f

text: with

text: equal

text: probability.

text: Figure

text: shows

text: the

text: performance

text: of

text: the

text: sequential

text: two-sided

text: change

text: detector

text: alongside

text: that

text: of

text: CUSUM,

text: the

text: optimal

text: one-sided

text: change

text: detection

text: algorithm

text: assuming

text: the

text: initial

text: state

text: were

text: known.

text: From

text: Figure

text: it

text: is

text: clear

text: that

text: the

text: sequential

text: two-sided

text: detection

text: scheme

text: has

text: delay

text: only

text: slightly

text: greater

text: than

text: that

text: of

text: the

text: optimal

text: one-sided

text: change

text: detector,

text: CUSUM,

text: whose

text: threshold

text: is

text: chosen

text: to

text: give

text: the

text: same

text: allowable

text: probability

text: of

text: false

text: alarm

text: to

text: within

text: statistical

text: conﬁdence.

text: It

text: must

text: be

text: noted

text: that

text: this

text: performance

text: comparison

text: is

text: unfair,

text: being

text: solutions

text: to

text: different

text: problems:

text: the

text: sequential

text: two-sided

text: change

text: detector

text: identiﬁes

text: a

text: change

text: in

text: state

text: given

text: a

text: random

text: initial

text: state,

text: whereas

text: CUSUM

text: identiﬁes

text: a

text: change

text: in

text: state

text: given

text: a

text: known

text: initial

text: state.

text: Rather,

text: the

text: intention

text: of

text: ##LTFigure##

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text: 3 reinforce the insights provided in Section as well as make comparisons to CUSUM, which has been shown to be optimal for one-sided quickest detection [4] [5]. II. SEQUENTIAL TWO SIDED CHANGE DETECTION At time n, without knowledge of the starting PDF (of X ), a sequence of independent random 1 ≡ · · · variables, Xn X , X , X , may assume 2n possible joint distributions corresponding to all 1 1 2 n possible single abrupt changes in distribution from f to f and vice versa. Classically, for a 0 1 ﬁxed number of samples, the problem may be formulated in terms of M-ary hypothesis testing [9]. Two of these 2n sequences, corresponding to no change, are described by the sequence Xn i ≤ ≤ ∈ { } − where all X , 1 i n are distributed according to f , where j 0, 1 . The other 2(n 1) i j ≤ possible sequences are described by the sequence Xn where X , 1 i < m, are distributed 1 i ≤ ≤ according to f , and the rest of the X , m i n, are distributed according to f , where j i k { } ∈ { } { } j, k 0, 1 or 1, 0 . The notation adopted to represent the 2n possible sequences is given by: { ∼ | ≤ ≤ } H = X f 1 i n (1) 1  i 0  { ∼ | ≤ ≤ }  H = X f 1 i n (2) 1 i 1  ∼ | ≤  X f 1 i < j ≤ ≤ H =  i 0  2 j n (3) j ∼ | ≤ ≤   X f j i n i 1  ∼ | ≤  X f 1 i < j ≤ ≤ H = i 1 2 j n. (4) j ∼ | ≤ ≤ X f j i n i 0 Suppose for now that n is ﬁxed. To formulate a decision rule to select among the 2n possible hypotheses, we adopt a Bayesian framework based on selecting the hypothesis with minimum risk according to Bayesian (2n)-ary hypothesis testing. Under equally likely prior probabilities of change time, a traditional Bayesian hypothesis testing scheme minimizes the posterior risk asso- ciated with each of the 2n hypotheses. However, when an inﬁnite-duration sequence is causally observed, the number of hypotheses grows with n, which results in increasing computational and ##LTLine## ##LTLine##

text: reinforce

text: the

text: insights

text: provided

text: in

text: Section

text: as

text: well

text: as

text: make

text: comparisons

text: to

text: CUSUM,

text: which

text: has

text: been

text: shown

text: to

text: be

text: optimal

text: for

text: one-sided

text: quickest

text: detection

text: II.

label: section

text: SEQUENTIAL

text: TWO

text: SIDED

text: CHANGE

text: DETECTION

text: At

text: time

text: n,

text: without

text: knowledge

text: of

text: the

text: starting

text: PDF

text: (of

text: X

text: a

text: sequence

text: of

text: independent

text: random

text: variables,

text: Xn

text: X

text: X

text: X

text: may

text: assume

text: possible

text: joint

text: distributions

text: corresponding

text: to

text: all

text: n

text: possible

text: single

text: abrupt

text: changes

text: in

text: distribution

text: from

text: f

text: to

text: f

text: and

text: vice

text: versa.

text: Classically,

text: for

text: a

text: ﬁxed

text: number

text: of

text: samples,

text: the

text: problem

text: may

text: be

text: formulated

text: in

text: terms

text: of

text: M-ary

text: hypothesis

text: testing

text: Two

text: of

text: these

text: sequences,

text: corresponding

text: to

text: no

text: change,

text: are

text: described

text: by

text: the

text: sequence

text: Xn

text: i

text: where

text: all

text: X

text: i

text: n

text: are

text: distributed

text: according

text: to

text: f

text: where

text: j

text: The

text: other

text: i

text: j

text: possible

text: sequences

text: are

text: described

text: by

text: the

text: sequence

text: Xn

text: where

text: X

text: i

text: m,

text: are

text: distributed

text: i

text: according

text: to

text: f

text: and

text: the

text: rest

text: of

text: the

text: X

text: m

text: i

text: n,

text: are

text: distributed

text: according

text: to

text: f

text: where

text: j

text: i

text: k

text: j,

text: k

text: or

text: The

text: notation

text: adopted

text: to

text: represent

text: the

text: possible

text: sequences

text: is

text: given

text: by:

text: H

text: X

text: f

text: i

text: n

text: i

text: H

text: X

text: f

text: i

text: n

text: i

text: X

text: f

text: i

text: j

text: H

text: i

text: j

text: n

text: j

text: X

text: f

text: j

text: i

text: n

text: i

text: X

text: f

text: i

text: j

text: H

text: i

text: j

text: n.

text: j

text: X

text: f

text: j

text: i

text: n

text: i

text: Suppose

text: for

text: now

text: that

text: n

text: is

text: ﬁxed.

text: To

text: formulate

text: a

text: decision

text: rule

text: to

text: select

text: among

text: the

text: possible

text: hypotheses,

text: we

text: adopt

text: a

text: Bayesian

text: framework

text: based

text: on

text: selecting

text: the

text: hypothesis

text: with

text: minimum

text: risk

text: according

text: to

text: Bayesian

text: hypothesis

text: testing.

text: Under

text: equally

text: likely

text: prior

text: probabilities

text: of

text: change

text: time,

text: a

text: traditional

text: Bayesian

text: hypothesis

text: testing

text: scheme

text: minimizes

text: the

text: posterior

text: risk

text: asso-

text: ciated

text: with

text: each

text: of

text: the

text: hypotheses.

text: However,

text: when

text: an

text: inﬁnite-duration

text: sequence

text: is

text: causally

text: observed,

text: the

text: number

text: of

text: hypotheses

text: grows

text: with

text: n,

text: which

text: results

text: in

text: increasing

text: computational

text: and

text: ##LTLine##

text: ##LTLine##


text: 21 90 CUSUM 80 Sequential Two−Sided Change Detector 70 60 y a Del 50 e g era 40 v A 30 20 10 0 −10 −5 0 5 10 SNR (dB) Fig. 2. Minimum average detection delay for a maximum probability of false alarm of 0.05. this comparison is to illustrate the performance penalty in not knowing the initial state. From this example, it can be concluded that the unknown initial state adds less than 10% to the average delay at an SNR as low as -6dB. In summary, this new approach is motivated by the fact that conventional approaches to change detection are one-sided and require a known initial state, and that creating a two-sided change detector from one-sided approaches has necessarily poor performance. Important features of the proposed approach are its efﬁcient recursive structure (but not provided in detail here) as well as its ability to effectively trade off delay and error probabilities under suitable parameter choices motivated by asymptotic analysis. REFERENCES [1] E. S. Page, “Continuous inspection schemes,” Biometrika, vol. 41, pp. 100–115, June 1954. [2] G. Lorden, “Procedures for reacting to a change in distribution,” Ann. Statist., vol. 42, no. 6, pp. 1897–1908, 1971. [3] A. N. Shiryaev, Optimal Stopping Rules. Springer, 1978. [4] G. V. Moustakides, “Optimal stopping times for detecting changes in distributions,” Ann. Statist., vol. 14, no. 4, pp. 1379– 1387, 1986. [5] H. V. Poor and O. Hadjiliadis, Quickest Detection. Cambridge University Press, 2008. [6] Q. Zhao and J. Ye, “Quickest Detection in Multiple On-Off Processes,” IEEE Trans. Signal Proc., vol. 58, no. 12, pp. 5994–6006, Dec. 2010. ##LTFigure## ##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: CUSUM

text: Sequential

text: Two−Sided

text: Change

text: Detector

text: y

text: a

text: Del

text: e

text: g

text: era

text: v

text: A

text: −5

text: SNR

text: (dB)

text: Fig.

text: Minimum

text: average

text: detection

text: delay

text: for

text: a

text: maximum

text: probability

text: of

text: false

text: alarm

text: of

text: this

text: comparison

text: is

text: to

text: illustrate

text: the

text: performance

text: penalty

text: in

text: not

text: knowing

text: the

text: initial

text: state.

text: From

text: this

text: example,

text: it

text: can

text: be

text: concluded

text: that

text: the

text: unknown

text: initial

text: state

text: adds

text: less

text: than

text: to

text: the

text: average

text: delay

text: at

text: an

text: SNR

text: as

text: low

text: as

text: In

text: summary,

text: this

text: new

text: approach

text: is

text: motivated

text: by

text: the

text: fact

text: that

text: conventional

text: approaches

text: to

text: change

text: detection

text: are

text: one-sided

text: and

text: require

text: a

text: known

text: initial

text: state,

text: and

text: that

text: creating

text: a

text: two-sided

text: change

text: detector

text: from

text: one-sided

text: approaches

text: has

text: necessarily

text: poor

text: performance.

text: Important

text: features

text: of

text: the

text: proposed

text: approach

text: are

text: its

text: efﬁcient

text: recursive

text: structure

text: (but

text: not

text: provided

text: in

text: detail

text: here)

text: as

text: well

text: as

text: its

text: ability

text: to

text: effectively

text: trade

text: off

text: delay

text: and

text: error

text: probabilities

text: under

text: suitable

text: parameter

text: choices

text: motivated

text: by

text: asymptotic

text: analysis.

text: REFERENCES

text: E.

text: S.

text: Page,

text: “Continuous

text: inspection

text: schemes,”

text: Biometrika,

text: vol.

text: pp.

text: June

text: G.

text: Lorden,

text: “Procedures

text: for

text: reacting

text: to

text: a

text: change

text: in

text: distribution,”

text: Ann.

text: Statist.,

text: vol.

text: no.

text: pp.

text: A.

text: N.

text: Shiryaev,

text: Optimal

text: Stopping

text: Rules.

text: Springer,

text: G.

text: V.

text: Moustakides,

text: “Optimal

text: stopping

text: times

text: for

text: detecting

text: changes

text: in

text: distributions,”

text: Ann.

text: Statist.,

text: vol.

text: no.

text: pp.

text: H.

text: V.

text: Poor

text: and

text: O.

text: Hadjiliadis,

text: Quickest

text: Detection.

text: Cambridge

text: University

text: Press,

text: Q.

text: Zhao

text: and

text: J.

text: Ye,

text: “Quickest

text: Detection

text: in

text: Multiple

text: On-Off

text: Processes,”

text: IEEE

text: Trans.

text: Signal

text: Proc.,

text: vol.

text: no.

text: pp.

text: Dec.

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text: 9 (cid:88) (cid:0) (cid:1) − n 1 − | Rn[1] = an jP (H Xn = xn) n j 1 1 j=2 | Rn[2] = bP (H Xn = xn)) n 1 1 1 (cid:88)− (cid:0) | (cid:1) Rn[3] = cn 1P (H Xn = xn) n 1 1 1 n − | Rn[4] = aj 1cP (H Xn = xn) , n j 1 1 j=2 where Rn[1] can be updated recursively as: n (cid:88) (cid:0) (cid:1) n − | Rnn++11[1] = (cid:88) (cid:0)a(n+1) jP (Hj X1n+1 = xn1+1)(cid:1) j=2 − n 1 − | | = (cid:88) (cid:0)a(n+1) jP (Hj X1n+1 =(cid:1) xn1+1) + aP (Hn X1n+1 = xn1+1) j=2 − n 1 − | | = an jP (H Xn = xn) af (x ) + aP (H Xn+1 = xn+1) j 1 1 1 n+1 n 1 1 j=2 | = Rn[1]af (x ) + aP (H Xn+1 = xn+1), n 1 n+1 n 1 1 Rn[2] can be updated recursively as: n | Rn+1[2] = bP (H Xn+1 = xn+1) n+1 1 1 1 | = (bP (H Xn = xn)) f (x ) 1 1 1 0 n+1 = Rn[2]f (x ), n 0 n+1 Rn[3] can be updated recursively as: n (cid:0) | (cid:1) Rn+1[3] = cnP (H Xn+1 = xn+1) n+1 1 1 1 − | = cn 1P (H Xn = xn) cf (x ) 1 1 1 1 n+1 = Rn[3]cf (x ), n 1 n+1 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: n

text: an

text: jP

text: (H

text: Xn

text: xn)

text: n

text: j

text: bP

text: (H

text: Xn

text: xn))

text: n

text: cn

text: (H

text: Xn

text: xn)

text: n

text: n

text: aj

text: (H

text: Xn

text: xn)

text: n

text: j

text: where

text: can

text: be

text: updated

text: recursively

text: as:

text: n

text: n

text: jP

text: (Hj

text: n

text: jP

text: (Hj

text: aP

text: (Hn

text: n

text: an

text: jP

text: (H

text: Xn

text: xn)

text: af

text: (x

text: aP

text: (H

text: j

text: n

text: (x

text: aP

text: (H

text: n

text: n

text: can

text: be

text: updated

text: recursively

text: as:

text: n

text: bP

text: (H

text: (bP

text: (H

text: Xn

text: xn))

text: f

text: (x

text: (x

text: n

text: can

text: be

text: updated

text: recursively

text: as:

text: n

text: cnP

text: (H

text: cn

text: (H

text: Xn

text: xn)

text: cf

text: (x

text: (x

text: n

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text: ##LTLine##

text: ##LTLine##


text: (cid:114) (cid:110) (cid:111) 1 µ − µ f = exp µ cos(φ ) φ2 (13) 0 2πI (µ) 2πα x 2 y 0 ∆ µ = 1 (14) 2k T b Where I (µ) is modiﬁed Bessel function of zero order. We can use this speciﬁc form of f to ﬁnd Fourier 0 0 coeﬃcients a(n0) (cid:114) (cid:110) (cid:111) σ(n)I (µ) (cid:40) µ −µ a(0) = n exp φ2 (15) n πI (µ) 2πα 2 y 0 1/2 : n = 0 σ(n) = (cid:54) (16) 1 : n = 1 where I (µ) are the modiﬁed Bessel functions of order n. n We also assume that initially at time t = 0 PDF is in equilibrium state f(φ , φ ; t = 0) = f (φ , φ ), x y 0 x y i.e. a (t = 0) = a(0) and b (t = 0) = 0. Equations (9) and (10) do not preclude time dependency of n n n both the electric E and magnetic B ﬁelds. However, keeping in line with the existing research in this ﬁeld [14], here we consider a magnetic ﬁeld B to be constant and the electric ﬁeld to be sum of a constant E dc and monochromatic ac E cos(ωt) components. Thus the total electric ﬁeld acting on electrons in SL is ω E = Edc + Eω cos(ωt). We are most interes(cid:28)ted in the property of(cid:29)absorption of ac electric ﬁeld, which we deﬁned as 2I (µ)v (t) A = 0 dr cos(ωt) (17) I (µ) 1 t (cid:104) (cid:105) Where v (t) is the instantaneous electron drift velocity along the x-axis and . . . means time averaging dr t over the period of 2π/ω. Negative absorption indicates ac ﬁeld ampliﬁcation, paving the way to a lasing medium. To compute absorption we have to let the system to relax to the steady state, which happens over time period of several relaxation time constants τ. In our case, since time is deﬁned in multiples of τ, see (4), in all numerical experiments we let system evolve up to time t = 10 and then compute averages, such as absorption (17). Instantaneous drift velocity along x-axis used in (17) is deﬁned as miniband velocity v x (2) averaged over PDF (cid:90)(cid:90) 2d(cid:90) ∂ε(cid:90) vdr(t) =∆ (cid:126) ∂p f(px, py; t)dpxdpy (18) √1 x ∞ π + = α dφ dφ sin(φ )f(φ , φ ; t) (19) − x −∞ y x x y π (cid:90) which in view of Fourier expansion (8) takes the form √ ∞ + v (t) = π α b (φ ; t)dφ (20) dr −∞ 1 y y 3. Numerical algorithm Naive application of method of ﬁnite diﬀerences to (9) and (10) leads to either unstable and/or compu- tationally intensive numerical system. To combat this problem we are using several methods at once. First, we discretize a and b along time and φ axes. n n y 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: µ

text: µ

text: f

text: exp

text: µ

text: cos(φ

text: φ2

text: (µ)

text: 2πα

text: x

text: y

text: µ

text: T

text: b

text: Where

text: I

text: (µ)

text: is

text: modiﬁed

text: Bessel

text: function

text: of

text: zero

text: order.

text: We

text: can

text: use

text: this

text: speciﬁc

text: form

text: of

text: f

text: to

text: ﬁnd

text: Fourier

text: coeﬃcients

text: σ(n)I

text: (µ)

text: µ

text: −µ

text: n

text: exp

text: φ2

text: n

text: πI

text: (µ)

text: 2πα

text: y

text: n

text: σ(n)

text: n

text: where

text: I

text: (µ)

text: are

text: the

text: modiﬁed

text: Bessel

text: functions

text: of

text: order

text: n.

text: n

text: We

text: also

text: assume

text: that

text: initially

text: at

text: time

text: t

text: PDF

text: is

text: in

text: equilibrium

text: state

text: f(φ

text: φ

text: t

text: f

text: (φ

text: φ

text: x

text: y

text: x

text: y

text: i.e.

text: a

text: (t

text: and

text: b

text: (t

text: Equations

text: and

text: do

text: not

text: preclude

text: time

text: dependency

text: of

text: n

text: n

text: n

text: both

text: the

text: electric

text: E

text: and

text: magnetic

text: B

text: ﬁelds.

text: However,

text: keeping

text: in

text: line

text: with

text: the

text: existing

text: research

text: in

text: this

text: ﬁeld

text: here

text: we

text: consider

text: a

text: magnetic

text: ﬁeld

text: B

text: to

text: be

text: constant

text: and

text: the

text: electric

text: ﬁeld

text: to

text: be

text: sum

text: of

text: a

text: constant

text: E

text: dc

text: and

text: monochromatic

text: ac

text: E

text: cos(ωt)

text: components.

text: Thus

text: the

text: total

text: electric

text: ﬁeld

text: acting

text: on

text: electrons

text: in

text: SL

text: is

text: ω

text: E

text: Edc

text: Eω

text: cos(ωt).

text: We

text: are

text: most

text: in

text: the

text: property

text: of

text: ac

text: electric

text: ﬁeld,

text: which

text: we

text: deﬁned

text: as

text: (µ)v

text: (t)

text: A

text: dr

text: cos(ωt)

text: I

text: (µ)

text: t

text: Where

text: v

text: (t)

text: is

text: the

text: instantaneous

text: electron

text: drift

text: velocity

text: along

text: the

text: x-axis

text: and

text: means

text: time

text: averaging

text: dr

text: t

text: over

text: the

text: period

text: of

text: Negative

text: absorption

text: indicates

text: ac

text: ﬁeld

text: ampliﬁcation,

text: paving

text: the

text: way

text: to

text: a

text: lasing

text: medium.

text: To

text: compute

text: absorption

text: we

text: have

text: to

text: let

text: the

text: system

text: to

text: relax

text: to

text: the

text: steady

text: state,

text: which

text: happens

text: over

text: time

text: period

text: of

text: several

text: relaxation

text: time

text: constants

text: τ.

text: In

text: our

text: case,

text: since

text: time

text: is

text: deﬁned

text: in

text: multiples

text: of

text: τ,

text: see

text: in

text: all

text: numerical

text: experiments

text: we

text: let

text: system

text: evolve

text: up

text: to

text: time

text: t

text: and

text: then

text: compute

text: averages,

text: such

text: as

text: absorption

text: Instantaneous

text: drift

text: velocity

text: along

text: x-axis

text: used

text: in

text: is

text: deﬁned

text: as

text: miniband

text: velocity

text: v

text: x

text: averaged

text: over

text: PDF

text: vdr(t)

text: ∂p

text: f(px,

text: py;

text: t)dpxdpy

text: √1

text: x

text: π

text: α

text: dφ

text: dφ

text: sin(φ

text: )f(φ

text: φ

text: t)

text: x

text: y

text: x

text: x

text: y

text: π

text: which

text: in

text: view

text: of

text: Fourier

text: expansion

text: takes

text: the

text: form

text: v

text: (t)

text: π

text: α

text: b

text: (φ

text: t)dφ

text: dr

text: y

text: y

text: Numerical

text: algorithm

text: Naive

text: application

text: of

text: method

text: of

text: ﬁnite

text: diﬀerences

text: to

text: and

text: leads

text: to

text: either

text: unstable

text: and/or

text: compu-

text: tationally

text: intensive

text: numerical

text: system.

text: To

text: combat

text: this

text: problem

text: we

text: are

text: using

text: several

text: methods

text: at

text: once.

text: First,

text: we

text: discretize

text: a

text: and

text: b

text: along

text: time

text: and

text: φ

text: axes.

text: n

text: n

text: y

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: ##LTLine##


text: 14 4 Optimality conditions and strong duality In this section,we turn our attention to the optimality conditions for MCFP. In par- ticular, we ﬁrst show that not only conditions (CS1)-(CS4) are suﬃcient for optimality, but also are necessary under Assumptions 1 and 2. We then extend the negative cycle optimality condition for MCFP and derive a strong duality result between (MCFP) ∗ and (MCFP ). We start with the following lemma which will be used in the proof of the subsequent theorems. Lemma 4 Let x be a piecewise analytic ﬂow. Then x is not optimal if there exists a negative augmenting cycle with respect to x. Proof Suppose that W : (v1, θ1), . . . , ((cid:48)vq, θq) is a ne(cid:48)gative augmentin(cid:48)g cycle under x. We show that there exists a cycle W with cap(W ) > 0 and c(W ) < 0. To prove this, let Q : (v(cid:96), θ(cid:96)), . . . , (vr, θr) be |a maximal arc-subpath of W. Then we know that there is some (cid:15) > 0 such that W (α) is also an augmenting cycle for each α in − Q | (θ (cid:15), θ ) or (θ , θ + (cid:15)). We assume without loss of generality that W (α) is an (cid:96) (cid:96) (cid:96) (cid:96) − ∈ − Q augmenting cycle for each α in (θ (cid:15), θ ). Then for some α (θ (cid:15), θ ), we have | − | (cid:96) (cid:96) (cid:96) (cid:96) c(Q(α)) c(Q) < (cid:15). Here Q(α) denotes the arc path (v(cid:96), v(cid:96)+1), . . . , (vr−1, vr) with starting time α. More precisely, we have Q(α)−: (v(cid:96), α(cid:96)), . . . , (vr, αr) where α(cid:96) = α and αk+1(α) = αk + τvk,vk+1 for k = (cid:96), . . . , r 1. Furth|er, α can be chosen in such a way that cap(Q(α)) > 0. Now we consider the cycle W (α) and repeat the above Q (cid:48) procedure for all remaining maximal arc-subpaths of W. Let W be the resulting cycle. (cid:48) (cid:48) It is easy to see that for suﬃciently small (cid:15) > 0, we get cap(W ) > 0 and c(W ) < 0. Based on the above observation, we assume that cap(W) > 0 and c(W) < 0. For ∈ every k = 2, . . . , q with e = (v − , v ) Er, there exist δ , γ > 0 such that k k 1 k k k ≥ ∀ ∈ ur (θ) δ θ [θ − , θ − + γ ) . ek k k 1 k 1 k Let δ and γ be the minimum of δ and γ , respectively, and deﬁne (cid:15) = 2δγ. k k k Moreover, for every k = 2, . . . , q with v − = v , there exist δ , γ > 0 such that k 1 k k k ≥ ∀ ∈ − Ur (θ) δ θ (θ − γ , θ + γ ) if θ − < θ , vk ≥ k ∀ ∈ k −1 k k k k 1 k Lr (θ) δ θ (θ γ , θ − + γ ) if θ − > θ . vk k k k k 1 k k 1 k (cid:40) Let δ and γ be the minimum of δ and γ , respectively. We then deﬁne k k (cid:54) 2δγ, if v − = v , (cid:15) := k 1 k k δ, if v − = v , k 1 k for k = 2, . . . , and let ∗ 1 { } z := min (cid:15)2, (cid:15)3, . . . , (cid:15)q . (7) 2γ  We now deﬁne ∗ ∈ ∈ − z if e = (v , v ) E, θ [θ , θ + γ) for k = 1, . . . , q 1, k k+1 k k − ∗ ∈ ∈ − ze(θ) := z if e = (vk+1, i = vk) E, θ [θk+1, θk+1 + γ), for k = 1, . . . , q 1, 0 otherwise. ##LTLine## ##LTLine##

text: Optimality

text: conditions

text: and

text: strong

text: duality

text: In

text: this

text: section,we

text: turn

text: our

text: attention

text: to

text: the

text: optimality

text: conditions

text: for

text: MCFP.

text: In

text: par-

text: ticular,

text: we

text: ﬁrst

text: show

text: that

text: not

text: only

text: conditions

text: are

text: suﬃcient

text: for

text: optimality,

text: but

text: also

text: are

text: necessary

text: under

text: Assumptions

text: and

text: We

text: then

text: extend

text: the

text: negative

text: cycle

text: optimality

text: condition

text: for

text: MCFP

text: and

text: derive

text: a

text: strong

text: duality

text: result

text: between

text: (MCFP)

text: and

text: (MCFP

text: We

text: start

text: with

text: the

text: following

text: lemma

text: which

text: will

text: be

text: used

text: in

text: the

text: proof

text: of

text: the

text: subsequent

text: theorems.

text: Lemma

text: Let

text: x

text: be

text: a

text: piecewise

text: analytic

text: ﬂow.

text: Then

text: x

text: is

text: not

text: optimal

text: if

text: there

text: exists

text: a

text: negative

text: augmenting

text: cycle

text: with

text: respect

text: to

text: x.

text: Proof

text: Suppose

text: that

text: W

text: θq)

text: is

text: a

text: cycle

text: under

text: x.

text: We

text: show

text: that

text: there

text: exists

text: a

text: cycle

text: W

text: with

text: cap(W

text: and

text: c(W

text: To

text: prove

text: this,

text: let

text: Q

text: (vr,

text: θr)

text: be

text: |a

text: maximal

text: arc-subpath

text: of

text: W.

text: Then

text: we

text: know

text: that

text: there

text: is

text: some

text: such

text: that

text: W

text: (α)

text: is

text: also

text: an

text: augmenting

text: cycle

text: for

text: each

text: α

text: in

text: Q

text: (θ

text: θ

text: or

text: (θ

text: θ

text: We

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: W

text: (α)

text: is

text: an

text: Q

text: augmenting

text: cycle

text: for

text: each

text: α

text: in

text: (θ

text: θ

text: Then

text: for

text: some

text: α

text: (θ

text: θ

text: we

text: have

text: c(Q(α))

text: c(Q)

text: Here

text: Q(α)

text: denotes

text: the

text: arc

text: path

text: vr)

text: with

text: starting

text: time

text: α.

text: More

text: precisely,

text: we

text: have

text: Q(α)−:

text: (vr,

text: αr)

text: where

text: α

text: and

text: αk

text: for

text: k

text: r

text: Furth|er,

text: α

text: can

text: be

text: chosen

text: in

text: such

text: a

text: way

text: that

text: cap(Q(α))

text: Now

text: we

text: consider

text: the

text: cycle

text: W

text: (α)

text: and

text: repeat

text: the

text: above

text: Q

text: procedure

text: for

text: all

text: remaining

text: maximal

text: arc-subpaths

text: of

text: W.

text: Let

text: W

text: be

text: the

text: resulting

text: cycle.

text: It

text: is

text: easy

text: to

text: see

text: that

text: for

text: suﬃciently

text: small

text: we

text: get

text: cap(W

text: and

text: c(W

text: Based

text: on

text: the

text: above

text: observation,

text: we

text: assume

text: that

text: cap(W)

text: and

text: c(W)

text: For

text: every

text: k

text: q

text: with

text: e

text: (v

text: v

text: Er,

text: there

text: exist

text: δ

text: γ

text: such

text: that

text: k

text: k

text: k

text: k

text: k

text: ur

text: (θ)

text: δ

text: θ

text: [θ

text: θ

text: γ

text: ek

text: k

text: k

text: k

text: k

text: Let

text: δ

text: and

text: γ

text: be

text: the

text: minimum

text: of

text: δ

text: and

text: γ

text: respectively,

text: and

text: deﬁne

text: 2δγ.

text: k

text: k

text: k

text: Moreover,

text: for

text: every

text: k

text: q

text: with

text: v

text: v

text: there

text: exist

text: δ

text: γ

text: such

text: that

text: k

text: k

text: k

text: k

text: Ur

text: (θ)

text: δ

text: θ

text: (θ

text: γ

text: θ

text: γ

text: if

text: θ

text: θ

text: vk

text: k

text: k

text: −1

text: k

text: k

text: k

text: k

text: k

text: Lr

text: (θ)

text: δ

text: θ

text: (θ

text: γ

text: θ

text: γ

text: if

text: θ

text: θ

text: vk

text: k

text: k

text: k

text: k

text: k

text: k

text: k

text: Let

text: δ

text: and

text: γ

text: be

text: the

text: minimum

text: of

text: δ

text: and

text: γ

text: respectively.

text: We

text: then

text: deﬁne

text: k

text: k

text: 2δγ,

text: if

text: v

text: v

text: k

text: k

text: k

text: δ,

text: if

text: v

text: v

text: k

text: k

text: for

text: k

text: and

text: let

text: z

text: min

text: 2γ

text: We

text: now

text: deﬁne

text: z

text: if

text: e

text: (v

text: v

text: E,

text: θ

text: [θ

text: θ

text: γ)

text: for

text: k

text: q

text: k

text: k

text: k

text: ze(θ)

text: z

text: if

text: e

text: i

text: vk)

text: E,

text: θ

text: γ),

text: for

text: k

text: q

text: otherwise.

text: ##LTLine##

text: ##LTLine##


text: 17 Acknowledgment We would like to thank two anonymous referees for many useful comments that helped to improve the presentation of the paper. We are grateful to Martin Skutella for valuable suggestions that enhanced this paper. References 1. E. J. Anderson. A Continuous Model for Job-Shop Scheduling. PhD thesis, University of Cambridge,, 1978. 2. E. J. Anderson. Extreme-points for continuous network programs with arc delays. Journal of Information and Optimization Sciences, 10:45–52, 1989. 3. E. J. Anderson and P. Nash. Linear Programming in Inﬁnite-Dimensional Spaces. Wiley, New York, 1987. 4. E. J. Anderson, P. Nash, and A. B. Philpott. A class of continuous network ﬂow problems. Mathematics of Operations Research, 7:501–514, 1982. 5. E. J. Anderson and A. B. Philpott. A continuous-time network simplex algorithm. Net- works, 19:395–425, 1989. 6. E. J. Anderson and A. B. Philpott. Optimisation of ﬂows in networks over time. In F. P. Kelly, editor, Probability, Statistics and Optimisation, chapter 27, pages 369–382. Wiley, New York, 1994. 7. T. M. Apostol. Mathematical Analysis. 2th Edition, Addison-Wesley, 1974. 8. X. Cai, T. Kloks, and C. K. Wong. Time-varying shortest path problems with constraints. Networks, 27:141–149, 1997. 9. X. Cai, D. Sha, and C. K. Wong. Time-varying minimum cost ﬂow problems. European Journal of Operational Research, 131:352–374, 2001. 10. L. Fleischer and M. Skutella. Quickest ﬂows over time. SIAM Journal on Computing, 36:1600–1630, 2007. 11. L. K. Fleischer and E´. Tardos. Eﬃcient continuous-time dynamic network ﬂow algorithms. Operations Research Letters, 23:71–80, 1998. 12. L. R. Ford and D. R. Fulkerson. Constructing maximal dynamic ﬂows from static ﬂows. Operations Research, 6:419–433, 1958. 13. L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, 1962. 14. S. M. Hashemi and E. Nasrabadi. On solving continuous-time dynamic network ﬂows. Journal of Global Optimization, 497–524:2012, 2011. 15. R. Koch and E. Nasrabadi. Continuous-time dynamic shortest paths with negative transit times. Technical Report 6-2010, Technische Universit¨at Berlin, 2010. under revision for SIAM Journal on Control and Optimization. 16. R. Koch, E. Nasrabadi, and M. Skutella. Continuous and discrete ﬂows over time. Math- ematical Methods of Operations Research, 73:301–337, 2010. 17. E. Miller-Hooks and S. S. Patterson. On solving quickest time problems in time-dependent, dynamic networks. Journal of Mathematical Modelling and Algorithms, 3:39–71, 2004. 18. E. D Miller-Hooks and H. S. Mahmassani. Least possible time paths in stochastic, time- varying networks. Computers & operations research, 25(12):1107–1125, 1998. 19. E. Nasrabadi and S. M. Hashemi. Minimum cost time-varying network ﬂow problems. Optimization Methods & Software, 25(3):429–447, 2010. 20. S. Opasanon and E. Miller-Hooks. Multicriteria adaptive paths in stochastic, time-varying networks. European Journal of Operational Research, 173(1):72–91, 2006. 21. A. Orda and R. Rom. Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length. Journal of the ACM, 37(3):607–625, 1990. 22. A. Orda and R. Rom. Minimum weight paths in time-dependent networks. Networks, 21(3):295–319, 1991. 23. A. Orda and R. Rom. On continuous network ﬂows. Operations Research Letters, 17:27–36, 1995. 24. A. B. Philpott. Algorithms For Continuous Network Flow Problems. PhD thesis, Univer- sity of Cambridge, UK, 1982. 25. A. B. Philpott. Network programming in continuous time with node storage. In E. J. An- derson and A. B. Philpott, editors, Inﬁnite Programming: Proceedings of an International Symposium on Inﬁnite Dimensional Linear Programming, volume 259 of Lecture notes in economics and mathematical systems, pages 136–153. Springer, Berlin, 1985. ##LTLine##

text: Acknowledgment

text: We

text: would

text: like

text: to

text: thank

text: two

text: anonymous

text: referees

text: for

text: many

text: useful

text: comments

text: that

text: helped

text: to

text: improve

text: the

text: presentation

text: of

text: the

text: paper.

text: We

text: are

text: grateful

text: to

text: Martin

text: Skutella

text: for

text: valuable

text: suggestions

text: that

text: enhanced

text: this

text: paper.

text: References

label: reference

text: E.

text: J.

text: Anderson.

text: A

text: Continuous

text: Model

text: for

text: Job-Shop

text: Scheduling.

text: PhD

text: thesis,

text: University

text: of

text: Cambridge,,

text: E.

text: J.

text: Anderson.

text: Extreme-points

text: for

text: continuous

text: network

text: programs

text: with

text: arc

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text: thesis,

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text: sity

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text: UK,

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text: with

text: node

text: storage.

text: In

text: E.

text: J.

text: An-

text: derson

text: and

text: A.

text: B.

text: Philpott,

text: editors,

text: Inﬁnite

text: Programming:

text: Proceedings

text: of

text: an

text: International

text: Symposium

text: on

text: Inﬁnite

text: Dimensional

text: Linear

text: Programming,

text: volume

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text: notes

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text: 3 tion. In particular, Anderson and Philpott [5] attempt to develop a simplex algorithm for MCFP with zero transit times and piecewise constant/linear input functions. They discuss how the simplex method can be developed for MCFP to directly produce an exact solution, rather than doing a discretization to get an approximation to the op- timal solution. But, there are no guarantees for the convergence of this algorithm and it often produces a sequence of solutions which converge to a suboptimal solution. In the most recent paper, Weiss [32] examines SCLP with piecewise linear problem data and develops a simplex algorithm that gives an exact solution after a ﬁnite number of iterations. He also characterizes the form of optimal solutions and establishes a strong duality result. So far we reviewed the literature on network ﬂows over time where transit times are assumed to be time-invariant. There is also a number of models that allow the transit times to vary over time in both discrete-time model [8,9,17,17,20,18,19,33] and continuous-time model [14,21–23]. In particular, Miller-Hooks and Patterson [17] present a pseudo-polynomial time algorithm for the problem of sending a given amount of ﬂow from a single source to a single sink at the shortest possible time in a time- varying network, where all parameters can change at discrete points in time. They also present a technique for converting a network with multiple sources and multiple sinks into an equivalent single source and sink network by adding a small number of nodes and arcs to the existing network. Cai et al. [9] study the problem of sending a given amount of ﬂow from a single source to a single sink with minimum cost in a time-varying network. They present a kind of the successive shortest path algorithm for solving the problem that runs in pseudo-polynomial-time. Nasrabadi and Hashemi [19] extent this model to multiple sources and multiple sinks in which the supplies at source nodes and demands at sink nodes may vary with time. They present a discrete-time version of the successive shortest path algorithm for solving the resulting problem. Our contribution. In this paper, we are concerned with the development of continuous- time analogues to those concepts and techniques which are the cornerstones of static network ﬂows. Speciﬁcally, we develop several network based optimality conditions analogous to that found in static network ﬂows for MCFP with piecewise analytic input functions and rational transit times. We derive a strong duality result from these optimality conditions. It is worth pointing out that previously, strong duality was developed by Pullan [28,30] for SCLP given piecewise analytic problem data and for SCLPTD with rational transit times and piecewise constant/linear input functions. The main result of his pa- per is a strong duality theorem for SCLP with piecewise analytic data. He ﬁrst showed that strong duality holds for SCLP under the conditions of piecewise constant/linear problem data in his original paper [27] as a consequence of his elegant algorithm. This result was the starting point of an extensive duality theory in [28]. In this paper, we do not follow the Pullan’s approach, but rather we make use of ideas from the area of static network ﬂows. The remainder of this paper is organized as follows. Section provides preliminaries and earlier results that are required for the purpose of the paper. In Section , we introduce the concept of augmenting paths and cycles, and prove the existence of least cost augmenting paths. We then establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for MCFP in Section . Section is devoted to our conclusions. Our results in Sections and are based on the assumption that the cost functions are continuous. In Appendix , we show that this ##LTLine##

text: tion.

text: In

text: particular,

text: Anderson

text: and

text: Philpott

text: attempt

text: to

text: develop

text: a

text: simplex

text: algorithm

text: for

text: MCFP

text: with

text: zero

text: transit

text: times

text: and

text: piecewise

text: constant/linear

text: input

text: functions.

text: They

text: discuss

text: how

text: the

text: simplex

text: method

text: can

text: be

text: developed

text: for

text: MCFP

text: to

text: directly

text: produce

text: an

text: exact

text: solution,

text: rather

text: than

text: doing

text: a

text: discretization

text: to

text: get

text: an

text: approximation

text: to

text: the

text: op-

text: timal

text: solution.

text: But,

text: there

text: are

text: no

text: guarantees

text: for

text: the

text: convergence

text: of

text: this

text: algorithm

text: and

text: it

text: often

text: produces

text: a

text: sequence

text: of

text: solutions

text: which

text: converge

text: to

text: a

text: suboptimal

text: solution.

text: In

text: the

text: most

text: recent

text: paper,

text: Weiss

text: examines

text: SCLP

text: with

text: piecewise

text: linear

text: problem

text: data

text: and

text: develops

text: a

text: simplex

text: algorithm

text: that

text: gives

text: an

text: exact

text: solution

text: after

text: a

text: ﬁnite

text: number

text: of

text: iterations.

text: He

text: also

text: characterizes

text: the

text: form

text: of

text: optimal

text: solutions

text: and

text: establishes

text: a

text: strong

text: duality

text: result.

text: So

text: far

text: we

text: reviewed

text: the

text: literature

text: on

text: network

text: ﬂows

text: over

text: time

text: where

text: transit

text: times

text: are

text: assumed

text: to

text: be

text: time-invariant.

text: There

text: is

text: also

text: a

text: number

text: of

text: models

text: that

text: allow

text: the

text: transit

text: times

text: to

text: vary

text: over

text: time

text: in

text: both

text: discrete-time

text: model

text: and

text: continuous-time

text: model

text: In

text: particular,

text: Miller-Hooks

text: and

text: Patterson

text: present

text: a

text: pseudo-polynomial

text: time

text: algorithm

text: for

text: the

text: problem

text: of

text: sending

text: a

text: given

text: amount

text: of

text: ﬂow

text: from

text: a

text: single

text: source

text: to

text: a

text: single

text: sink

text: at

text: the

text: shortest

text: possible

text: time

text: in

text: a

text: time-

text: varying

text: network,

text: where

text: all

text: parameters

text: can

text: change

text: at

text: discrete

text: points

text: in

text: time.

text: They

text: also

text: present

text: a

text: technique

text: for

text: converting

text: a

text: network

text: with

text: multiple

text: sources

text: and

text: multiple

text: sinks

text: into

text: an

text: equivalent

text: single

text: source

text: and

text: sink

text: network

text: by

text: adding

text: a

text: small

text: number

text: of

text: nodes

text: and

text: arcs

text: to

text: the

text: existing

text: network.

text: Cai

text: et

text: al.

text: study

text: the

text: problem

text: of

text: sending

text: a

text: given

text: amount

text: of

text: ﬂow

text: from

text: a

text: single

text: source

text: to

text: a

text: single

text: sink

text: with

text: minimum

text: cost

text: in

text: a

text: time-varying

text: network.

text: They

text: present

text: a

text: kind

text: of

text: the

text: successive

text: shortest

text: path

text: algorithm

text: for

text: solving

text: the

text: problem

text: that

text: runs

text: in

text: pseudo-polynomial-time.

text: Nasrabadi

text: and

text: Hashemi

text: extent

text: this

text: model

text: to

text: multiple

text: sources

text: and

text: multiple

text: sinks

text: in

text: which

text: the

text: supplies

text: at

text: source

text: nodes

text: and

text: demands

text: at

text: sink

text: nodes

text: may

text: vary

text: with

text: time.

text: They

text: present

text: a

text: discrete-time

text: version

text: of

text: the

text: successive

text: shortest

text: path

text: algorithm

text: for

text: solving

text: the

text: resulting

text: problem.

text: Our

text: contribution.

text: In

text: this

text: paper,

text: we

text: are

text: concerned

text: with

text: the

text: development

text: of

text: continuous-

text: time

text: analogues

text: to

text: those

text: concepts

text: and

text: techniques

text: which

text: are

text: the

text: cornerstones

text: of

text: static

text: network

text: ﬂows.

text: Speciﬁcally,

text: we

text: develop

text: several

text: network

text: based

text: optimality

text: conditions

text: analogous

text: to

text: that

text: found

text: in

text: static

text: network

text: ﬂows

text: for

text: MCFP

text: with

text: piecewise

text: analytic

text: input

text: functions

text: and

text: rational

text: transit

text: times.

text: We

text: derive

text: a

text: strong

text: duality

text: result

text: from

text: these

text: optimality

text: conditions.

text: It

text: is

text: worth

text: pointing

text: out

text: that

text: previously,

text: strong

text: duality

text: was

text: developed

text: by

text: Pullan

text: for

text: SCLP

text: given

text: piecewise

text: analytic

text: problem

text: data

text: and

text: for

text: SCLPTD

text: with

text: rational

text: transit

text: times

text: and

text: piecewise

text: constant/linear

text: input

text: functions.

text: The

text: main

text: result

text: of

text: his

text: pa-

text: per

text: is

text: a

text: strong

text: duality

text: theorem

text: for

text: SCLP

text: with

text: piecewise

text: analytic

text: data.

text: He

text: ﬁrst

text: showed

text: that

text: strong

text: duality

text: holds

text: for

text: SCLP

text: under

text: the

text: conditions

text: of

text: piecewise

text: constant/linear

text: problem

text: data

text: in

text: his

text: original

text: paper

text: as

text: a

text: consequence

text: of

text: his

text: elegant

text: algorithm.

text: This

text: result

text: was

text: the

text: starting

text: point

text: of

text: an

text: extensive

text: duality

text: theory

text: in

text: In

text: this

text: paper,

text: we

text: do

text: not

text: follow

text: the

text: Pullan’s

text: approach,

text: but

text: rather

text: we

text: make

text: use

text: of

text: ideas

text: from

text: the

text: area

text: of

text: static

text: network

text: ﬂows.

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: provides

text: preliminaries

text: and

text: earlier

text: results

text: that

text: are

text: required

text: for

text: the

text: purpose

text: of

text: the

text: paper.

text: In

text: Section

text: we

text: introduce

text: the

text: concept

text: of

text: augmenting

text: paths

text: and

text: cycles,

text: and

text: prove

text: the

text: existence

text: of

text: least

text: cost

text: augmenting

text: paths.

text: We

text: then

text: establish

text: a

text: reduced

text: cost

text: optimality

text: condition,

text: a

text: negative

text: cycle

text: optimality

text: condition,

text: and

text: a

text: strong

text: duality

text: result

text: for

text: MCFP

text: in

text: Section

text: Section

text: is

text: devoted

text: to

text: our

text: conclusions.

text: Our

text: results

text: in

text: Sections

text: and

text: are

text: based

text: on

text: the

text: assumption

text: that

text: the

text: cost

text: functions

text: are

text: continuous.

text: In

text: Appendix

text: we

text: show

text: that

text: this

text: ##LTLine##