text: 23 As discussed in the Minkowski example, if one is only interested in is the shape of the curve and doesn’t care whether the parameter is aﬃne or not, then by rescaling(cid:32)(cid:18)with (cid:19) (cid:33) − d ± A 1 2d − s = − 1 τ 1 (A64) a (1 2d)A d 0 we recover − x (τ) = s (u cos ϕ r ω sin ϕ ) + r cos ϕ c rc 0 c0 0 0 c0 0 (A65) y (τ) = s (u sin ϕ + r ω cos ϕ ) + r sin ϕ , c rc 0 c0 0 0 c0 0 which is exactly the same as the spatial parametrizations in Minkowski spacetime. With an additional rescaling of t we have explicitly recovered the conformal ﬂatness of FLRW in comoving spherical coordinates. This may seem like a very roundabout way to get to a result that could have been found more or less immediately from conformal arguments, however, the point of this approach is that we can ﬁnd an explicit aﬃne parametrization of the geodesics in these coordinates, and the fact that we explicitly recover the conformality of the solutions is an important check on the calculations. 5. Null geodesics in FLRW: physical coordinates It is also advantageous to work in “physical coordinates” centered on the cloak, rather than co-moving coordinates. The reason for this is that for the cloak to remain of ﬁxed size, the individual elements making up the cloak must overcome the Hubble ﬂow by whatever internal forces that are keeping the cloak together. This means that although we may consider the center of the cloak to be comoving, other points on the cloak are not comoving. Therefore, a cloak of ﬁxed physical size must actually be shrinking when described in comoving coordinates. Despite the fact that any coordinate system should be equally adequate for describing null geodesics, physical coordinates are may be more advantageous for describing a cloaking device of ﬁxed physical size, and for describing the cloak parameters in a system in which all parts of the cloak are seen to be at rest with respect to the observer. Physical coordinates are reﬂective of the distances actually measured by an observer. A ﬁxed comoving observer measures the physical distance from himself (at r = 0) to a galaxy at ﬁxed comoving coordinate r to be d = a(t)r . p p p We let this increasing physical distance be the physical radial coordinate, so r = a(t)r, where r refers to the comoving p coordinates. From this relation we have (cid:48) (cid:48) a dr = a r dt + adr = r dt + adr , (A66) p c c a p c and therefore (cid:48) 1 − a dr = dr r dt. (A67) c a p a2 p Plugging this into the FLRW line el(cid:18)ement Eq. (18(cid:19)), there results − − a˙(t)2 − a˙(t) ds2 = c2 r2 dt2 2 r dr dt + dr2 + r2dΩ2 (A68) a(t)2 p a(t) p p p p Since the transformation from comoving to physical coordinates only involves transforming the r coordinate, it is expected that the equations for t and ϕ˙ should remain essentially unchanged (we again assume equatorial motion). Calculating the geodesic equations conﬁrms that expectation, with (cid:18) (cid:19) u a t˙ = t 0 , a r 2 (A69) ϕ˙ = ω p0 . 0 r p ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: As

text: discussed

text: in

text: the

text: Minkowski

text: example,

text: if

text: one

text: is

text: only

text: interested

text: in

text: is

text: the

text: shape

text: of

text: the

text: curve

text: and

text: doesn’t

text: care

text: whether

text: the

text: parameter

text: is

text: aﬃne

text: or

text: not,

text: then

text: by

text: d

text: A

text: s

text: τ

text: a

text: d

text: we

text: recover

text: x

text: (τ)

text: s

text: (u

text: cos

text: ϕ

text: r

text: ω

text: sin

text: ϕ

text: r

text: cos

text: ϕ

text: c

text: rc

text: y

text: (τ)

text: s

text: (u

text: sin

text: ϕ

text: r

text: ω

text: cos

text: ϕ

text: r

text: sin

text: ϕ

text: c

text: rc

text: which

text: is

text: exactly

text: the

text: same

text: as

text: the

text: spatial

text: parametrizations

text: in

text: Minkowski

text: spacetime.

text: With

text: an

text: additional

text: rescaling

text: of

text: t

text: we

text: have

text: explicitly

text: recovered

text: the

text: conformal

text: ﬂatness

text: of

text: FLRW

text: in

text: comoving

text: spherical

text: coordinates.

text: This

text: may

text: seem

text: like

text: a

text: very

text: roundabout

text: way

text: to

text: get

text: to

text: a

text: result

text: that

text: could

text: have

text: been

text: found

text: more

text: or

text: less

text: immediately

text: from

text: conformal

text: arguments,

text: however,

text: the

text: point

text: of

text: this

text: approach

text: is

text: that

text: we

text: can

text: ﬁnd

text: an

text: explicit

text: aﬃne

text: parametrization

text: of

text: the

text: geodesics

text: in

text: these

text: coordinates,

text: and

text: the

text: fact

text: that

text: we

text: explicitly

text: recover

text: the

text: conformality

text: of

text: the

text: solutions

text: is

text: an

text: important

text: check

text: on

text: the

text: calculations.

text: geodesics

text: in

text: FLRW:

text: physical

text: coordinates

text: It

text: is

text: also

text: advantageous

text: to

text: work

text: in

text: “physical

text: coordinates”

text: centered

text: on

text: the

text: cloak,

text: rather

text: than

text: co-moving

text: coordinates.

text: The

text: reason

text: for

text: this

text: is

text: that

text: for

text: the

text: cloak

text: to

text: remain

text: of

text: ﬁxed

text: size,

text: the

text: individual

text: elements

text: making

text: up

text: the

text: cloak

text: must

text: overcome

text: the

text: Hubble

text: ﬂow

text: by

text: whatever

text: internal

text: forces

text: that

text: are

text: keeping

text: the

text: cloak

text: together.

text: This

text: means

text: that

text: although

text: we

text: may

text: consider

text: the

text: center

text: of

text: the

text: cloak

text: to

text: be

text: comoving,

text: other

text: points

text: on

text: the

text: cloak

text: are

text: not

text: comoving.

text: Therefore,

text: a

text: cloak

text: of

text: ﬁxed

text: physical

text: size

text: must

text: actually

text: be

text: shrinking

text: when

text: described

text: in

text: comoving

text: coordinates.

text: Despite

text: the

text: fact

text: that

text: any

text: coordinate

text: system

text: should

text: be

text: equally

text: adequate

text: for

text: describing

text: geodesics,

text: physical

text: coordinates

text: are

text: may

text: be

text: more

text: advantageous

text: for

text: describing

text: a

text: cloaking

text: device

text: of

text: ﬁxed

text: physical

text: size,

text: and

text: for

text: describing

text: the

text: cloak

text: parameters

text: in

text: a

text: system

text: in

text: which

text: all

text: parts

text: of

text: the

text: cloak

text: are

text: seen

text: to

text: be

text: at

text: rest

text: with

text: respect

text: to

text: the

text: observer.

text: Physical

text: coordinates

text: are

text: reﬂective

text: of

text: the

text: distances

text: actually

text: measured

text: by

text: an

text: observer.

text: A

text: ﬁxed

text: comoving

text: observer

text: measures

text: the

text: physical

text: distance

text: from

text: himself

text: (at

text: r

text: to

text: a

text: galaxy

text: at

text: ﬁxed

text: comoving

text: coordinate

text: r

text: to

text: be

text: d

text: a(t)r

text: p

text: p

text: p

text: We

text: let

text: this

text: increasing

text: physical

text: distance

text: be

text: the

text: physical

text: radial

text: coordinate,

text: so

text: r

text: a(t)r,

text: where

text: r

text: refers

text: to

text: the

text: comoving

text: p

text: coordinates.

text: From

text: this

text: relation

text: we

text: have

text: a

text: dr

text: a

text: r

text: dt

text: adr

text: r

text: dt

text: adr

text: p

text: c

text: c

text: a

text: p

text: c

text: and

text: therefore

text: a

text: dr

text: dr

text: r

text: dt.

text: c

text: a

text: p

text: p

text: Plugging

text: this

text: into

text: the

text: FLRW

text: line

text: Eq.

text: there

text: results

text: a˙(t)

text: r

text: dr

text: dt

text: p

text: a(t)

text: p

text: p

text: p

text: p

text: Since

text: the

text: transformation

text: from

text: comoving

text: to

text: physical

text: coordinates

text: only

text: involves

text: transforming

text: the

text: r

text: coordinate,

text: it

text: is

text: expected

text: that

text: the

text: equations

text: for

text: t

text: and

text: ϕ˙

text: should

text: remain

text: essentially

text: unchanged

text: (we

text: again

text: assume

text: equatorial

text: motion).

text: Calculating

text: the

text: geodesic

text: equations

text: conﬁrms

text: that

text: expectation,

text: with

text: u

text: a

text: t˙

text: t

text: a

text: r

text: ϕ˙

text: ω

text: r

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 yp yc xc xp (a) Comoving coordinates (b) Physical coordinates FIG. 2. Null curves of FLRW spacetime plotted in (a) comoving coordinates, and (b) physical coordinates. Despite the non- zero curvature of FLRW spacetime its conformal ﬂatness is apparent in comoving coordinates, where initially parallel geodesics remain parallel as in Minkowski spacetime. The expansion of an initially parallel congruence of null geodesics is made clear in physical coordinates. T − Also recall that to calculate the transformation medium χ˜ with Eq. (17) we actually require the inverse = T 1 rather than T itself. This transformation maps the origin to the cloak’s inner surface at ﬁxed radius r (1), but in comoving coordinates c the ﬁxed radius r (1) deﬁnes the surface of a physically expanding sphere. To a local observer, such a comoving cloak c would appear to be expanding with the expansion of the universe. Although one could, with all validity in the context of transformation optics, use this corporeal transformation and calculate the corresponding transformation medium, a physically expanding cloak is not what we had in mind. Instead, the picture we have in mind is of a cloak of ﬁxed physical size relative to a real, physical observer. Every element of a cloak of ﬁxed physical size with inner radius R and outer radius R would actually have some non-zero 1 2 velocity in comoving coordinates. This means that the cloak medium would not be comoving, being instead held together by whatever electrostatic forces are present to overcome the universal expansion and keep the cloak from breaking apart. In other words, a cloak of ﬁxed physical size would actually appear to be shrinking if described in comoving coordinates. On the other hand, in physical coordinates a cloak of ﬁxed physical size would not appear unordinary to a real, physical observer, and there the transformation − → R R r R + 2 1 r (29) p 1 R p 2 would be appropriate. It is not immediately obvious whether the Pendry transformation will work in this setting of universal expansion, but the action of a cloak derived from this corporeal transformation in physical coordinates is depicted in Fig. 3, and it may be observed that the downstream ray trajectories are perfectly preserved. yp xp FIG. 3. Equatorial slice of a cloak in physical coordinates of FLRW spacetime. The universal expansion of a set of initially parallel null curves is accommodated by the cloak transformation. ##LTFigure## ##LTFigure## ##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##

label: paragraph

text: yp

font: default

text: yc

text: xc

text: xp

text: (a)

text: Comoving

text: coordinates

text: (b)

text: Physical

text: coordinates

text: FIG.

label: caption

text: curves

text: of

text: FLRW

text: spacetime

text: plotted

text: in

text: (a)

text: comoving

text: coordinates,

text: and

text: (b)

text: physical

text: coordinates.

text: Despite

text: the

text: non-

text: zero

text: curvature

text: of

text: FLRW

text: spacetime

text: its

text: conformal

text: ﬂatness

text: is

text: apparent

text: in

text: comoving

text: coordinates,

text: where

text: initially

text: parallel

text: geodesics

text: remain

text: parallel

text: as

text: in

text: Minkowski

text: spacetime.

text: The

text: expansion

text: of

text: an

text: initially

text: parallel

text: congruence

text: of

text: geodesics

text: is

text: made

text: clear

text: in

text: physical

text: coordinates.

text: T

text: Also

text: recall

text: that

text: to

text: calculate

text: the

text: transformation

text: medium

text: χ˜

text: with

text: Eq.

text: we

text: actually

text: require

text: the

text: inverse

text: T

text: rather

text: than

text: T

text: itself.

text: This

text: transformation

text: maps

text: the

text: origin

text: to

text: the

text: cloak’s

text: inner

text: surface

text: at

text: ﬁxed

text: radius

text: r

text: but

text: in

text: comoving

text: coordinates

text: c

text: the

text: ﬁxed

text: radius

text: r

text: deﬁnes

text: the

text: surface

text: of

text: a

text: physically

text: expanding

text: sphere.

text: To

text: a

text: local

text: observer,

text: such

text: a

text: comoving

text: cloak

text: c

text: would

text: appear

text: to

text: be

text: expanding

text: with

text: the

text: expansion

text: of

text: the

text: universe.

text: Although

text: one

text: could,

text: with

text: all

text: validity

text: in

text: the

text: context

text: of

text: transformation

text: optics,

text: use

text: this

text: corporeal

text: transformation

text: and

text: calculate

text: the

text: corresponding

text: transformation

text: medium,

text: a

text: physically

text: expanding

text: cloak

text: is

text: not

text: what

text: we

text: had

text: in

text: mind.

text: Instead,

text: the

text: picture

text: we

text: have

text: in

text: mind

text: is

text: of

text: a

text: cloak

text: of

text: ﬁxed

text: physical

text: size

text: relative

text: to

text: a

text: real,

text: physical

text: observer.

text: Every

text: element

text: of

text: a

text: cloak

text: of

text: ﬁxed

text: physical

text: size

text: with

text: inner

text: radius

text: R

text: and

text: outer

text: radius

text: R

text: would

text: actually

text: have

text: some

text: non-zero

text: velocity

text: in

text: comoving

text: coordinates.

text: This

text: means

text: that

text: the

text: cloak

text: medium

text: would

text: not

text: be

text: comoving,

text: being

text: instead

text: held

text: together

text: by

text: whatever

text: electrostatic

text: forces

text: are

text: present

text: to

text: overcome

text: the

text: universal

text: expansion

text: and

text: keep

text: the

text: cloak

text: from

text: breaking

text: apart.

text: In

text: other

text: words,

text: a

text: cloak

text: of

text: ﬁxed

text: physical

text: size

text: would

text: actually

text: appear

text: to

text: be

text: shrinking

text: if

text: described

text: in

text: comoving

text: coordinates.

text: On

text: the

text: other

text: hand,

text: in

text: physical

text: coordinates

text: a

text: cloak

text: of

text: ﬁxed

text: physical

text: size

text: would

text: not

text: appear

text: unordinary

text: to

text: a

text: real,

text: physical

text: observer,

text: and

text: there

text: the

text: transformation

text: R

text: R

text: r

text: R

text: r

text: p

text: R

text: p

text: would

text: be

text: appropriate.

text: It

text: is

text: not

text: immediately

text: obvious

text: whether

text: the

text: Pendry

text: transformation

text: will

text: work

text: in

text: this

text: setting

text: of

text: universal

text: expansion,

text: but

text: the

text: action

text: of

text: a

text: cloak

text: derived

text: from

text: this

text: corporeal

text: transformation

text: in

text: physical

text: coordinates

text: is

text: depicted

text: in

text: Fig.

text: and

text: it

text: may

text: be

text: observed

text: that

text: the

text: downstream

text: ray

text: trajectories

text: are

text: perfectly

text: preserved.

text: yp

text: xp

text: FIG.

text: Equatorial

text: slice

text: of

text: a

text: cloak

text: in

text: physical

text: coordinates

text: of

text: FLRW

text: spacetime.

text: The

text: universal

text: expansion

text: of

text: a

text: set

text: of

text: initially

text: parallel

text: curves

text: is

text: accommodated

text: by

text: the

text: cloak

text: transformation.

text: ##LTFigure##

text: ##LTFigure##

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: ##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: 10 9 20 8 40 7 60 6 5 80 4 100 3 2 120 1 20 40 60 80 100 120 Figure 9: Parameter n(x). 1 −0.1 20 0.8 20 −0.2 0.6 −0.3 40 40 0.4 −0.4 60 0.2 60 −0.5 −0.6 0 80 −0.2 80 −0.7 −0.8 −0.4 100 100 −0.9 −0.6 −1 120 −0.8 120 −1.1 20 40 60 80 100 120 20 40 60 80 100 120 Figure 10: Reference solution u, Left: Real part of the solution; Right: Imaginary part of the solution. 1 −0.1 20 0.8 20 −0.2 0.6 −0.3 40 40 0.4 −0.4 60 0.2 60 −0.5 0 −0.6 80 −0.2 80 −0.7 −0.8 −0.4 100 100 −0.9 −0.6 −1 120 −0.8 120 −1.1 20 40 60 80 100 120 20 40 60 80 100 120 Figure 11: Snapshot solution (i.e., when using all snapshot vectors), u, Left: Real part; Right: Imaginary part. 1 with length h = H/16, therefore, the ﬁne mesh size h = . We consider Ω = 8 and n(x) is shown 128 in ﬁgure 13. The parameter κ, source function f, and boundary condition g, are the same as the previous example. Because of higher value of Ω, we take the ﬁne grid 2 times ﬁner. Discussions of numerical results. We will compare our results with the multiscale approach with the reference solution, which calculated on the ﬁne grid and shown in Figure 14. Notice that, within each coarse grid block, the reference solution has few dominant propagating directions, which 19 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## ##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## ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: Parameter

text: n(x).

text: −1

text: Figure

text: Reference

text: solution

text: u,

text: Left:

text: Real

text: part

text: of

text: the

text: solution;

text: Right:

text: Imaginary

text: part

text: of

text: the

text: solution.

text: −1

text: Figure

text: Snapshot

text: solution

text: (i.e.,

text: when

text: using

text: all

text: snapshot

text: vectors),

text: u,

text: Left:

text: Real

text: part;

text: Right:

text: Imaginary

text: part.

text: with

text: length

text: h

text: therefore,

text: the

text: ﬁne

text: mesh

text: size

text: h

text: We

text: consider

text: Ω

text: and

text: n(x)

text: is

text: shown

text: in

text: ﬁgure

text: The

text: parameter

text: κ,

text: source

text: function

text: f,

text: and

text: boundary

text: condition

text: g,

text: are

text: the

text: same

text: as

text: the

text: previous

text: example.

text: Because

text: of

text: higher

text: value

text: of

text: Ω,

text: we

text: take

text: the

text: ﬁne

text: grid

text: times

text: ﬁner.

text: Discussions

text: of

text: numerical

text: results.

text: We

text: will

text: compare

text: our

text: results

text: with

text: the

text: multiscale

text: approach

text: with

text: the

text: reference

text: solution,

text: which

text: calculated

text: on

text: the

text: ﬁne

text: grid

text: and

text: shown

text: in

text: Figure

text: Notice

text: that,

text: within

text: each

text: coarse

text: grid

text: block,

text: the

text: reference

text: solution

text: has

text: few

text: dominant

text: propagating

text: directions,

text: which

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: L (ψj) = 0 in τ (1) 0 i j subject to some boundary conditions, where these boundary conditions play an important role in deﬁning snapshot functions. One option is to choose all possible boundary conditions considering all unit vectors on the boundary of τ . More precisely, ψj(x) = δ (x) on ∂τ , where δ (x) is 1 at the j i i j i node i and zero elsewhere. The computations of these snapshot functions are expensive. Instead, we use the boundary conditions, which are randomly distributed numbers on the ﬁne-grid nodes of the boundary ∂τ (see the left plot in Figure 2). The random boundary conditions allow extracting the j essential information provided we choose several more snapshot vectors than the number of modes, we would like to use. For parameter-dependent problems, the snapshot vectors are deﬁned as above (1) for each pre-selected value of µ . For example, for one-dimensional case τ = [x , x ], for (cid:0) m (cid:1) j j j+1 parameter-independent problem, the snapshot space in τ consists of two solutions ψj and ψj+1, j such that ψn(x ) = δ , n = j, j + 1, l = j, j + 1, where δ is the Kronecker symbol and ψn l nl jl (n = j, j + 1) is a solution of d κ((cid:18)x; µ = 0) d ψn =(cid:19) 0 in τ (see Figure 1 for illustration). For dx dx j parameter-dependent problems, the snapshot vectors in τ = [x , x ] are the solutions of j j j+1 d d κ(x; µ ) ψn = 0 in τ , dx m dx m j ψn (x ) = δ , n = j, j + 1, l = j, j + 1. For multi-dimensional examples, we can construct the m l nl snapshots similarly for each τ and for using diﬀerent boundary conditions and use one index to j represent the snapshot vectors as ψj. For the second example, Lu = −div(κ(x)∇u) − Ω2n(x)u, we i · choose the snapshot vectors to be functions eiΩki x for a set of pre-deﬁned values of ki on a unit circle (see the right plot in Figure 2). 1.5 ψj ψj+1 1 0.5 0 x x j j+1 fine grid τ j −0.5 −1 0 0.2 0.4 0.6 0.8 1 Figure 1: Illustration of snapshot concepts in one dimensional example. The multiscale basis functions are constructed in the snapshot space. Our earlier approaches seek a small dimensional subspace of the snapshot space by performing a local spectral decom- 3 ##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##

text: L

label: equation

text: (ψj)

text: in

text: τ

text: i

text: j

text: subject

text: to

text: some

text: boundary

text: conditions,

text: where

text: these

text: boundary

text: conditions

text: play

text: an

text: important

text: role

text: in

text: deﬁning

text: snapshot

text: functions.

text: One

text: option

text: is

text: to

text: choose

text: all

text: possible

text: boundary

text: conditions

text: considering

text: all

text: unit

text: vectors

text: on

text: the

text: boundary

text: of

text: τ

text: More

text: precisely,

text: ψj(x)

text: δ

text: (x)

text: on

text: ∂τ

text: where

text: δ

text: (x)

text: is

text: at

text: the

text: j

text: i

text: i

text: j

text: i

text: node

text: i

text: and

text: zero

text: elsewhere.

text: The

text: computations

text: of

text: these

text: snapshot

text: functions

text: are

text: expensive.

text: Instead,

text: we

text: use

text: the

text: boundary

text: conditions,

text: which

text: are

text: randomly

text: distributed

text: numbers

text: on

text: the

text: ﬁne-grid

text: nodes

text: of

text: the

text: boundary

text: ∂τ

text: (see

text: the

text: left

text: plot

text: in

text: Figure

text: The

text: random

text: boundary

text: conditions

text: allow

text: extracting

text: the

text: j

text: essential

text: information

text: provided

text: we

text: choose

text: several

text: more

text: snapshot

text: vectors

text: than

text: the

text: number

text: of

text: modes,

text: we

text: would

text: like

text: to

text: use.

text: For

text: parameter-dependent

text: problems,

text: the

text: snapshot

text: vectors

text: are

text: deﬁned

text: as

text: above

text: for

text: each

text: pre-selected

text: value

text: of

text: µ

text: For

text: example,

text: for

text: one-dimensional

text: case

text: τ

text: [x

text: x

text: for

text: m

text: j

text: j

text: parameter-independent

text: problem,

text: the

text: snapshot

text: space

text: in

text: τ

text: consists

text: of

text: two

text: solutions

text: ψj

text: and

text: j

text: such

text: that

text: ψn(x

text: δ

text: n

text: j,

text: j

text: l

text: j,

text: j

text: where

text: δ

text: is

text: the

text: Kronecker

text: symbol

text: and

text: ψn

text: l

text: nl

text: jl

text: (n

text: j,

text: j

text: is

text: a

text: solution

text: of

text: d

text: µ

text: d

text: ψn

text: in

text: τ

text: (see

text: Figure

text: for

text: illustration).

text: For

text: dx

text: dx

text: j

text: parameter-dependent

text: problems,

text: the

text: snapshot

text: vectors

text: in

text: τ

text: [x

text: x

text: are

text: the

text: solutions

text: of

text: j

text: j

text: d

text: d

text: κ(x;

text: µ

text: ψn

text: in

text: τ

text: dx

text: m

text: dx

text: m

text: j

text: ψn

text: (x

text: δ

text: n

text: j,

text: j

text: l

text: j,

text: j

text: For

text: multi-dimensional

text: examples,

text: we

text: can

text: construct

text: the

text: m

text: l

text: nl

text: snapshots

text: similarly

text: for

text: each

text: τ

text: and

text: for

text: using

text: diﬀerent

text: boundary

text: conditions

text: and

text: use

text: one

text: index

text: to

text: j

text: represent

text: the

text: snapshot

text: vectors

text: as

text: ψj.

text: For

text: the

text: second

text: example,

text: Lu

text: −div(κ(x)∇u)

text: we

text: i

text: choose

text: the

text: snapshot

text: vectors

text: to

text: be

text: functions

text: eiΩki

text: x

text: for

text: a

text: set

text: of

text: pre-deﬁned

text: values

text: of

text: ki

text: on

text: a

text: unit

text: circle

text: (see

text: the

text: right

text: plot

text: in

text: Figure

text: ψj

text: x

text: x

text: j

text: fine

text: grid

text: τ

text: j

text: −1

text: Figure

text: Illustration

text: of

text: snapshot

text: concepts

text: in

text: one

text: dimensional

text: example.

text: The

text: multiscale

text: basis

text: functions

text: are

text: constructed

text: in

text: the

text: snapshot

text: space.

text: Our

text: earlier

text: approaches

text: seek

text: a

text: small

text: dimensional

text: subspace

text: of

text: the

text: snapshot

text: space

text: by

text: performing

text: a

text: local

text: spectral

text: decom-

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: ##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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: position (based on analysis). However, these approaches use all snapshot vectors when seeking (cid:88) multiscale basis functions. In a number of applications, the solution is sparse in the snapshot space. I.e., in the expansion u = c ψj, i,j i i,j many coeﬃcients c are zeros. In this case, one can save computational eﬀort by employing sparsity i,j techniques. In this paper, our main goal is to discuss how GMsFEM can be designed if the solution is sparse in the snapshot space. We describe two classes of approaches and present a framework for constructing sparse GMsFEM. The main challenge in these applications is to construct a snapshot space, where the solution is sparse. In our ﬁrst example, this can be achieved, because an online parameter value µ can be close to some of the pre-selected oﬄine values of µ’s, and thus, the multiscale basis functions (and the solution) can have a sparse representation in the snapshot space. In our second example, we select cases where the solution u contains only a few snapshot vectors corresponding to directions k . We i note that if the snapshot space is not chosen carefully, one may not have the sparsity. In general, there can be many other examples and our goal is to show how local multiscale model reduction techniques can be used for such problems. Figure 2: Illustration of snapshot concepts. Left: For the ﬁrst problem; Right: For the second problem. 1.4 Approaches for identifying sparse solutions in snapshot spaces The main goal of this paper is to discuss how to explore sparsity ideas within GMsFEM for constructing local multiscale basis functions. We consider two distinct cases. • First approach: “Local-Sparse Snapshot Subspace Approach”. Determining the online sparse space locally via local spectral sparse decomposition in the snapshot space (motivated by parameter-dependent problems). • Second approach: “Sparse Snapshot Subspace Approach”. Determining the online space globally via a global solve (motivated by using plane wave snapshot vectors and the Helmholtz equation). See Figure 3 for illustration. We use sparsity techniques (e.g., [6, 5, 7, 38, 44]) to identify local multiscale basis functions and solve the global problem. 4 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: position

text: (based

text: on

text: analysis).

text: However,

text: these

text: approaches

text: use

text: all

text: snapshot

text: vectors

text: when

text: seeking

text: multiscale

text: basis

text: functions.

text: In

text: a

text: number

text: of

text: applications,

text: the

text: solution

text: is

text: sparse

text: in

text: the

text: snapshot

text: space.

text: I.e.,

text: in

text: the

text: expansion

text: u

text: c

text: ψj,

text: i,j

text: i

text: i,j

text: many

text: coeﬃcients

text: c

text: are

text: zeros.

text: In

text: this

text: case,

text: one

text: can

text: save

text: computational

text: eﬀort

text: by

text: employing

text: sparsity

text: i,j

text: techniques.

text: In

text: this

text: paper,

text: our

text: main

text: goal

text: is

text: to

text: discuss

text: how

text: GMsFEM

text: can

text: be

text: designed

text: if

text: the

text: solution

text: is

text: sparse

text: in

text: the

text: snapshot

text: space.

text: We

text: describe

text: two

text: classes

text: of

text: approaches

text: and

text: present

text: a

text: framework

text: for

text: constructing

text: sparse

text: GMsFEM.

text: The

text: main

text: challenge

text: in

text: these

text: applications

text: is

text: to

text: construct

text: a

text: snapshot

text: space,

text: where

text: the

text: solution

text: is

text: sparse.

text: In

text: our

text: ﬁrst

text: example,

text: this

text: can

text: be

text: achieved,

text: because

text: an

text: online

text: parameter

text: value

text: µ

text: can

text: be

text: close

text: to

text: some

text: of

text: the

text: pre-selected

text: oﬄine

text: values

text: of

text: µ’s,

text: and

text: thus,

text: the

text: multiscale

text: basis

text: functions

text: (and

text: the

text: solution)

text: can

text: have

text: a

text: sparse

text: representation

text: in

text: the

text: snapshot

text: space.

text: In

text: our

text: second

text: example,

text: we

text: select

text: cases

text: where

text: the

text: solution

text: u

text: contains

text: only

text: a

text: few

text: snapshot

text: vectors

text: corresponding

text: to

text: directions

text: k

text: We

text: i

text: note

text: that

text: if

text: the

text: snapshot

text: space

text: is

text: not

text: chosen

text: carefully,

text: one

text: may

text: not

text: have

text: the

text: sparsity.

text: In

text: general,

text: there

text: can

text: be

text: many

text: other

text: examples

text: and

text: our

text: goal

text: is

text: to

text: show

text: how

text: local

text: multiscale

text: model

text: reduction

text: techniques

text: can

text: be

text: used

text: for

text: such

text: problems.

text: Figure

text: Illustration

text: of

text: snapshot

text: concepts.

text: Left:

text: For

text: the

text: ﬁrst

text: problem;

text: Right:

text: For

text: the

text: second

text: problem.

text: Approaches

text: for

text: identifying

text: sparse

text: solutions

text: in

text: snapshot

text: spaces

text: The

text: main

text: goal

text: of

text: this

text: paper

text: is

text: to

text: discuss

text: how

text: to

text: explore

text: sparsity

text: ideas

text: within

text: GMsFEM

text: for

text: constructing

text: local

text: multiscale

text: basis

text: functions.

text: We

text: consider

text: two

text: distinct

text: cases.

text: First

text: approach:

text: “Local-Sparse

text: Snapshot

text: Subspace

text: Approach”.

text: Determining

text: the

text: online

text: sparse

text: space

text: locally

text: via

text: local

text: spectral

text: sparse

text: decomposition

text: in

text: the

text: snapshot

text: space

text: (motivated

text: by

text: parameter-dependent

text: problems).

text: Second

text: approach:

text: “Sparse

text: Snapshot

text: Subspace

text: Approach”.

text: Determining

text: the

text: online

text: space

text: globally

text: via

text: a

text: global

text: solve

text: (motivated

text: by

text: using

text: plane

text: wave

text: snapshot

text: vectors

text: and

text: the

text: Helmholtz

text: equation).

text: See

text: Figure

text: for

text: illustration.

text: We

text: use

text: sparsity

text: techniques

text: (e.g.,

text: to

text: identify

text: local

text: multiscale

text: basis

text: functions

text: and

text: solve

text: the

text: global

text: problem.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: through multiplication of a random matrix T τ . Denote T τ as a matrix of size (L × J)by q random random (q << (L×J)) with rows of i.i.d standard Gaussian random vectors. Then the test space is deﬁned as Ψτ T τ . snap random Speciﬁcally, the local problem is arranged as follows. Find U , such that, ψτ,on = Ψτ U , and l l snap l 1 U = argmin (cid:107)U(cid:107) subject to A (µ)U = F . (9) l ν 1 c l l Here, A (µ) = (Ψτ T τ )T A(µ)Ψτ and F = (Ψτ T τ )T R with A(µ) being the local c snap random snap l snap random l stiﬀness matrix and R being the right hand side for the local problem with Dirichlet boundary l condition r . Namely, we are solving the following local problems in V τ with the l minimized l (cid:40) snap 1 coeﬃcient vector U for the testing space equal to Ψτ T τ , l snap random −div(κ(x; µ)∇ψτ+,on) = 0 in τ+ l (10) ψτ+,on = r on ∂τ+. l l Later, we will brieﬂy introduce the algorithm to solve Equation (9). Note that we impose the same random vectors as the boundary conditions in Equation (7) to guarantee the sparse solution in Equation (10). We can obtain the local online snapshot functions on the target domain τ by restricting the solution of the local problem, ψτ+,on to τ (which is denoted by ψτ,on). Now we are ready to present l l the local online snapshot space as follows, V τ = Span{ψτ,on : 1 ≤ l ≤ L}, on l for each coarse subdomain τ. Then we denote Ψτ = [ψτ,on, · · · , ψτ,on]. on 1 L Next we will select the dominant modes from V τ by the following eigenvalue problem, on Aτ,on(µ)zτ,on = λτ,onSτ,on(µ)zτ,on, (11) k k k (cid:90) where (cid:90) Aτ,on(µ) = [aon(µ) ] = [ κ(x; µ)∇ψτ,on · ∇ψτ,on] = Ψτ T A(µ)Ψτ mn m n on on τ Sτ,on(µ) = [son(µ) ] = [ κ(x; µ)ψτ,onψτ,on] = Ψτ T S(µ)Ψτ , mn m n on on τ and κ(x;(cid:80)µ) is now parameter dependent. To generate the coarse space we then choose the smallest Mτ eigenvalues from Equation (11) and form the corresponding eigenvectors in Rτ by setting on on L φτ,on = ψτ,onzτ,on (for k = 1, . . . , Mτ ), where zτ,on are the coordinates of the vector φτ,on. k j k,j on k,j k j=1 Above, we presented an algorithm for solving an eigenvalue problem in the space which has a sparse representation in the snapshot space. One can attempt to obtain sparse spectral basis from Equation (10) using l -minimization method in the local snapshot space V τ ([31, 47]). Here, we 1 snap can use the algorithm proposed in [41]. The general basis pursuit problem is as follows 10 ##LTLine##

text: through

text: multiplication

text: of

text: a

text: random

text: matrix

text: T

text: τ

text: Denote

text: T

text: τ

text: as

text: a

text: matrix

text: of

text: size

text: (L

text: J)by

text: q

text: random

text: random

text: (q

text: (L×J))

text: with

text: rows

text: of

text: i.i.d

text: standard

text: Gaussian

text: random

text: vectors.

text: Then

text: the

text: test

text: space

text: is

text: deﬁned

text: as

text: Ψτ

text: T

text: τ

text: snap

text: random

text: Speciﬁcally,

text: the

text: local

text: problem

text: is

text: arranged

text: as

text: follows.

text: Find

text: U

text: such

text: that,

text: ψτ,on

text: Ψτ

text: U

text: and

text: l

text: l

text: snap

text: l

text: U

text: argmin

text: subject

text: to

text: A

text: (µ)U

text: F

text: l

text: ν

text: c

text: l

text: l

text: Here,

text: A

text: (µ)

text: (Ψτ

text: T

text: τ

text: )T

text: A(µ)Ψτ

text: and

text: F

text: (Ψτ

text: T

text: τ

text: )T

text: R

text: with

text: A(µ)

text: being

text: the

text: local

text: c

text: snap

text: random

text: snap

text: l

text: snap

text: random

text: l

text: stiﬀness

text: matrix

text: and

text: R

text: being

text: the

text: right

text: hand

text: side

text: for

text: the

text: local

text: problem

text: with

text: Dirichlet

text: boundary

text: l

text: condition

text: r

text: Namely,

text: we

text: are

text: solving

text: the

text: following

text: local

text: problems

text: in

text: V

text: τ

text: with

text: the

text: l

text: minimized

text: l

text: snap

text: coeﬃcient

text: vector

text: U

text: for

text: the

text: testing

text: space

text: equal

text: to

text: Ψτ

text: T

text: τ

text: l

text: snap

text: random

text: −div(κ(x;

text: µ)∇ψτ+,on)

text: in

text: τ+

text: l

text: ψτ+,on

text: r

text: on

text: ∂τ+.

text: l

text: l

text: Later,

text: we

text: will

text: brieﬂy

text: introduce

text: the

text: algorithm

text: to

text: solve

text: Equation

text: Note

text: that

text: we

text: impose

text: the

text: same

text: random

text: vectors

text: as

text: the

text: boundary

text: conditions

text: in

text: Equation

text: to

text: guarantee

text: the

text: sparse

text: solution

text: in

text: Equation

text: We

text: can

text: obtain

text: the

text: local

text: online

text: snapshot

text: functions

text: on

text: the

text: target

text: domain

text: τ

text: by

text: restricting

text: the

text: solution

text: of

text: the

text: local

text: problem,

text: ψτ+,on

text: to

text: τ

text: (which

text: is

text: denoted

text: by

text: ψτ,on).

text: Now

text: we

text: are

text: ready

text: to

text: present

text: l

text: l

text: the

text: local

text: online

text: snapshot

text: space

text: as

text: follows,

text: V

text: τ

text: Span{ψτ,on

text: l

text: L},

text: on

text: l

text: for

text: each

text: coarse

text: subdomain

text: τ.

text: Then

text: we

text: denote

text: Ψτ

text: [ψτ,on,

text: ψτ,on].

text: on

text: L

text: Next

text: we

text: will

text: select

text: the

text: dominant

text: modes

text: from

text: V

text: τ

text: by

text: the

text: following

text: eigenvalue

text: problem,

text: on

text: Aτ,on(µ)zτ,on

text: λτ,onSτ,on(µ)zτ,on,

text: k

text: k

text: k

text: where

text: Aτ,on(µ)

text: [aon(µ)

text: κ(x;

text: µ)∇ψτ,on

text: ∇ψτ,on]

text: Ψτ

text: T

text: A(µ)Ψτ

text: mn

text: m

text: n

text: on

text: on

text: τ

text: Sτ,on(µ)

text: [son(µ)

text: κ(x;

text: µ)ψτ,onψτ,on]

text: Ψτ

text: T

text: S(µ)Ψτ

text: mn

text: m

text: n

text: on

text: on

text: τ

text: and

text: is

text: now

text: parameter

text: dependent.

text: To

text: generate

text: the

text: coarse

text: space

text: we

text: then

text: choose

text: the

text: smallest

text: Mτ

text: eigenvalues

text: from

text: Equation

text: and

text: form

text: the

text: corresponding

text: eigenvectors

text: in

text: Rτ

text: by

text: setting

text: on

text: on

text: L

text: φτ,on

text: ψτ,onzτ,on

text: (for

text: k

text: Mτ

text: where

text: zτ,on

text: are

text: the

text: coordinates

text: of

text: the

text: vector

text: φτ,on.

text: k

text: j

text: k,j

text: on

text: k,j

text: k

text: Above,

text: we

text: presented

text: an

text: algorithm

text: for

text: solving

text: an

text: eigenvalue

text: problem

text: in

text: the

text: space

text: which

text: has

text: a

text: sparse

text: representation

text: in

text: the

text: snapshot

text: space.

text: One

text: can

text: attempt

text: to

text: obtain

text: sparse

text: spectral

text: basis

text: from

text: Equation

text: using

text: l

text: -minimization

text: method

text: in

text: the

text: local

text: snapshot

text: space

text: V

text: τ

text: Here,

text: we

text: snap

text: can

text: use

text: the

text: algorithm

text: proposed

text: in

text: The

text: general

text: basis

text: pursuit

text: problem

text: is

text: as

text: follows

text: ##LTLine##


text: – 21 – REFERENCES Sawyer R. F., Phys. Rev. Lett. 29 (1972) 382; erratum: ibid., 29 823. Sokolov A. I., JETP 52 (1980) 575. R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, (Springer-Verlang, Berlin Heidelberg, New York, 1990). Chattopadhyay P. C. and Paul B. C., Pramana- J. of phys. 74 (2010) 513. Tikekar R. and Jotania K., Int. J. Mod. Phys. D 14 (2005) 1037. Tikekar R. and Jotania K., Pramana- J. of phys. 68 (2007) 397. Tikekar R. , J. Math. Phys. 31, (1990) 2454. Vaidya P. C. and Tikekar R. , J. Astrophys. Astron. 3, (1982) 325. Pandya D. M., Thomas V. O., Sharma R., Astrophys. Space Sci. 356 (2015) 285. Finch M. R. and Skea J. E. F., Class. Quantum Grav. 6 (1989) 467. Hansraj S. and Maharaj S. D., Int. J. Mod. Phys. D. 8 (2006) 1311. Tikekar R. and Jotania K., Gravitation and Cosmology 15 (2009) 129. Banerjee A., Rahaman F., Jotania K., Sharma R. and Karar I., Gen. Revativ. Grav. 45 (2013) 717. Ruderman R., Astro. Astrophys. 10 (1972) 427. Canuto V., Annu. Rev. Astron. Astrophys. 12 (1974) 167. Bowers R. and Liang E., Astrophys. J. 188 (1974) 657.

text: REFERENCES

text: Sawyer

label: reference

text: R.

text: F.,

text: Phys.

text: Rev.

text: Lett.

text: erratum:

text: ibid.,

text: Sokolov

text: A.

text: I.,

text: JETP

text: R.

text: Kippenhahn

text: and

text: A.

text: Weigert,

text: Stellar

text: Structure

text: and

text: Evolution,

text: (Springer-Verlang,

text: Berlin

text: Heidelberg,

text: New

text: York,

text: Chattopadhyay

text: P.

text: C.

text: and

text: Paul

text: B.

text: C.,

text: Pramana-

text: J.

text: of

text: phys.

text: Tikekar

text: R.

text: and

text: Jotania

text: K.,

text: Int.

text: J.

text: Mod.

text: Phys.

text: D

text: Tikekar

text: R.

text: and

text: Jotania

text: K.,

text: Pramana-

text: J.

text: of

text: phys.

text: Tikekar

text: R.

text: J.

text: Math.

text: Phys.

text: Vaidya

text: P.

text: C.

text: and

text: Tikekar

text: R.

text: J.

text: Astrophys.

text: Astron.

text: Pandya

text: D.

text: M.,

text: Thomas

text: V.

text: O.,

text: Sharma

text: R.,

text: Astrophys.

text: Space

text: Sci.

text: Finch

text: M.

text: R.

text: and

text: Skea

text: J.

text: E.

text: F.,

text: Class.

text: Quantum

text: Grav.

text: Hansraj

text: S.

text: and

text: Maharaj

text: S.

text: D.,

text: Int.

text: J.

text: Mod.

text: Phys.

text: D.

text: Tikekar

text: R.

text: and

text: Jotania

text: K.,

text: Gravitation

text: and

text: Cosmology

text: Banerjee

text: A.,

text: Rahaman

text: F.,

text: Jotania

text: K.,

text: Sharma

text: R.

text: and

text: Karar

text: I.,

text: Gen.

text: Revativ.

text: Grav.

text: Ruderman

text: R.,

text: Astro.

text: Astrophys.

text: Canuto

text: V.,

text: Annu.

text: Rev.

text: Astron.

text: Astrophys.

text: Bowers

text: R.

text: and

text: Liang

text: E.,

text: Astrophys.

text: J.


text: – 23 – Malaver M., American Journal of Astronomy and Astrophysics 1 (2013) 41. Malaver M., World Applied Programming 3 (2013) 309. Maharaj S. D. and Takisa P. M., Gen. Relativ. Grav. 45 (2013b) 1951. Paul B. C., Chattopadhyay P. K., Karmakar S. and Tikekar R., Mod. Phys. Lett. A 26 (2011) 575. Sharma R. and Ratanpal B. S., Int. J. Mod. Phys. D 13 (2013) 1350074. Gangopadhyay T., Ray S., Li X-D., Dey J. and Dey M., Mon. Not. R. Astron. Soc. 431 (2013) 3216. This manuscript was prepared with the AAS LATEX macros v5.2. ##LTLine##

text: Malaver

text: M.,

text: American

text: Journal

text: of

text: Astronomy

text: and

text: Astrophysics

text: Malaver

text: M.,

text: World

text: Applied

text: Programming

text: Maharaj

text: S.

text: D.

text: and

text: Takisa

text: P.

text: M.,

text: Gen.

text: Relativ.

text: Grav.

text: Paul

text: B.

text: C.,

text: Chattopadhyay

text: P.

text: K.,

text: Karmakar

text: S.

text: and

text: Tikekar

text: R.,

text: Mod.

text: Phys.

text: Lett.

text: A

text: Sharma

text: R.

text: and

text: Ratanpal

text: B.

text: S.,

text: Int.

text: J.

text: Mod.

text: Phys.

text: D

text: Gangopadhyay

text: T.,

text: Ray

text: S.,

text: Li

text: X-D.,

text: Dey

text: J.

text: and

text: Dey

text: M.,

text: Mon.

text: Not.

text: R.

text: Astron.

text: Soc.

text: This

text: manuscript

text: was

text: prepared

text: with

text: the

text: AAS

text: LATEX

text: macros

text: ##LTLine##


text: – 4 – model developed by Maharaj and Marteens (1989) have uniform density for its matter content. Gokhroo and Mehra (1994) gave a more realistic model incorporating non-uniform density. Tikekar and Thomas (1998, 2005), Thomas et al (2005) developed superdense anisotropic distributions on pseudo-spheroidal spacetime. The spacetime used to study the non-adiabatic gravitational collapse of anisotropic distribution has a pseudo spheroidal geometry. Dev and Gleiser (2002, 2003, 2004) have study the impact of anisotropy on the stability of stars. Anisotropic distributions with linear equation of state have been studied by Sharma and Maharaj (2007), Thirukkanesh and Maharaj (2008). Komathiraj and Maharaj (2007) studied charged distributions using linear equation of state. Sunzu et al (2014) used linear equation of state for describing charged quark star models. Feroze and Siddiqui (2011) and Maharaj and Takisa (2012) developed models of anisotropic distributions using quadratic equation of state. In the MIT bag model of quark stars, Paul et al (2011) have shown that introduction of anisotropy can aﬀect bag constant. Thirukkanesh and Ragel (2012), Maharaj and Takisa (2013b) used polytropic equation of state for developing anisotropic models. The anisotropic charged star models given by Malaver (2013a,b, 2014) and Thirukkanesh and Ragel (2014) have Van der Waals equation of state. Pandya et al (2014) have given anisotropic compact stars compatible with observational data by generalizing Finch and Skea (1989) model. The anisotropic model given by Sharma and Ratanpal (2013) is a particular case of this model. It is found that the models given by Pandya et al (2014) agree with the recent observational data given by Gangopadhyay et al (2013). In this paper we have studied anisotropic stellar models on pseudo-spheroidal spacetime. The geometric parameter R plays the role of radius of the star. The geometric parameter K appearing in the model is bounded between two limits to comply with the physical requirements. It is found that the anisotropic model developed can accommodate wide range of pulsars like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230,

text: model

text: developed

text: by

text: Maharaj

text: and

text: Marteens

text: have

text: uniform

text: density

text: for

text: its

text: matter

text: content.

text: Gokhroo

text: and

text: Mehra

text: gave

text: a

text: more

text: realistic

text: model

text: incorporating

text: non-uniform

text: density.

text: Tikekar

text: and

text: Thomas

text: Thomas

text: et

text: al

text: developed

text: superdense

text: anisotropic

text: distributions

text: on

text: pseudo-spheroidal

text: spacetime.

text: The

text: spacetime

text: used

text: to

text: study

text: the

text: non-adiabatic

text: gravitational

text: collapse

text: of

text: anisotropic

text: distribution

text: has

text: a

text: pseudo

text: spheroidal

text: geometry.

text: Dev

text: and

text: Gleiser

text: have

text: study

text: the

text: impact

text: of

text: anisotropy

text: on

text: the

text: stability

text: of

text: stars.

text: Anisotropic

text: distributions

text: with

text: linear

text: equation

text: of

text: state

text: have

text: been

text: studied

text: by

text: Sharma

text: and

text: Maharaj

text: Thirukkanesh

text: and

text: Maharaj

text: Komathiraj

text: and

text: Maharaj

text: studied

text: charged

text: distributions

text: using

text: linear

text: equation

text: of

text: state.

text: Sunzu

text: et

text: al

text: used

text: linear

text: equation

text: of

text: state

text: for

text: describing

text: charged

text: quark

text: star

text: models.

text: Feroze

text: and

text: Siddiqui

text: and

text: Maharaj

text: and

text: Takisa

text: developed

text: models

text: of

text: anisotropic

text: distributions

text: using

text: quadratic

text: equation

text: of

text: state.

text: In

text: the

text: MIT

text: bag

text: model

text: of

text: quark

text: stars,

text: Paul

text: et

text: al

text: have

text: shown

text: that

text: introduction

text: of

text: anisotropy

text: can

text: aﬀect

text: bag

text: constant.

text: Thirukkanesh

text: and

text: Ragel

text: Maharaj

text: and

text: Takisa

text: used

text: polytropic

text: equation

text: of

text: state

text: for

text: developing

text: anisotropic

text: models.

text: The

text: anisotropic

text: charged

text: star

text: models

text: given

text: by

text: Malaver

text: and

text: Thirukkanesh

text: and

text: Ragel

text: have

text: Van

text: der

text: Waals

text: equation

text: of

text: state.

text: Pandya

text: et

text: al

text: have

text: given

text: anisotropic

text: compact

text: stars

text: compatible

text: with

text: observational

text: data

text: by

text: generalizing

text: Finch

text: and

text: Skea

text: model.

text: The

text: anisotropic

text: model

text: given

text: by

text: Sharma

text: and

text: Ratanpal

text: is

text: a

text: particular

text: case

text: of

text: this

text: model.

text: It

text: is

text: found

text: that

text: the

text: models

text: given

text: by

text: Pandya

text: et

text: al

text: agree

text: with

text: the

text: recent

text: observational

text: data

text: given

text: by

text: Gangopadhyay

text: et

text: al

text: In

text: this

text: paper

text: we

text: have

text: studied

text: anisotropic

text: stellar

text: models

text: on

text: pseudo-spheroidal

text: spacetime.

text: The

text: geometric

text: parameter

text: R

text: plays

text: the

text: role

text: of

text: radius

text: of

text: the

text: star.

text: The

text: geometric

text: parameter

text: K

text: appearing

text: in

text: the

text: model

text: is

text: bounded

text: between

text: two

text: limits

text: to

text: comply

text: with

text: the

text: physical

text: requirements.

text: It

text: is

text: found

text: that

text: the

text: anisotropic

text: model

text: developed

text: can

text: accommodate

text: wide

text: range

text: of

text: pulsars

text: like

text: PSR

text: Vela

text: PSR


text: – 6 – (cid:0) (cid:1) − − Here S = S(r) is the magnitude of the anisotropic stress and ci = 0, e λ/2, 0, 0 denotes a radial vector. We have now the following expressions for radial and transverse pressures − √2S p = T 1 = p + , (4) r 1 3 and − − √S p⊥ = T 2 = p . (5) 2 3 The diﬀerence in radial and transverse pressures is taken as the measure of anisotropy and is given by − p √ p⊥ r S = . (6) 3 The Einstein’s ﬁeld equations − 1 R Rg = 8πT , (7) ij ij ij 2 are equivalent to following set of three equations − − − (cid:48) 1 e λ e λλ 8πρ = + , (8) r2 r ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Here

text: S

text: S(r)

text: is

text: the

text: magnitude

text: of

text: the

text: anisotropic

text: stress

text: and

text: ci

text: e

text: denotes

text: a

text: radial

text: vector.

text: We

text: have

text: now

text: the

text: following

text: expressions

text: for

text: radial

text: and

text: transverse

text: pressures

text: p

text: T

text: p

text: r

text: and

text: √S

text: p⊥

text: T

text: p

text: The

text: diﬀerence

text: in

text: radial

text: and

text: transverse

text: pressures

text: is

text: taken

text: as

text: the

text: measure

text: of

text: anisotropy

text: and

text: is

text: given

text: by

text: p

text: p⊥

text: r

text: S

text: The

text: Einstein’s

text: ﬁeld

text: equations

text: R

text: Rg

text: ij

text: ij

text: ij

text: are

text: equivalent

text: to

text: following

text: set

text: of

text: three

text: equations

text: e

text: λ

text: e

text: λλ

text: 8πρ

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 7 – − − − (cid:48) e λ 1 − e λν 8πp = , (9) r r2 r (cid:18) (cid:19) (cid:48)(cid:48) (cid:48) (cid:48) (cid:48) (cid:48) − (cid:48) − ν ν 2 − ν λ ν λ 8πp⊥ = e λ + + . (10) 2 4 4 2r Equation (8)-(10) can be further modiﬁed to the following form, − − 2m e λ = 1 , (11) r (cid:18) (cid:19) − 2m (cid:48) 2m 1 ν = 8πp r + , (12) r r r2 (cid:16) (cid:17) √ −4 (cid:48) (cid:48) 8π 3S = (8πρ + 8πp ) ν + 2 (8πp ) , (13) r r r where, (cid:90) r m(r) = 4π u2ρ(u)du. (14) 0 Using ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: e

text: λ

text: e

text: λν

text: r

text: r

text: ν

text: ν

text: ν

text: λ

text: ν

text: λ

text: 8πp⊥

text: e

text: λ

text: Equation

text: can

text: be

text: further

text: modiﬁed

text: to

text: the

text: following

text: form,

text: e

text: λ

text: r

text: ν

text: r

text: r

text: r

text: −4

text: 8π

text: (8πρ

text: ν

text: r

text: r

text: r

text: where,

text: r

text: m(r)

text: 4π

text: Using

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 10 – For a physically acceptable relativistic distribution of matter, the interior spacetime metric (23) should continuously match with Schwarzschild exterior metric (cid:18) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) − − 2M − − 2M 1 − ds2 = 1 dt2 1 dr2 r2 dθ2 + sin2 θdφ2 , (24) r r across the boundary r = R. This gives the constants M and C as − R (K 1) M = , (25) 2 (K + 1) and (cid:16) (cid:17) − K2−2K+1 −(3K+1) (5−K ) C = R K (1 + K) 2K 2 2 . (26) The expression for anisotropy S can be obtained using (16), (17), (18) and (19) in (13). We have (cid:104) (cid:105) (cid:0) (cid:1) (cid:0) (cid:1) √ − − − − − (K 1)r2 12 + ( 6K2 + 16K 2) r2 + ( K3 + 3K2 7K + 1) r2 R2 R2 8π 3S = , (27) 4R4 1 + K r2 3 1 + r2 R2 R2 √ − which takes the value zero at r = 0. The expression for 8πp⊥ = 8πp 8π 3S, now r takes the form ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: For

text: a

text: physically

text: acceptable

text: relativistic

text: distribution

text: of

text: matter,

text: the

text: interior

text: spacetime

text: metric

text: should

text: continuously

text: match

text: with

text: Schwarzschild

text: exterior

text: metric

text: θdφ2

text: r

text: r

text: across

text: the

text: boundary

text: r

text: R.

text: This

text: gives

text: the

text: constants

text: M

text: and

text: C

text: as

text: R

text: (K

text: M

text: (K

text: and

text: C

text: R

text: K

text: K)

text: The

text: expression

text: for

text: anisotropy

text: S

text: can

text: be

text: obtained

text: using

text: and

text: in

text: We

text: have

text: (K

text: 8π

text: K

text: which

text: takes

text: the

text: value

text: zero

text: at

text: r

text: The

text: expression

text: for

text: 8πp⊥

text: 8π

text: now

text: r

text: takes

text: the

text: form

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: approach. In force-based models the macroscopic limit can be cohesion and aversion — this means they try to follow and derived using the so called BBGKY hierarchies, see for exam- stay close to individuals moving in the same direction and to ple [11], initially developed in the ﬁeld of statistical physics. step aside when being approached by an individual moving in In the case of stochastic underlying dynamics the derivation the opposite direction. We expect that these minimal dynamics of the mean-ﬁeld description has been studied rigorously for will lead to the formation of directional lanes, a phenomenon simpler models by considering the hydrodynamic limit, see that has been observed in crowded corridors, pedestrian walks [12]. Simpler models, such as the Patlak-Keller-Segel model or experiments. for chemotaxis or reaction-diffusion equations, were rigorously A. The microscopic model derived for a stochastic many-particle system, see [13] and [14] respectively. We start with the underlying microscopic model, i.e. a We would like to mention a related work by Penington and co- lattice-based approach in which we consider, for the sake of workers [15] on a systematic construction method to determine simplicity, a rectangle Ω ⊆ R2 such as a corridor, partitioned the continuum limit of nonlinear PDEs from discrete lattice into a square lattice of grid size h. This can be generalized based models. Their approach is based on representing the to higher dimensions. Each lattice site (x , y ) = (ih, jh), i j transition rates using appropriate rotation operators as well i = 0, . . . , N and j = 0, . . . , M can be occupied by as symmetry conditions to derive general expressions for an individual. We consider two groups moving in opposite the transportation coefﬁcients in the corresponding nonlin- direction — one to the right (called the reds) and one to the ear PDEs. This technique can be used for a large class of left (called the blues). The probability to ﬁnd a red individual problems (including multi-species dynamics in various space at time t at location (x , y ) is given by: i j dimension), but assumes that the transition rate of a species depends only on the average occupancy of a site by any of the r (t) = P(red individual is at position (x , y ) at time t), i,j i j different species and not if the site is occupied by a particular where P denotes the probability. The probability for the blue subpopulation or not. This approach cannot be applied to the T { }→{ } individuals is deﬁned analogously. We denote by i,j k,l pedestrian model presented later on, since the transition rates c depend on the afﬁliation to either group. the rate at which an individual of color c moves from (xi, yj) to (x , y ). The transition rates for the red and blue individuals k l In the case of a lattice-based model we respectively are given by: 1) replace the probability to ﬁnd a particle at a lattice site by a formal Taylor expansion (up to a certain T { }→{ } − order) of the corresponding density, r i,j i+1,j = (1 ρi+1,j)(1 + α ri+2,j), T { }→{ − } − 2) pass to an appropriate limit as the lattice size and i,j i,j 1 = (1 ρ − )(γ + γ b ), (1a) r i,j 1 0 1 i+1,j time tends to zero (dropping higher-order terms). T { }→{ } − i,j i,j+1 = (1 ρ )(γ + γ b ), r i,j+1 0 2 i+1,j In this paper we present an algorithmic approach to derive the corresponding continuum equations from a lattice-based model using tools from symbolic computation. T { }→{ − } − While it is relatively straightforward to perform the formal i,j i 1,j = (1 ρ − )(1 + α b − ), b i 1,j i 2,j Taylor expansions and the corresponding limit, it is a more T { }→{ } − i,j i,j+1 = (1 ρ )(γ + γ r − ), (1b) challenging task to rewrite the PDEs obtained this way in b i,j+1 0 1 i 1,j T { }→{ − } − a conservative form. For this purpose, we employ symbolic i,j i,j 1 = (1 ρ − )(γ + γ r − ), b i,j 1 0 2 i 1,j integration methods that can deal with unspeciﬁed functions in several variables. Our approach allows us to deduce the ≤ ≤ where we write ρ = r +b , and with 0 γ , γ , γ mean-ﬁeld equations for a general class of transportation ≤ ≤ i,j i,j −i,j 0 1 2 1, 0 α 1. The prefactor (1 ρ) in all terms of (1) corre- processes in multiple space dimension, including the 2 sponds to the so-called size exclusion, i.e. an individual cannot dynamics of multiple species that may have different size or jump into the neighboring cell if it is occupied. We assume shape. We illustrate our approach for a minimal model of that the transition rates only depend on lattice sites in direction pedestrian dynamics, which includes cohesion and aversion of movement, a reasonable assumption when modeling the in bidirectional pedestrian ﬂows. movement of pedestrians. The second factors in (1) correspond to cohesion and aversion. Cohesion is modelled in the ﬁrst line We have produced a prototype implementation in Math- in each case, by introducing a factor α > 0 which increases the ematica of the methods described in this paper, which is probability to move in the walking direction if the individual in available at http://www.koutschan.de/data/meanﬁeld/ together front, i.e. in (1a) at position (x , y ), is moving in the same i+2 j with a demo notebook. direction. The second and third line in each case account for aversion via sidestepping. If an individual, i.e. in (1a) a blue II. FORMAL DERIVATION OF A MEAN-FIELD PDE MODEL particle located at (x , y ), is approaching, the red particle i+1 j FOR BIDIRECTIONAL PEDESTRIAN DYNAMICS jumps up or down (with rates γ resp. γ ). If γ > γ , the 1 2 1 2 preference is to jump to the right with respect to the direction We start with a speciﬁc example to illustrate the derivation of movement, if γ < γ , to the left. The parameter γ > 0 of a mean-ﬁeld model from a discrete lattice-based approach 1 2 0 corresponds to diffusion in the y-direction. in the case of bidirectional pedestrian ﬂows. We consider two groups of individuals — one moving to the right, the other to Then the evolution of the red particles is given by the so- the left. The dynamics of each individual are determined by called master equation ##LTLine##

text: approach.

text: In

text: force-based

text: models

text: the

text: macroscopic

text: limit

text: can

text: be

text: cohesion

text: and

text: aversion

text: this

text: means

text: they

text: try

text: to

text: follow

text: and

text: derived

text: using

text: the

text: so

text: called

text: BBGKY

text: hierarchies,

text: see

text: for

text: exam-

text: stay

text: close

text: to

text: individuals

text: moving

text: in

text: the

text: same

text: direction

text: and

text: to

text: ple

text: initially

text: developed

text: in

text: the

text: ﬁeld

text: of

text: statistical

text: physics.

text: step

text: aside

text: when

text: being

text: approached

text: by

text: an

text: individual

text: moving

text: in

text: In

text: the

text: case

text: of

text: stochastic

text: underlying

text: dynamics

text: the

text: derivation

text: the

text: opposite

text: direction.

text: We

text: expect

text: that

text: these

text: minimal

text: dynamics

text: of

text: the

text: mean-ﬁeld

text: description

text: has

text: been

text: studied

text: rigorously

text: for

text: will

text: lead

text: to

text: the

text: formation

text: of

text: directional

text: lanes,

text: a

text: phenomenon

text: simpler

text: models

text: by

text: considering

text: the

text: hydrodynamic

text: limit,

text: see

text: that

text: has

text: been

text: observed

text: in

text: crowded

text: corridors,

text: pedestrian

text: walks

text: Simpler

text: models,

text: such

text: as

text: the

text: Patlak-Keller-Segel

text: model

text: or

text: experiments.

text: for

text: chemotaxis

text: or

text: reaction-diffusion

text: equations,

text: were

text: rigorously

text: A.

text: The

text: microscopic

text: model

text: derived

text: for

text: a

text: stochastic

text: many-particle

text: system,

text: see

text: and

text: respectively.

text: We

text: start

text: with

text: the

text: underlying

text: microscopic

text: model,

text: i.e.

text: a

text: We

text: would

text: like

text: to

text: mention

text: a

text: related

text: work

text: by

text: Penington

text: and

text: co-

text: lattice-based

text: approach

text: in

text: which

text: we

text: consider,

text: for

text: the

text: sake

text: of

text: workers

text: on

text: a

text: systematic

text: construction

text: method

text: to

text: determine

text: simplicity,

text: a

text: rectangle

text: Ω

text: such

text: as

text: a

text: corridor,

text: partitioned

text: the

text: continuum

text: limit

text: of

text: nonlinear

text: PDEs

text: from

text: discrete

text: lattice

text: into

text: a

text: square

text: lattice

text: of

text: grid

text: size

text: h.

text: This

text: can

text: be

text: generalized

text: based

text: models.

text: Their

text: approach

text: is

text: based

text: on

text: representing

text: the

text: to

text: higher

text: dimensions.

text: Each

text: lattice

text: site

text: (x

text: y

text: (ih,

text: jh),

text: i

text: j

text: transition

text: rates

text: using

text: appropriate

text: rotation

text: operators

text: as

text: well

text: i

text: N

text: and

text: j

text: M

text: can

text: be

text: occupied

text: by

text: as

text: symmetry

text: conditions

text: to

text: derive

text: general

text: expressions

text: for

text: an

text: individual.

text: We

text: consider

text: two

text: groups

text: moving

text: in

text: opposite

text: the

text: transportation

text: coefﬁcients

text: in

text: the

text: corresponding

text: nonlin-

text: direction

text: one

text: to

text: the

text: right

text: (called

text: the

text: reds)

text: and

text: one

text: to

text: the

text: ear

text: PDEs.

text: This

text: technique

text: can

text: be

text: used

text: for

text: a

text: large

text: class

text: of

text: left

text: (called

text: the

text: blues).

text: The

text: probability

text: to

text: ﬁnd

text: a

text: red

text: individual

text: problems

text: (including

text: multi-species

text: dynamics

text: in

text: various

text: space

text: at

text: time

text: t

text: at

text: location

text: (x

text: y

text: is

text: given

text: by:

text: i

text: j

text: dimension),

text: but

text: assumes

text: that

text: the

text: transition

text: rate

text: of

text: a

text: species

text: depends

text: only

text: on

text: the

text: average

text: occupancy

text: of

text: a

text: site

text: by

text: any

text: of

text: the

text: r

text: (t)

text: P(red

text: individual

text: is

text: at

text: position

text: (x

text: y

text: at

text: time

text: t),

text: i,j

text: i

text: j

text: different

text: species

text: and

text: not

text: if

text: the

text: site

text: is

text: occupied

text: by

text: a

text: particular

text: where

text: P

text: denotes

text: the

text: probability.

text: The

text: probability

text: for

text: the

text: blue

text: subpopulation

text: or

text: not.

text: This

text: approach

text: cannot

text: be

text: applied

text: to

text: the

text: T

text: individuals

text: is

text: deﬁned

text: analogously.

text: We

text: denote

text: by

text: i,j

text: k,l

text: pedestrian

text: model

text: presented

text: later

text: on,

text: since

text: the

text: transition

text: rates

text: c

text: depend

text: on

text: the

text: afﬁliation

text: to

text: either

text: group.

text: the

text: rate

text: at

text: which

text: an

text: individual

text: of

text: color

text: c

text: moves

text: from

text: (xi,

text: yj)

text: to

text: (x

text: y

text: The

text: transition

text: rates

text: for

text: the

text: red

text: and

text: blue

text: individuals

text: k

text: l

text: In

text: the

text: case

text: of

text: a

text: lattice-based

text: model

text: we

text: respectively

text: are

text: given

text: by:

text: replace

text: the

text: probability

text: to

text: ﬁnd

text: a

text: particle

text: at

text: a

text: lattice

text: site

text: by

text: a

text: formal

text: Taylor

text: expansion

text: (up

text: to

text: a

text: certain

text: T

text: order)

text: of

text: the

text: corresponding

text: density,

text: r

text: i,j

text: α

text: T

text: pass

text: to

text: an

text: appropriate

text: limit

text: as

text: the

text: lattice

text: size

text: and

text: i,j

text: i,j

text: ρ

text: )(γ

text: γ

text: b

text: r

text: i,j

text: time

text: tends

text: to

text: zero

text: (dropping

text: higher-order

text: terms).

text: T

text: i,j

text: ρ

text: )(γ

text: γ

text: b

text: r

text: In

text: this

text: paper

text: we

text: present

text: an

text: algorithmic

text: approach

text: to

text: derive

text: the

text: corresponding

text: continuum

text: equations

text: from

text: a

text: lattice-based

text: model

text: using

text: tools

text: from

text: symbolic

text: computation.

text: T

text: While

text: it

text: is

text: relatively

text: straightforward

text: to

text: perform

text: the

text: formal

text: i,j

text: i

text: ρ

text: α

text: b

text: b

text: i

text: i

text: Taylor

text: expansions

text: and

text: the

text: corresponding

text: limit,

text: it

text: is

text: a

text: more

text: T

text: i,j

text: ρ

text: )(γ

text: γ

text: r

text: challenging

text: task

text: to

text: rewrite

text: the

text: PDEs

text: obtained

text: this

text: way

text: in

text: b

text: i

text: T

text: a

text: conservative

text: form.

text: For

text: this

text: purpose,

text: we

text: employ

text: symbolic

text: i,j

text: i,j

text: ρ

text: )(γ

text: γ

text: r

text: b

text: i,j

text: i

text: integration

text: methods

text: that

text: can

text: deal

text: with

text: unspeciﬁed

text: functions

text: in

text: several

text: variables.

text: Our

text: approach

text: allows

text: us

text: to

text: deduce

text: the

text: where

text: we

text: write

text: ρ

text: r

text: +b

text: and

text: with

text: γ

text: γ

text: γ

text: mean-ﬁeld

text: equations

text: for

text: a

text: general

text: class

text: of

text: transportation

text: i,j

text: i,j

text: −i,j

text: α

text: The

text: prefactor

text: ρ)

text: in

text: all

text: terms

text: of

text: corre-

text: processes

text: in

text: multiple

text: space

text: dimension,

text: including

text: the

text: sponds

text: to

text: the

text: so-called

text: size

text: exclusion,

text: i.e.

text: an

text: individual

text: cannot

text: dynamics

text: of

text: multiple

text: species

text: that

text: may

text: have

text: different

text: size

text: or

text: jump

text: into

text: the

text: neighboring

text: cell

text: if

text: it

text: is

text: occupied.

text: We

text: assume

text: shape.

text: We

text: illustrate

text: our

text: approach

text: for

text: a

text: minimal

text: model

text: of

text: that

text: the

text: transition

text: rates

text: only

text: depend

text: on

text: lattice

text: sites

text: in

text: direction

text: pedestrian

text: dynamics,

text: which

text: includes

text: cohesion

text: and

text: aversion

text: of

text: movement,

text: a

text: reasonable

text: assumption

text: when

text: modeling

text: the

text: in

text: bidirectional

text: pedestrian

text: ﬂows.

text: movement

text: of

text: pedestrians.

text: The

text: second

text: factors

text: in

text: correspond

text: to

text: cohesion

text: and

text: aversion.

text: Cohesion

text: is

text: modelled

text: in

text: the

text: ﬁrst

text: line

text: We

text: have

text: produced

text: a

text: prototype

text: implementation

text: in

text: Math-

text: in

text: each

text: case,

text: by

text: introducing

text: a

text: factor

text: α

text: which

text: increases

text: the

text: ematica

text: of

text: the

text: methods

text: described

text: in

text: this

text: paper,

text: which

text: is

text: probability

text: to

text: move

text: in

text: the

text: walking

text: direction

text: if

text: the

text: individual

text: in

text: available

text: at

text: together

text: front,

text: i.e.

text: in

text: at

text: position

text: (x

text: y

text: is

text: moving

text: in

text: the

text: same

text: j

text: with

text: a

text: demo

text: notebook.

text: direction.

text: The

text: second

text: and

text: third

text: line

text: in

text: each

text: case

text: account

text: for

text: aversion

text: via

text: sidestepping.

text: If

text: an

text: individual,

text: i.e.

text: in

text: a

text: blue

text: II.

text: FORMAL

text: DERIVATION

text: OF

text: A

text: MEAN-FIELD

text: PDE

text: MODEL

text: particle

text: located

text: at

text: (x

text: y

text: is

text: approaching,

text: the

text: red

text: particle

text: j

text: FOR

text: BIDIRECTIONAL

text: PEDESTRIAN

text: DYNAMICS

text: jumps

text: up

text: or

text: down

text: (with

text: rates

text: γ

text: resp.

text: γ

text: If

text: γ

text: γ

text: the

text: preference

text: is

text: to

text: jump

text: to

text: the

text: right

text: with

text: respect

text: to

text: the

text: direction

text: We

text: start

text: with

text: a

text: speciﬁc

text: example

text: to

text: illustrate

text: the

text: derivation

text: of

text: movement,

text: if

text: γ

text: γ

text: to

text: the

text: left.

text: The

text: parameter

text: γ

text: of

text: a

text: mean-ﬁeld

text: model

text: from

text: a

text: discrete

text: lattice-based

text: approach

text: corresponds

text: to

text: diffusion

text: in

text: the

text: y-direction.

text: in

text: the

text: case

text: of

text: bidirectional

text: pedestrian

text: ﬂows.

text: We

text: consider

text: two

text: groups

text: of

text: individuals

text: one

text: moving

text: to

text: the

text: right,

text: the

text: other

text: to

text: Then

text: the

text: evolution

text: of

text: the

text: red

text: particles

text: is

text: given

text: by

text: the

text: so-

text: the

text: left.

text: The

text: dynamics

text: of

text: each

text: individual

text: are

text: determined

text: by

text: called

text: master

text: equation

text: ##LTLine##