Fast multiscale Gaussian wavepacket transforms and ...gunther/MSRI_IPSeminar_files/Qian.pdfHigh frequency waves and geometrical optics Gaussian-beam setup for wave equations Multiscale

High frequency waves and geometrical optics Gaussian-beam setup for wave equations Multiscale Gaussian wavepacket transforms

Fast multiscale Gaussian wavepackettransforms and multiscale Gaussian beams

for wave equations

Jianliang Qian1

1Department of Mathematics, Michigan State University, Michigan

Joint work with Shingyu Leung (HKUST)

Joint work with Lexing Ying (UT Austin)

2010 MSRI Inverse Problems and Applications


Outline

1 High frequency waves and geometrical optics

2 Gaussian-beam setup for wave equationsBeam ingredientsBeam summationComputational challenges in Gaussian beams

3 Multiscale Gaussian wavepacket transformsContinuous transformsTransforms of discrete signals

4 Multiscale Gaussian beams for the wave equationTwo polarized wave modesMultiscale Gaussian beams

5 Numerical results


The wave equation

Utt − V 2(x)∆U = 0, x ∈ Rd , t > 0,

U|t=0 = f1(x), Ut |t=0 = f2(x).

V is smooth, positive, and bounded away from zero.f1(x) and f2(x) are compactly supported L2-functions,presumably highly oscillatory.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

x

u

Figure: Initial data: f1 = 0 and f2 = 6400 cos(128πx).


An example

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−50

0

50

100

x

rea

l u

Figure: Wave equation. Solution at T = 8.0.


The problem

The difficultiesWhen the initial condition contains oscillations of a smallwavelength, the wave equation propagates theseoscillations in space and time.

Resolving such small oscillations by direct numericalmethods requires an enormous computational grid and isvery costly in practice.

Methods based on geometrical optics are sought as analternative for capturing such highly oscillatoryphenomena.


Geometrical optics

The traditional geometrical-optics method starts from theWKBJ ansatz consisting of a real phase and a realamplitude, (the frequency ω is a large parameter),

U(x , t) ≈ A(x , t)eıωτ(x ,t).

The WKBJ ansatz breaks down at the caustics, in theneighborhood of which the phase function is multivaluedand the amplitude function blows up.


Geometrical optics for Schrodinger (Cont.)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

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t

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t


Gaussian beams

Gaussian beam (Babich’72, Hörmander’71, Ralston’83,Tanushev-Q-Ralston’07) constructs a global complexphase and a global complex amplitude that satisfy theeikonal equation and the transport equation approximatelynear a specified ray path.Away from the ray path, the quadratic imaginary part of thephase provides a rapidly decaying Gaussian profile.This gives rise to a single Gaussian-like asymptoticsolution, which is accurate near the underlying ray path.Since the wave equation is linear, a superposition yields aglobal asymptotic solution.

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Beam ingredients

Eikonal and transport equations

Utt − V 2(x)∆U = 0, x ∈ Rd , t > 0, (1)

U|t=0 = f1(x), Ut |t=0 = f2(x). (2)

Asymptotic solutions: A(x , t)eıωτ(x ,t), so that the waveequation (1) and its associated initial conditions (2) aresatisfied approximately with a small error when ω is large.

Consider the leading orders in inverse powers of ω:

τ2t − V 2(x)|∇xτ(x , t)|2 = 0,

2Atτt − 2V 2∇xA · ∇xτ + A(τtt − V 2trace(τxx)) = 0.


Beam ingredients

Eikonal and transport equations (cont.)

Factorizing the eikonal equation gives

τ±t + G±(x ,∇xτ(x , t)) = 0, (3)

where G±(x ,∇xτ(x , t)) = ±V (x)|∇xτ(x , t)| correspond totwo polarized modes in the 2nd-order wave equation.

Define the Hamiltonians, G±(x ,p) = ±V (x)|p|, whereG±(x ,p) is homogeneous of degree one in the momentumvariable p.

Consider the following eikonal equation:

τt + G(x ,∇xτ(x , t)) = 0, (4)

where G can be taken to be either G+ or G− and τ to beeither τ+ or τ−.


Beam ingredients

A single Gaussian beam (1)

A single Gaussian beam is concentrated near a ray pathwhich is the x-projection of a certain bicharacteristic.

Apply the method of characteristics to the eikonal (4):

x =dxdt

= Gp, x |t=0 = x0,

p =dpdt

= −Gx , p|t=0 = p0, (5)

where t is time parameterizing bicharacteristics.

This yields the bicharacteristic (x(t),p(t)) : t ≥ 0, whichemanates from the initial point (x0,p0) in phase space att = 0. The corresponding ray path is γ = (x(t), t) : t ≥ 0,which is defined in the (x , t)-space.


Beam ingredients


Along the ray path γ = (x(t), t) : t ≥ 0, we have byconstruction p(t) = τx(x(t), t) due to the method ofcharacteristics.

The phase function τ(x(t), t) along the ray path satisfies

dτ(x(t), t)dt

= τt(x(t), t) + G(x(t), τx(x(t), t)) = 0,

so the phase function τ(x(t), t) does not change along γbecause the Hamiltonian G is homogeneous of degreeone. We will take τ(x(t), t) = 0.

To construct a 2nd-order Taylor expansion for the phasealong the ray path, one needs to compute the Hessian ofthe phase along the ray.


Beam ingredients


M(t) = τxx(x(t), t) satisfies the Riccati equation,

dM(t)dt

+Gxx +M(t)Gxp +GTxpM(t)+M(t)GppM(t) = 0. (6)

M|t=0 = M0 = ıǫI; ǫ > 0 is of order O(1).

Although the Riccati equation (6) does not admit a globalsmooth solution in general, it turns out that complexifyingthe equation by specifying a complex initial value willguarantee that a global smooth solution exists.

This is because of the underlying symplectic structureassociated with the related Hamiltonian system (Babich’72,Ralston’83, Maslov’94).


Beam ingredients


Lemma (Babich’72, Ralston’83, Maslov’94)

If the Hamiltonian G is smooth enough, then the Hessian M(t)along the ray path γ has a positive-definite imaginary partprovided that it initially does.

Generally, there is a high probability for so-calledtransmission caustics to occur in inhomogeneous media(White’84); therefore, it is critical for a numerical method tobe capable of treating caustics automatically.

Lemma 2.1 is significant for the Gaussian beamconstruction because it guarantees that an initial Gaussianprofile with small variances will propagate along the raypath if such an initial Gaussian profile is chosen properly topeak at the initial point of the ray path.


Beam ingredients


With the Hessian at our disposal, we may solve thetransport equation for A(t) = A(x(t), t) along the ray pathγ:

dAdt

+A

2G

(

V 2(x(t))trace(M(t)) − Gx · Gp − GTp M(t)Gp

)

= 0,

with an initial condition A|t=0 = A0.

We define two global, smooth approximate functions forthe phase and amplitude along the ray path γ:

τ(x , t) ≡ p(t) · (x − x(t)) +12(x − x(t))T M(t)(x − x(t)),

A(x , t) ≡ A(x(t), t) = A(t),

which are accurate near the ray path γ = (x(t), t) : t ≥ 0.


Beam ingredients

Taylor expansion for beams

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

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γ=(t,x(t))

(x(t),p(t),M(t),A(t))τ(t)=τ(t,x(t))=0

p(t)=τx(t,x(t))

M(t)=τxx

(t,x(t))

τ(t,x)=τx(t,x(t)) (x−x(t))+0.5 τ

xx(t,x(t)) (x−x(t)) 2

A(t,x)=A(t)=A(t,x(t))


Beam ingredients


We have a single-beam asymptotic solution,

Φ(x , t) = A(x , t) exp(ıωτ(x , t)). (7)

This beam solution is concentrated on a single smoothcurve γ = (x(t), t) : t ≥ 0.Because Im(τ(x , t)) = 1

2(x − x(t))T Im(M(t))(x − x(t)),Φ(x , t) has a Gaussian profile concentrated on γ:

exp(

−ω2

(x − x(t))T Im(M(t))(x − x(t)))

.

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t

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Beam summation

Two polarized wave modes

Applying the above construction to the two polarizedmodes with G = G± results in two sets of solutions x±(t),p±(t), M±(t), A±(t), τ±(x , t), A±(x , t), and Φ±(x , t).

These functions are uniquely determined by the initial datax0, p0, M0, and A0.

We denote these initial data collectively by a tupleα = (x0,p0,M0,A0). The solutions are denoted,respectively, by x±

α (t), p±α (t), M±

α (t), A±α (t), τ±α (x , t),

A±α (x , t), and Φ±

α (x , t).


Beam summation

Beam summation

For a given tuple α = (x0,p0,M0,A0), the Gaussian beamsΦ±

α (x , t) have a simple Gaussian envelope.

For a general initial condition (U(x ,0),Ut(x ,0)), one needsto find two sets I+ and I− of tuples such that at time t = 0

U(x ,0) ≈∑

α∈I+Φ+

α (x ,0) +∑

α∈I−

Φ−α (x ,0),

Ut(x ,0) ≈∑

α∈I+Φ+

α,t(x ,0) +∑

α∈I−

Φ−α,t(x ,0).

Once this initial decomposition is given, the linearity of thewave equation gives the Gaussian beam solution

U(x , t) ≈∑

α∈I+Φ+

α (x , t) +∑

α∈I−

Φ−α (x , t).


Computational challenges in Gaussian beams

The first challenge: how to generate a beam decomposition

Related worksBeam decompositions for wave equations:Leung-Q-Burridge (Geophysics’07), Tanushev-Q-Ralston(SIAM MMS’07), Tanushev-Engquist-Tsai’09.

FBI-transform based beam decompositions forSchrödinger equations: Leung-Q (JCP’09,’10).

For Schrödinger equations the most efficient beamdecomposition method is based on single-scale Gaussianwavepacket transforms: Q-Ying (JCP’10).

For wave equations, the most efficient beamdecomposition method is based on multiscale Gaussianwavepacket transforms: Q-Ying (SIAM MMS’10). This isinspired by the parabolic scaling principle (Smith’98,Candes-Demanet’05).



The second challenge: long-term beam propagation

Beam width and reinitializationA simple analysis reveals that the width of a Gaussianbeam in the case of the wave equation is related to thederivatives of the underlying velocity, and a beam widthmay grow exponentially during the evolution process.

So the beam loses its localized significance, leading todeteriorating accuracy in the Taylor expansion for thephase function and high cost in the beam summation.

Following Q-Ying (JCP’10), we propose to reinitialize thebeam propagation using the fast multiscale Gaussianwavepacket transforms when the beams become too wide.

This controls the widths of the Gaussian beams andenables us to propagate the wave efficiently and accuratelyfor a long period of time.



Beam initialization

To justify that the beam solution is a valid asymptoticsolution for equation (1) with initial conditions (2), we haveto consider initial conditions in the beam construction aswell.However, this depends on how the initial conditions aredecomposed into Gaussian profiles and how the beampropagation is initialized.Q-Ying (JCP’10): an approach based on single-scale fastGaussian wavepacket transforms to initialize beampropagation for the Schrödinger equation.For the wave equation, since the Hamiltonian G±(x ,p) ishomogeneous of degree one, the initialization requires newmultiscale transforms with basis functions satisfying theparabolic scaling principle (Smith’98,Candes-Demanet’05).


Motivation

At time t = 0 each Gaussian beam takes the form

A0 exp(

ıω

(

p0 · (x − x0) +12(x − x0)

T M0(x − x0)

))

,

where p0 = O(1) and the Hessian M0 is purely imaginaryand of order O(1).

This is a modulated Gaussian function that oscillates at awavelength of order O(1/ω) and has an effective supportof width O(1/

√ω) in space.

The initialization step is equivalent to decomposing U(x ,0)and Ut(x ,0) into a linear combination of such Gaussianfunctions. → Multiscale Gaussian wavepacket transforms.


Continuous transforms

Partitioning the Fourier domain

Start by partitioning the Fourier domain Rd into Cartesian

coronae Cℓ for ℓ ≥ 1: C1 = [−4,4]d ,

Cℓ = ξ = (ξ1, ξ2, . . . , ξd) : max1≤s≤d |ξs| ∈ [4ℓ−1,4ℓ], ℓ ≥ 2.

ξ ∈ Cℓ implies that |ξ| = O(4ℓ).Each corona Cℓ is further partitioned into boxes

Bℓ,i =

d∏

s=1

[2ℓ · is,2ℓ · (is + 1)],

where the integer multiindex i = (i1, i2, . . . , id) ranges overall possible choices that satisfy Bℓ,i ⊂ Cℓ.All boxes in a fixed Cℓ have the same length W ℓ = 2ℓ ineach dimension and the center of the box Bℓ,i is denotedby ξℓ,i = (ξℓ,i,1, ξℓ,i,2, . . . , ξℓ,i,d ).



Illustrations



Gaussian windows in the Fourier domain

To each box Bℓ,i , we associate a smooth function gℓ,i(ξ),which is compactly supported in a box centered at ξℓ,i withsize Lℓ = 2Wℓ in each dimension.

gℓ,i(ξ) is also required to approximate a Gaussian profile

gℓ,i(ξ) ≈ e−

„

|ξ−ξℓ,i |

σℓ

«2

with σℓ = Wℓ/2.



A partition of unity

Choose gℓ,i(ξ) to satisfy three admissible conditions(Q-Ying MMS’10).

For each Bℓ,i define the conjugate filter hℓ,i(ξ):

hℓ,i(ξ) =gℓ,i(ξ)

∑

ℓ,i(gℓ,i(ξ))2 .

By construction, the products of gℓ,i(ξ) and hℓ,i(ξ) form apartition of unity:

∑

ℓ,i gℓ,i(ξ)hℓ,i(ξ) = 1.



Two sets of wavepacket functions

We introduce two sets of functions φℓ,i,k (x), ψℓ,i,k (x),which are defined in the Fourier domain by

φℓ,i,k (ξ) = 1Ld/2

ℓ

e−2πı k·ξ

Lℓ gℓ,i(ξ), ∀k ∈ Zd ,

ψℓ,i,k (ξ) = 1Ld/2

ℓ

e−2πı k·ξ

Lℓ hℓ,i(ξ), ∀k ∈ Zd .

Taking the inverse Fourier transforms gives their definitionsin the spatial domain:

φℓ,i,k (x) = 1Ld/2

ℓ

∫

Rd e2πı(x− k

Lℓ)·ξ

gℓ,i(ξ)dξ,∀k ∈ Zd

ψℓ,i,k (x) = 1Ld/2

ℓ

∫

Rd e2πı(x− k

Lℓ)·ξ

hℓ,i(ξ)dξ,∀k ∈ Zd



Two sets of wavepacket functions (Cont.)

The definitions of gℓ,i(ξ) and φℓ,i,k (x) imply that

φℓ,i,k (x) ≈(√

π

Lℓσℓ

)d

· e2πı(x− k

Lℓ)·

ξℓ,i|ξℓ,i |

|ξℓ,i | · e−σ2

ℓπ2|x− kLℓ

|2.

φℓ,i,k (x) is approximately a Gaussian function that isspatially centered at k/Lℓ, oscillates at frequency ξℓ,i with|ξℓ,i | = O(4ℓ), and has an O(σℓ) = O(Wℓ) = O(2ℓ) effectivewidth in the Fourier domain and an O(1/σℓ) = O(2−ℓ)effective width in the spatial domain.

The functions φℓ,i,k (x) fit exactly into the profile of aGaussian beam with ω = O(4ℓ): ω ≈ |ξℓ,i |.Qualitatively, ψℓ,i,k (x) is also a wavepacket with slightlylarger support in x compared to φℓ,i,k (x).



Two sets of wavepacket functions (Cont.)

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Figure: Typical profiles of φℓ,i,k (x) and ψℓ,i,k (x).



Two frames

φℓ,i,k (x) and ψℓ,i,k (x) are two frames of L2(Rd ).

Lemma (Q-Ying (MMS’10))

There exists constants C1 and C2 such that for any f ∈ L2(Rd )

C1‖f‖22 ≤

∑

ℓ,i,k

|〈φℓ,i,k , f 〉|2 ≤ C2‖f‖22,

C1‖f‖22 ≤

∑

ℓ,i,k

|〈ψℓ,i,k , f 〉|2 ≤ C2‖f‖22.



Dual frames

φℓ,i,k (x) and ψℓ,i,k (x) are dual frames.

Lemma (Q-Ying (MMS’10))

For any f ∈ L2(Rd ),

f (x) =∑

ℓ,i,k

〈ψℓ,i,k , f 〉φℓ,i,k (x).

Lemma 3.2 offers a way to decompose any functionf ∈ L2(Rd ) into a sum of Gaussian-like functions.



Forward and inverse Gaussian wavepacket transforms

Given f , the forward multiscale Gaussian wavepackettransform computes the coefficients cℓ,i,k defined by

cℓ,i,k = 〈ψℓ,i,k , f 〉 = 〈ψℓ,i,k , f 〉 (8)

where 〈, 〉 is the usual L2(Rd ) inner product and f denotesthe Fourier transform of f .By Lemma 3.1 each coefficient cℓ,i,k is at most of orderO(1) and many coefficients are negligible.Given a set of coefficients cℓ,i,k, the inverse multiscaleGaussian wavepacket transform synthesizes a functionu(x) defined by

u(x) =∑

ℓ,i,k

cℓ,i,kφℓ,i,k (x). (9)


Transforms of discrete signals

Algorithmic setup

Work with the periodized d-dimensional cube [0,1]d . Thespatial grid and Fourier grid are defined respectively by

X = (n1/N, . . . ,nd/N) : 0 ≤ n1, . . . ,nd < N,n1, . . . ,nd ∈ Z,Ω = (k1, . . . , kd) : −N

2 ≤ k1, . . . , kd <N2 , k1, . . . , kd ∈ Z.

The regions Cℓ and the intervals Bℓ,i are defined in thesame way as in the continuous setting.

For a given signal f defined on the spatial grid X , we definethe discrete Fourier transforms as usual.



Discrete forward multiscale transform

Algorithm

(Forward multiscale Gaussian wavepacket transform.) Given asignal f defined at x ∈ X, compute the coefficients cD

ℓ,i,k.

1 Compute f (ξ) for ξ ∈ Ω using a d-dimensional forward FFTof size N in each dimension.

2 For each level ℓ and each box Bℓ,i , form hℓ,i(ξ)f (ξ) at thesupport of hℓ,i(ξ), wrap the result modulus Lℓ to the domain[−Lℓ/2,Lℓ/2)d , apply a d-dimensional inverse FFT of sizeLℓ in each dimension to the wrapped result to obtain cD

ℓ,i,kfor all k .

The cost of Algorithm 3.3 is O(Nd log N).



Discrete inverse multiscale transform

Given a set of coefficients cDℓ,i,k, the discrete version of

the inverse multiscale Gaussian wavepacket transform isdefined by:

f (x) =∑

ℓ,i,k

cDℓ,i,kφ

Dℓ,i,k (x)

or equivalently

f (ξ) =∑

ℓ,i,k

cDℓ,i,k φ

Dℓ,i,k (ξ) =

∑

ℓ,i

(

∑

k

1Ld/2

e−2πı k·ξ

Lℓ cDℓ,i,k

)

gℓ,i(ξ).

The inner summation in k of the last formula is a forwardFourier transform of size Lℓ.



Discrete inverse multiscale transform (cont.)

Algorithm

(Inverse multiscale Gaussian wavepacket transform.) Givencoefficients cD

ℓ,i,k, reconstruct the function f (x) for x ∈ X.

1 For each level ℓ and each box Bℓ,i , apply a d-dimensionalforward FFT of size L in each dimension to the coefficientscD

ℓ,i,k , unwrap the result modulus Lℓ to the support of gℓ,i(ξ),multiply the unwrapped data with gℓ,i(ξ), and add theproduct to f (ξ).

2 Compute f (x) for x ∈ X using a d-dimensional inverse FFTof size N in each dimension.

The cost of Algorithm 3.4 is O(Nd log N).


Motivation

Equipped with the fast transforms presented above: how toinitialize the Gaussian beam representation of a generalinitial condition (U(x ,0),Ut(x ,0)).

The number of samples N in each direction should beproportional to the highest frequency in U(x ,0) andUt(x ,0), as one uses a finite number of samples per unitwavelength.

The accuracy of the Gaussian beam ansatz gets better asthe large parameter ω increases.

For a Gaussian wavepacket φℓ,i,k (x), ω is on the order ofO(|ξℓ,i |), where ξℓ,i is the center of frequency of the packet.

This implies that we should not use the Gaussian beamrepresentation for the low frequency part of the initial data,as it will introduce error of O(1).


Low and high frequency parts

We first introduce the low frequency band pass filter wL(ξ)and the high frequency band pass filter wH(ξ) such thatwL(ξ) + wH(ξ) = 1 for all ξ.Decompose the initial data U(x ,0) and Ut(x ,0) into thelow frequency part by

UL(ξ,0) = U(ξ,0)wL(ξ), UtL(ξ,0) = Ut(ξ,0)wL(ξ),

and the high frequency part UH(x ,0) and UHt (x ,0) by

UH(ξ,0) = U(ξ,0)wH(ξ), UtH(ξ,0) = Ut(ξ,0)wH(ξ).

For the low frequency part UL(x ,0) and ULt (x ,0), use finite

difference, finite element, or spectral methods.For the high frequency part UH(x ,0) and UH

t (x ,0), useGaussian beams.



Multiscale wavepacket decomposition of initial data

Assume that initial conditions (2) belong to the highfrequency part.Applying the multiscale wavepacket transform to the initialconditions yields the following decompositions:

U|t=0 = f1(x) =∑

ℓ,i,k

aℓ,i,kφℓ,i,k (x), (10)

Ut |t=0 = f2(x) =∑

ℓ,i,k

bℓ,i,kφℓ,i,k (x). (11)

To obtain correct initial conditions for the two polarizedwave modes, we consider

Utt − V 2(x)∆U = 0, x ∈ Rd , t > 0, (12)

U|t=0 = aℓ,i,kφℓ,i,k (x), (13)

Ut |t=0 = bℓ,i,kφℓ,i,k (x). (14)



Multiscale Gaussian beam ingredients

Motivated by the approximation

φℓ,i,k (x) ≈(√

π

Lℓσℓ

)d

·eı|ξℓ,i |(x−k

Lℓ)·

2πξℓ,i|ξℓ,i | ·e

−|ξℓ,i |

„

σ2ℓπ2

|ξℓ,i ||x− k

Lℓ|2

«

,

where |ξℓ,i | behaves like a large parameter ω, constructone Gaussian beam for each wave mode,

x = Gp, x |t=0 =kLℓ

; p = −Gx , p|t=0 = 2πξℓ,i|ξℓ,i |

;

M = −(Gxp)T M − MGpx − MGppM − Gxx ,

A = − A2G

(

V 2trace(M) − Gx · Gp − GTp MGp

)

,

M|t=0 = ı · (2π2σ2ℓ /|ξℓ,i |), A|t=0 =

(√

π

Lℓσℓ

)d

,(15)

where we take G = G+ and G = G−, respectively.



A multiscale Gaussian beam

The solutions: x±ℓ,i,k (t), p±

ℓ,i,k (t), M±ℓ,i,k (t), A±

ℓ,i,k (t).

Assume that τ±ℓ,i,k (x , t) and A±ℓ,i,k (x , t) are defined as,

τ±ℓ,i,k (x , t) = p±ℓ,i,k (t) · (x − x±

ℓ,i,k (t)) +

+12(x − x±

ℓ,i,k (t))T M±ℓ,i,k (t)(x − x±

ℓ,i,k (t)),

A±ℓ,i,k (x , t) = A±

ℓ,i,k (t). (16)

The corresponding Gaussian beams are given by

Φ±ℓ,i,k (x , t) = A±

ℓ,i,k (x , t) exp(

ı|ξℓ,i | · τ±ℓ,i,k (x , t))

.



A multiscale Gaussian beam (Cont.)

Given these two Gaussian beams, we propose a globalasymptotic solution to the wave equation (12) in thefollowing form:

U(x , t) ≈ c+ℓ,i,kΦ+

ℓ,i,k (x , t) + c−ℓ,i,kΦ−

ℓ,i,k (x , t), (17)

where the coefficients c±ℓ,i,k are to be determined by

matching the beam asymptotic solution with the initialconditions given in (13) and (14).



Polarized coefficients

We have

c+ℓ,i,k =

12

aℓ,i,k − bℓ,i,k

ı · G+(

kLℓ,2πξℓ,i

)

, (18)

c−ℓ,i,k =

12

aℓ,i,k +bℓ,i,k

ı · G+(

kLℓ,2πξℓ,i

)

. (19)

The initial conditions (10) and (11) are polarized into thetwo wave modes U(x ,0) ≈ U+(x ,0) + U−(x ,0) in theframes of the multiscale Gaussian wavepacket transformswith

U±(x ,0) ≈∑

ℓ,i,k

c±ℓ,i,kΦ±

ℓ,i,k (x ,0).



Multiscale Gaussian beams

In turn, these two polarized wave modes allow us to designmultiscale Gaussian beam methods to compute asymptoticsolutions in the high frequency regime.

The global asymptotic solution at time t is given by

U(x , t) ≈∑

ℓ,i,k

c+ℓ,i,kΦ+

ℓ,i,k (x , t) +∑

ℓ,i,k

c−ℓ,i,kΦ−

ℓ,i,k (x , t).

Moreover, for a typical initial condition (U(x ,0),Ut(x ,0)),most of the coefficients c±

ℓ,i,k have small norms.

Therefore, in light of computation efficiency, one onlyneeds to keep the coefficients c±

ℓ,i,k for which theabsolute value is greater than a certain prescribedthreshold η. For all other coefficients, we simply set thevalue to zero.




Define the (index) sets of non-negligible coefficients S±η by

S±η =

(ℓ, i , k) : |c±ℓ,i,k | > η

.

Thus, for each (l , i , k) ∈ S±η we solve equations (15) to

obtain Gaussian beam ingredients and superpose eachbeam accordingly,

Uη(x , t) ≡∑

(ℓ,i,k)∈S+η

c+ℓ,i,kΦ+

ℓ,i,k (x , t)+∑

(ℓ,i,k)∈S−η

c−ℓ,i,kΦ−

ℓ,i,k (x , t),

(20)which yields a global asymptotic solution to the waveequation (1).



Multiscale Gaussian beams: algorithms

According to the above setup, one can easily design analgorithm to implement the above ideas: Q-Ying (MMS’10).

Beam initialization: the overall complexity is O(Nd log N).

The overall cost of tracing the Gaussian beams isproportional to the cardinalities of S+

η and S−η .

As the support of each Gaussian beam is of size O(N1/2)in each dimension, each beam at time T covers aboutO(Nd/2) points. Therefore, the overall cost of beamsummation is of order O((|S+

η | + |S−η |) · Nd/2).

Summing these estimates together shows that the overallcost of the multiscale Gaussian beam isO(

Nd log N + (|S+η | + |S−

η |) · Nd/2)

. The efficiency of theproposed algorithm evidently depends on the cardinalitiesof S+

η and S−η .



Multiscale Gaussian beams: efficiency

For all usual initial conditions, such as point sources, planewaves, and curvilinear wavefronts, the multiscale Gaussianwavepackets provide theoretically sparse approximationsto these initial conditions. Most importantly, such sparsityis preserved throughout the time evolution.

For those initial conditions, the sizes of S±η would always

be small.

Then the O(

Nd log N + (|S+η | + |S−

η |) · Nd/2)

cost of themultiscale Gaussian beams is much more efficientcompared to the O(Nd+1) cost of standard finite differenceor finite element methods.



Long time propagation by reinitialization

By a simple analysis we can reveal the exponential growthof Gaussian beam width (Q-Ying (MMS’10)).

If the beam width exponentially grows, the beam loses itslocalized significance, leading to deteriorating accuracy inthe Taylor expansion for the phase function and high costin beam summation.

One natural solution to this issue is to monitor the widths ofthe Gaussian beams and reinitialize the Gaussian beamrepresentation before any one of the beams becomes toowide.

This reinitialization strategy effectively avoids the twodifficulties mentioned above.



Long time propagation by reinitialization: cont.

Although the reinitialization idea is rather straightforward, itis rather difficult to combine with existing methods of beaminitialization for reasons related to representation andefficiency.

The reinitialization idea fits perfectly with our multiscaleGaussian beam algorithm (Q-Ying (MMS’10)).



Global asymptotic solutions

One can show that the wavepacket-transform-basedGaussian beam solution (20) is an asymptotic solution tothe evolution equation (1): (Q-Ying (MMS’10)).


1-D example: Case 1.

U|t=0 = 2 sin(512πx), Ut |t=0 = 256 cos(256πx).

In this case, we take V (x , y) = 0.25 + 0.1 sin(2πx),η = 0.1, N = 8192 and T = 8.0.

If we accept the solution by the pseudo-spectral method asthe exact solution, the relative L2-error in the Gaussianbeam solution is 4.6% and the relative maximum error is5.8%.

Here |S+η | = 640 and |S−

η | = 640.



U|t=0 = 0, Ut |t=0 = 6400 cos(128πx).

We take V (x , y) = 0.25 + 0.1 sin(2πx), N = 4096,T = 8.0, and η = 10−3.

(a)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.15

0.2

0.25

0.3

0.35

x

Ve

locity V

(b)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

x

u



(c) 0 1 2 3 4 5 6 7 8−2

−1

0

1

2

3

t

x

(d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−50

0

50

100

x

rea

l u

(e) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−50

0

50

100

x

rea

l u

(f) 0.7 0.75 0.8 0.85 0.9 0.95−100

−50

0

50

100

x

rea

l u


2-D example

V (x , y) = 1.0 + 0.5 sin(2πx) cos(2πy), and the initialconditions are given by

U|t=0 = 2.0 sin(256πx) cos(256πy),

Ut |t=0 = 200.0 cos(256πx) sin(256πy). (21)

We use N × N = 1024 × 1024 to discretize [0,1] × [0,1].η = 0.005.

We compute the solution up to the final time T = 0.60without reinitialization and with reinitialization.


2-D example: without reinitialization

(c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

u

(d) 0.35 0.4 0.45 0.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

u

(e) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

y

u

(f) 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6−0.5

0

0.5

y

u


2-D example: with reinitialization

(c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

u

(d) 0.35 0.4 0.45 0.5−1.5

−1

−0.5

0

0.5

1

1.5

2

y

u

(e) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

y

u

(f) 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

y

u


3-D example: with reinitialization T = 0.6

(a) y

z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

(b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

z

u

(c) x

z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

(d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

x

u

(e) x

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

(f) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

x

u


Conclusion

Two main computational aspects of the Gaussian beammethods for the wave equation: beam initialization forgeneral initial condition and long time propagation.

For the beam initialization problem, we proposed fastmultiscale Gaussian wavepacket transforms anddeveloped based on them a new efficient algorithm forbeam initialization for general initial data.

Long time propagation was addressed by reinitializing thebeam representation using this algorithm when the beamwidths get too wide.


Conclusion: cont.

Smith’98, Candes-Demanet’05 represented the solutionoperator of the wave equation in the curvelet frame.However, numerical tests show that these representationsoften have a significant constant factor, which limits theirpractical application.

We can also view our Gaussian beam method as a way torepresent the solution operator of the wave equation.

Since the Gaussian beams (indexed by continuousparameters) are much more flexible and general than themultiscale Gaussian wavepackets (indexed by the discrete(ℓ, i , k) parameters), what we gain in the Gaussian beammethod is a representation of the solution operator that isone-to-one and more efficient.


Acknowledgement

Many thanks to the organizers ...

Documents

Fast multiscale Gaussian wavepacket transforms and ...gunther/MSRI_IPSeminar_files/Qian.pdfHigh frequency waves and geometrical optics Gaussian-beam setup for wave equations Multiscale