Gaussian beams for high frequency waves: some recent developments

Transcription

1 Gaussian beams for high frequency waves: some recent developments Jianliang Qian Department of Mathematics, Michigan State University, Michigan International Symposium on Geophysical Imaging with Localized Waves Sanya, China. July, 2.

2 Outline High frequency waves and geometrical optics 2 Single-scale Gaussian beams for Schrödinger Beam ingredients Beam width Gaussian wavepacket transforms Numerical results 3 Multiscale Gaussian beams for wave equations 4 Eulerian Gaussian beams 5 Conclusion

3 The parabolic wave equation (Schrödinger) ı U t + H(, ı )U =, R d, t >, U t= = U (). H(, p) is a real principal symbol: H(, p) = V() + p 2 2. is a small positive parameter; V is a smooth potential. U is a compactly supported L 2 -function, presumably highly oscillatory. real(u)

4 An eample real(u) Figure: Schrödinger under a potential hill. = T = π. Solution at

5 The problem The difficulties When the initial condition contains oscillations of a small wavelength, the Schrödinger equation propagates these oscillations in space and time. Resolving such small oscillations by direct numerical methods requires an enormous computational grid and is very costly in practice. Methods based on geometrical optics are sought as an alternative for capturing such highly oscillatory phenomena.

6 Geometrical optics The traditional geometrical-optics method starts from the WKBJ ansatz consisting of a real phase and a real amplitude, U(, t) A(, t)e ı τ(,t). The WKBJ ansatz breaks down at the caustics, in the neighborhood of which the phase function is multivalued and the amplitude function blows up.

7 High frequency waves and geometrical optics Single-scale Gaussian beams for Schrödinger Multiscale Gaussian beams for wave e t t Geometrical optics for Schrödinger (Cont.)

8 t Gaussian beams Gaussian beam (Babich 72, Hörmander 7, Cerveny-Popov-Psencik 82, Popov 82, Ralston 83, Tanushev-Q-Ralston 7) constructs a global comple phase and a global comple amplitude that satisfy the eikonal equation and the transport equation approimately near a specified ray path. Away from the ray path, the quadratic imaginary part of the phase provides a rapidly decaying Gaussian profile. This gives rise to a single Gaussian-like asymptotic solution, which is accurate near the underlying ray path. Since the equation is linear, a superposition yields a global asymptotic solution

9 Beam ingredients Eikonal and transport equations ı U t + H(, ı )U =, R d, t >, () U(, ) = U () (2) Asymptotic solutions: A(, t)e ı τ(,t), so that the equation () and its associated initial conditions (2) are satisfied approimately with a small error when is small. Consider the leading orders in powers of : τ t (, t) + H(, τ (, t)) =, (3) A t (, t) + H p A + A 2 (trace(h ppτ + H p )) =. (4)

10 Beam ingredients A single Gaussian beam () A single Gaussian beam is concentrated near a ray path which is the -projection of a certain bicharacteristic. The beam ingredients are computed by solving: ẋ = d = H p ((t), p(t)), dt t= =, ṗ = dp = H ((t), p(t)), dt p t= = p, dτ dt = p(t) H p ((t), p(t)) H((t), p(t)), τ t= = τ, dm(t) + HpM T + MH p + MH pp M + H =, M t= = M = ıǫi dt da dt + A 2 (trace(h ppm + H p )) =, A t= = A, τ(t) = τ((t), t), p(t) = τ ((t), t), and M(t) = τ ((t), t).

11 Beam ingredients A single Gaussian beam (2) We define two global, smooth approimate functions for the phase and amplitude along the ray path γ: τ(, t) τ(t) + p(t) ( (t)) + 2 ( (t))t M(t)( (t)) A(, t) A((t), t) = A(t), which are accurate near the ray path γ = {((t), t) : t }.

12 Beam ingredients Taylor epansion for beams τ(t)=τ(t,(t)) p(t)=τ (t,(t)) γ={(t,(t))} ((t),p(t),m(t),a(t)) M(t)=τ (t,(t)).5 τ(t,)=τ(t,(t))+τ (t,(t)) ( (t))+.5τ (t,(t)) ( (t)) 2 A(t,)=A(t)=A(t,(t)) t

13 t Beam ingredients A single Gaussian beam (3) We have a single-beam asymptotic solution concentrated on γ = {((t), t) : t }, Φ(, t) = A(, t) ep(ı τ(, t) ). (5) Because Im(τ(, t)) = 2 ( (t))t Im(M(t))( (t)), Φ(, t) has a Gaussian profile concentrated on γ: ( ep ) 2 ( (t))t Im(M(t))( (t))

14 Beam width Beam width and long-term beam propagation Beam width and reinitialization A simple analysis reveals that the width of a Gaussian beam in the case of the Schrödinger equation is related to the derivatives of the underlying potentials, and a beam width may grow eponentially during the evolution process. So the beam loses its localized significance, leading to deteriorating accuracy in the Taylor epansion for the phase function and high cost in the beam summation.

15 Beam width Beam width and Hessian of potential A single Gaussian beam centered at a ray trajectory {((t), t) : t } propagates according to a quadratic approimation to potential centered at = (t) for t : V() = V((t))+V ((t))( (t))+ 2 ( (t))t V ((t))( (t) The properties of V ((t)) control how the beam width grows. Assume that the quadratic potential function is given at : V() = V( ) + V ( )( ) + 2 ( ) t V ( )( ). The Riccati equation for the Hessian of the phase function can be written as: Ṁ = M 2 V ( ), M t= = ıǫi, (6) where ǫ is a given positive number, and I is the identity.

16 Beam width Hessian of potential: V ( ). Solving the resulting Riccati equation for M yields M(t) = tǫ2 + ıǫ + t 2 ǫ 2. Thus the beam decays like ep ( 2 ) ( Im(M(t))( (t))2 = ep ) ǫ 2 + t 2 ǫ2( (t))2 (+t The beam width is roughly proportional to 2 ǫ 2 ) ǫ, which grows almost linearly as time t increases.

17 Beam width Hessian of potential: V ( ) >. Solving the Riccati equation for M yields (ω = V ( )) M(t) = ω(ǫ2 ω 2 ) sin ωt cos ωt + ıǫω 2 ω 2 cos 2 ωt + ǫ 2 sin 2. ωt Thus the beam decays like ( ) ǫω 2 ep 2 (ω 2 cos 2 ωt + ǫ 2 sin 2 ( (t))2. ωt) The beam width is roughly proportional to (ω 2 cos 2 ωt + ǫ 2 sin 2 ωt) ǫω 2, which is bounded from above by ǫ if ǫ ω or by ǫ if ω 2 ǫ ω. In particular, when ǫ = ω, the beam width does not change.

18 Beam width Hessian of potential: V ( ) < Solving the Riccati equation for M yields (ω = V ( )) M(t) = ǫ2 ω( + )( e 4ωt ) + ı4ǫe 2ωt ω 2 ( + e 2ωt ) 2 + ǫ2 ( e ω 2ωt ). 2 2 The beam decays very slowly like ( ep 4ǫe 2ωt ( 2 [( + e 2ωt ) 2 + ǫ2 ( e ω 2ωt ) 2 (t))2 ] 2 The beam width is roughly proportional to e ωt [ω 2 ( + e 2ωt ) 2 + ǫ 2 ( e 2ωt ) 2 ] 4ǫω 2, which grows eponentially as t increases. ).

19 Beam width Hessian of potential: V ( ) < (Cont.) This implies that the beam loses its localized significance, leading to deteriorating accuracy in the Taylor epansion for the phase function and high cost in beam summation. In practice, along a ray trajectory the Hessian of a potential function may change from positive definite to negative definite, so that the beam width may grow eponentially unepectedly. In the multi-dimensional case, the growth of the beam width depends on the Hessian of the potential in more complicated ways. However, some general features will be the same as in the one-dimensional case.

20 Beam width Long time propagation by reinitialization One natural solution to this issue is to monitor the widths of the Gaussian beams and reinitialize the Gaussian beam representation before any one of the beams becomes too wide. Although the reinitialization idea is rather straightforward, it is rather difficult to combine with eisting methods of beam initialization for reasons related to representation and efficiency. The reinitialization idea fits perfectly with the fast Gaussian wavepacket transform based beam algorithm (Q-Ying (JCP )).

21 Beam width Long time propagation by reinitialization: cont. As an eample, we take the case V < to demonstrate the benefit of reinitialization. Suppose that V ( ) = 4π 2, ω = V ( ) = 2π, and the final time T = 2.. Without reinitialization the beam width at the final time T is roughly e ωt e 4π = π 2π If we choose t such that e ω t 2π, and we divide [, T] into subintervals, then the beam width in each T t subinterval is roughly. In this particular eample, t.25, and = 8. T t This simple case analysis demonstrates that it is critical to carry out reinitialization.

22 Beam width Beam ingredients The functions, (t), p(t), τ(t), M(t), A(t), τ(, t), A(, t), and Φ(, t), are uniquely determined by the initial data, p, τ, M, and A. We denote these initial data collectively by a tuple α = (, p, τ, M, A ). The solutions are denoted, respectively, by α (t), p α (t), M α (t), A α (t), τ α (, t), A α (, t), and Φ α (, t).

23 Beam width Beam summation For a given tuple α = (, p, τ, M, A ), the Gaussian beams Φ α (, t) have a simple Gaussian envelope. For a general initial condition U(, ), one needs to find a set I of tuples such that at time t = U(, ) α I Φ α (, ). Once this initial decomposition is given, the linearity of the equation gives the Gaussian beam solution U(, t) α I Φ α (, t).

24 Beam width How to generate a beam decomposition Related works Beam decompositions for wave equations: Cerveny 82, Hill (Geop 9), Leung-Q-Burridge (Geop 7), Tanushev-Q-Ralston (SIAM MMS 7). FBI-transform based beam decompositions for Schrödinger equations: Leung-Q (JCP 9, ). For Schrödinger equations the most efficient beam decomposition method is based on single-scale Gaussian wavepacket transforms: Q-Ying (JCP ).

25 Gaussian wavepacket transforms Motivation At time t = each Gaussian beam takes the form ( ( ı A ep p ( ) + )) 2 ( ) T M ( ), where p = O() and the Hessian M is purely imaginary and of order O(). This is a modulated Gaussian function that oscillates at a wavelength of order O( ) and has an effective support of width O( /2 ) in space. The initialization step is equivalent to decomposing U(, ) into a linear combination of such Gaussian functions. Gaussian wavepacket transforms.

26 Gaussian wavepacket transforms Partitioning the Fourier domain Start by partitioning the Fourier domain R d into d-dimensional boes of size W in each dimension. Denote these boes by B i where i = (i, i 2,...,i d ) is a multiinde with integer components and the center of each B i by ξ i = (ξ i,, ξ i,2,...,ξ i,d ) R d.

27 Gaussian wavepacket transforms Illustrations

28 Gaussian wavepacket transforms Gaussian windows in the Fourier domain Each B i is associated with a smooth window function g i (ξ) that is compactly supported in a bo centered at ξ i with width L = 2W (i.e., d s= [ξ i,s W, ξ i,s + W]). g i (ξ) is also required to approimate a Gaussian profile with σ = W/2. g i (ξ) e ξ ξi 2 σ

29 Gaussian wavepacket transforms A partition of unity Choose g i (ξ) to satisfy three admissible conditions (Q-Ying JCP ). For each B i define the conjugate filter h i (ξ): h i (ξ) = g i (ξ) i (g i(ξ)) 2. By construction, the products of g i (ξ) and h i (ξ) form a partition of unity: i g i(ξ)h i (ξ) =.

30 Gaussian wavepacket transforms Two sets of wavepacket functions We introduce two sets of functions {φ i,k ()}, {ψ i,k ()}, which are defined in the Fourier domain by ˆφ i,k (ξ) = k ξ 2πı e L d/2 L g i (ξ), k Z d, ˆψ i,k (ξ) = k ξ 2πı e L d/2 L h i (ξ), k Z d. Taking the inverse Fourier transforms gives their definitions in the spatial domain: φ i,k () = L d/2 R d e 2πı( k L ) ξ g i (ξ)dξ, k Z d ψ i,k () = L d/2 R d e 2πı( k L ) ξ h i (ξ)dξ, k Z d

31 Gaussian wavepacket transforms Two sets of wavepacket functions (Cont.) The definitions of g i (ξ) and φ i,k () imply that φ i,k () ( ) d π L σ e 2πı( k L ) ξ i e σ2 π 2 k L 2. φ i,k () is approimately a Gaussian function that is spatially centered at k/l, oscillates at frequency ξ i, and has an O(σ) = O( 2) effective width in the Fourier domain and an O(/σ) = O( 2) effective width in the spatial domain. The functions {φ i,k ()} fit eactly into the profile of a Gaussian beam with: ξ i. Qualitatively, ψ i,k () is also a wavepacket with slightly larger support in compared to φ i,k ().

32 Gaussian wavepacket transforms Two sets of wavepacket functions (Cont.) Figure: Typical profiles of φ i,k () and ψ i,k ().

33 Gaussian wavepacket transforms Two frames {φ i,k ()} and {ψ i,k ()} are two frames of L 2 (R d ). Lemma (Q-Ying (JCP )) There eists constants C and C 2 such that for any f L 2 (R d ) C f 2 2 i,k φ i,k, f 2 C 2 f 2 2, C f 2 2 i,k ψ i,k, f 2 C 2 f 2 2.

34 Gaussian wavepacket transforms Dual frames {φ i,k ()} and {ψ i,k ()} are dual frames. Lemma (Q-Ying (JCP )) For any f L 2 (R d ), f() = ψ i,k, f φ i,k (). i,k Lemma 2.2 offers a way to decompose any function f L 2 (R d ) into a sum of Gaussian-like functions.

35 Gaussian wavepacket transforms Forward and inverse Gaussian wavepacket transforms Given f, the forward Gaussian wavepacket transform computes the coefficients {c i,k } defined by c i,k = ψ i,k, f = ˆψ i,k,ˆf (7) where, is the usual L 2 (R d ) inner product and ˆf denotes the Fourier transform of f. By Lemma 2. each coefficient c i,k is at most of order O() and many coefficients are negligible. Given a set of coefficients {c i,k }, the inverse Gaussian wavepacket transform synthesizes a function u() defined by u() = c i,k φ i,k (). (8) i,k

36 Numerical results -D eample: potential hill () Initial condition: ( U(, t = ) = ep 25(.5) 2) ep ( ıτ () ), τ () = ln (ep(5(.5)) + ep( 5(.5))). 5 In this case, we take V() = cos(2π( +.5)), = 256π, η =., N = 24 and T = 2..

37 Numerical results potential hill (2): no reinitialization.5.5 real u real u.5.5 (a) (b) real u.2 real u (c) (d)

38 Numerical results potential hill (3): reinitialization.5.5 real u real u.5.5 (a) (b) real u real u (c) (d)

39 Numerical results potential hill (4): = 2 4 π, T =., N = 27, and Rein = real u real(u) (a) (b) real (u).5.5 position density.5.5 (c) (d)

40 Numerical results 2-D eample V(, y) = +.5 sin(2π) cos(2πy), and the initial condition is given by ( U(t = ) = ep 25(.5) 2) ( ) ıτ () ep, τ () = ln (ep(5(.5)) + ep( 5(.5))). 5 We use N N = = 52π. η =.. We compute the solution up to the final time T =.56 without reinitialization and with reinitialization.

41 y y Numerical results 2-D eample: without reinitialization () (a) (b) real u real u (c) (d)

42 Numerical results 2-D eample: without reinitialization (2) real u real u (e) (f) real u real u (g) y (h) y

43 y y Numerical results 2-D eample: with reinitialization () (a) (b) real u real u (c) (d)

44 Numerical results 2-D eample: with reinitialization (2) real u real u (e) (f) real u real u (g) y (h) y

45 z z y u u u Numerical results 3-D eample: with reinitialization T =.56 (a) y (b) z (c) (d) y (e) (f) y

46 Multiscale Gaussian beams for wave equations U tt V 2 () U =, R d, t >, U t= = f (), U t t= = f 2 (). V is smooth, positive, and bounded away from zero. f () and f 2 () are compactly supported L 2 -functions, presumably highly oscillatory.

47 Multiscale Gaussian-beam setup Beam ingredients can be computed according to the eikonal and transport equations. Two polarized wave modes for the wave equation. For wave equations, the most efficient beam decomposition method is based on fast multiscale Gaussian wavepacket transforms: Q-Ying (SIAM MMS ). This is inspired by the parabolic scaling principle (Smith 98, Candes-Demanet 5).

48 Partitioning the Fourier domain Start by partitioning the Fourier domain R d into Cartesian coronae {C l } for l : C = [ 4, 4] d, C l = {ξ = (ξ, ξ 2,...,ξ d ) : ma s d ξ s [4 l, 4 l ]}, l 2. ξ C l implies that ξ = O(4 l ). Each corona C l is further partitioned into boes B l,i = d [2 l i s, 2 l (i s + )], s= where the integer multiinde i = (i, i 2,...,i d ) ranges over all possible choices that satisfy B l,i C l. All boes in a fied C l have the same length W l = 2 l in each dimension and the center of the bo B l,i is denoted by ξ l,i = (ξ l,i,, ξ l,i,2,...,ξ l,i,d ).

49 Illustrations

50 Gaussian windows in the Fourier domain To each bo B l,i, we associate a smooth function g l,i (ξ), which is compactly supported in a bo centered at ξ l,i with size L l = 2W l in each dimension. g l,i (ξ) is also required to approimate a Gaussian profile with σ l = W l /2. «2 g l,i (ξ) e ξ ξl,i σ l

51 Multiscale Gaussian wavepacket transforms Different from Single-scale Gaussian wavepacket transforms (Q-Ying JCP), we introduce angular resolution as well in Multiscale Gaussian wavepacket transforms (Q-Ying MMS). Define a partition of unity based on these associated Gaussian windows. We introduce two sets of functions {φ l,i,k ()}, {ψ l,i,k ()}, ˆφ l,i,k (ξ) = L d/2 l ˆψ l,i,k (ξ) = L d/2 l k ξ 2πı L e l g l,i (ξ), k Z d, k ξ 2πı L e l h l,i (ξ), k Z d. These define dual frames and consist of multiscale Forward and inverse Gaussian wavepacket transforms.

52 2-D eample V(, y) =. +.5 sin(2π) cos(2πy), and the initial conditions are given by U t= = 2. sin(256π) cos(256πy), U t t= = 2. cos(256π) sin(256πy). (9) We use N N = to discretize [, ] [, ]. η =.5. We compute the solution up to the final time T =.6 without reinitialization and with reinitialization.

53 High frequency waves and geometrical optics Single-scale Gaussian beams for Schrödinger Multiscale Gaussian beams for wave e 2-D eample: without reinitialization () y y

54 2-D eample: without reinitialization (2) u u (c) y (d) y u u 2 (e) y (f) y

55 High frequency waves and geometrical optics Single-scale Gaussian beams for Schrödinger Multiscale Gaussian beams for wave e 2-D eample: with reinitialization () y y

56 2-D eample: with reinitialization (2) u.5.5 u (c) y (d) y u 2 u..2.3 (e) y (f) y

57 Eulerian Gaussian beams: Methodology Based on phase-space PDE formulation of geometrical optics, we can reformulate the ODEs of beam ingredients into PDEs in the full phase space by using the material derivative along ray trajectories (Leung-Q-Burridge Geop7, Leung-Q JCP9, JCP). The initial beam decomposition is achieved by using the Fourier-Biros-Iagnitzer (FBI) transform (Leung-Q JCP9,).

58 Conclusions Two main computational aspects of the Gaussian beam methods for high frequency waves: beam initialization for general initial condition and long time propagation. For the beam initialization problem, we proposed fast (multiscale) Gaussian wavepacket transforms and developed based on them a new efficient algorithm for beam initialization for general initial data. Long time propagation was addressed by reinitializing the beam representation using this algorithm when the beam widths get too wide.

59 Conclusions: cont. Smith 98, Candes-Demanet 5 represented the solution operator of the wave equation in the curvelet frame. However, numerical tests show that these representations often have a significant constant factor, which limits their practical application. We can also view our Gaussian beam method as a way to represent the solution operator of the Schrödinger equation or the wave equation. Since the Gaussian beams (indeed by continuous parameters) are much more fleible and general than the Gaussian wavepackets (indeed by the discrete (i, k) parameters), what we gain in the Gaussian beam method is a representation of the solution operator that is one-to-one and more efficient.

60 Acknowledgement Many thanks to the organizers for the invitation.