Gaussian Beam Approximations

Transcription

1 Gaussian Beam Approximations Olof Runborg CSC, KTH Joint with Hailiang Liu, Iowa, and Nick Tanushev, Austin Technischen Universität München München, January 2011 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

2 High frequency waves Cauchy problem for scalar wave equation u tt c(x) 2 u = 0, (t, x) R + R n, u(0, x) = A(x)e iφ(x)/ε, u t (0, x) = 1 ε B(x)eiφ(x)/ε, where c(x) (variable) speed of propagation. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

3 High frequency waves Cauchy problem for scalar wave equation u tt c(x) 2 u = 0, (t, x) R + R n, u(0, x) = A(x)e iφ(x)/ε, u t (0, x) = 1 ε B(x)eiφ(x)/ε, where c(x) (variable) speed of propagation. High frequency short wave length highly oscillatory solutions many gridpoints. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

4 High frequency waves Cauchy problem for scalar wave equation u tt c(x) 2 u = 0, (t, x) R + R n, u(0, x) = A(x)e iφ(x)/ε, u t (0, x) = 1 ε B(x)eiφ(x)/ε, where c(x) (variable) speed of propagation. High frequency short wave length highly oscillatory solutions many gridpoints. Direct numerical solution resolves wavelength: #gridpoints ε n at least cost ε n 1 at least Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

5 High frequency waves Cauchy problem for scalar wave equation u tt c(x) 2 u = 0, (t, x) R + R n, u(0, x) = A(x)e iφ(x)/ε, u t (0, x) = 1 ε B(x)eiφ(x)/ε, where c(x) (variable) speed of propagation. High frequency short wave length highly oscillatory solutions many gridpoints. Direct numerical solution resolves wavelength: #gridpoints ε n at least cost ε n 1 at least Often unrealistic approach for applications in e.g. optics, electromagnetics, geophysics, acoustics,... Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

6 Geometrical optics Wave equation u tt c(x) 2 u = 0. Write solution on the form u(t, x) = a(t, x, ε)e iφ(t,x)/ε. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

7 Geometrical optics Wave equation u tt c(x) 2 u = 0. Write solution on the form u(t, x) = a(t, x, ε)e iφ(t,x)/ε. (a) Amplitude a(x) (b) Phase φ(x) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

8 Geometrical optics a, φ vary on a much coarser scale than u. (And varies little with ε.) Geometrical optics approximation considers a and φ as ε 0. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

9 Geometrical optics a, φ vary on a much coarser scale than u. (And varies little with ε.) Geometrical optics approximation considers a and φ as ε 0. Phase and amplitude satisfy eikonal and transport equations φ 2 t c(y) 2 φ 2 = 0, φ a a t + c φ + c2 φ φ tt a = 0. 2c φ Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

10 Geometrical optics a, φ vary on a much coarser scale than u. (And varies little with ε.) Geometrical optics approximation considers a and φ as ε 0. Phase and amplitude satisfy eikonal and transport equations φ 2 t c(y) 2 φ 2 = 0, φ a a t + c φ + c2 φ φ tt a = 0. 2c φ Good accuracy for small ε. Computational cost ε-independent. u(t, x) = a(t, x)e iφ(t,x)/ε + O(ε). (#gridpoints and cost independent of ε) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

11 Geometrical optics a, φ vary on a much coarser scale than u. (And varies little with ε.) Geometrical optics approximation considers a and φ as ε 0. Phase and amplitude satisfy eikonal and transport equations φ 2 t c(y) 2 φ 2 = 0, φ a a t + c φ + c2 φ φ tt a = 0. 2c φ Good accuracy for small ε. Computational cost ε-independent. u(t, x) = a(t, x)e iφ(t,x)/ε + O(ε). (#gridpoints and cost independent of ε) Waves propagate as rays, c.f. visible light. Not all wave effects captured correctly. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

12 Numerical approaches Eikonal and transport equation (PDEs) φ 2 t c(y) 2 φ 2 = 0, φ a a t + c φ + c2 φ φ tt a = 0. 2c φ Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

13 Numerical approaches Eikonal and transport equation (PDEs) φ 2 t c(y) 2 φ 2 = 0, φ a a t + c φ + c2 φ φ tt a = 0. 2c φ Raytracing (ODEs) Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, dx dt = c(x) 2 p, dp dt = c(x) c(x), φ(t, x(t)) = φ(0, x(0)). p(t) is local ray direction, "slowness" vector ( p = 1/c and p = φ(x))). (Also ODEs for amplitude.) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

14 Numerical approaches Eikonal and transport equation (PDEs) φ 2 t c(y) 2 φ 2 = 0, φ a a t + c φ + c2 φ φ tt a = 0. 2c φ Raytracing (ODEs) Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, dx dt = c(x) 2 p, dp dt = c(x) c(x), φ(t, x(t)) = φ(0, x(0)). p(t) is local ray direction, "slowness" vector ( p = 1/c and p = φ(x))). (Also ODEs for amplitude.) Difficulties: Nonlinearity of eikonal equation, viscosity solution, kinks Multiphase solutions, crossing rays Diverging rays Breakdown of geometrical optics (boundaries, caustics) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

15 Example Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

16 Caustics Concentration of rays. Amplitude a(t, y) but should be a(t, y) ε α, 0 < α < 1. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

17 Schrödinger Case Same conclusions also for the time-dependent Schrödinger equation iεu t + ε 2 u V (x)u = 0. Geometrical optics with u(t, x) = a(t, x)e iφ(t,x)/ε + O(ε), Eikonal and transport equation φ t + φ 2 V (x)φ = 0, a t + 2 φ a + a φ = 0. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

18 Schrödinger Case Same conclusions also for the time-dependent Schrödinger equation iεu t + ε 2 u V (x)u = 0. Geometrical optics with u(t, x) = a(t, x)e iφ(t,x)/ε + O(ε), Eikonal and transport equation φ t + φ 2 V (x)φ = 0, a t + 2 φ a + a φ = 0. Raytracing (p(t) is momentum) dx dt = p, dp dt = V (x), dφ dt = 1 2 p 2 V (x). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

19 Schrödinger Case Same conclusions also for the time-dependent Schrödinger equation iεu t + ε 2 u V (x)u = 0. Geometrical optics with u(t, x) = a(t, x)e iφ(t,x)/ε + O(ε), Eikonal and transport equation φ t + φ 2 V (x)φ = 0, a t + 2 φ a + a φ = 0. Raytracing (p(t) is momentum) Difficulties: dx dt = p, dp dt = V (x), dφ dt = 1 2 p 2 V (x). Nonlinearity of eikonal equation, viscosity solution, kinks Multiphase solutions, crossing rays, diverging rays Breakdown of geometrical optics (boundaries, caustics) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

20 Outline Improved high frequency asymptotic approximations based on Gaussian beams 1 Construction and derivation 2 Error estimates in terms of ε 3 Numerics (some) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

21 Constructing asymptotic solutions 1 Make an ansatz for the form the solution ũ(t, y) with some unknown coefficients/functions, e.g. ũ(t, y) = a(t, y)e iφ(t,y)/ε, Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

22 Constructing asymptotic solutions 1 Make an ansatz for the form the solution ũ(t, y) with some unknown coefficients/functions, e.g. ũ(t, y) = a(t, y)e iφ(t,y)/ε, 2 Find the coefficients/functions in the ansatz such that ũ satisfies the Schrödinger equation as well as possible, iεũ t + ε 2 ũ V (x)ũ 1. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

23 Constructing asymptotic solutions 1 Make an ansatz for the form the solution ũ(t, y) with some unknown coefficients/functions, e.g. ũ(t, y) = a(t, y)e iφ(t,y)/ε, 2 Find the coefficients/functions in the ansatz such that ũ satisfies the Schrödinger equation as well as possible, iεũ t + ε 2 ũ V (x)ũ and such that the coefficients can be numerically computed much easier than the full solution u itself. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

24 Example: Geometrical optics 1 Use WKB expansion as ansatz (a(t, y) a power series in iε) K 1 u(t, y) = e iφ(t,y)/ε k=0 a k (t, y)(iε) k. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

25 Example: Geometrical optics 1 Use WKB expansion as ansatz (a(t, y) a power series in iε) K 1 u(t, y) = e iφ(t,y)/ε 2 Find the coefficients in the ansatz. k=0 iεu t + ε 2 u Vu = E[Φ]u + iεp[a 0 ]e iφ/ε K 2 + a k (t, y)(iε) k. (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 where E and P are the eikonal and transport operators E[Φ] := Φ t + Φ 2 + V Φ, P[a] := a t + 2 φ a + a φ. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

26 Example: Geometrical optics 1 Use WKB expansion as ansatz (a(t, y) a power series in iε) K 1 u(t, y) = e iφ(t,y)/ε 2 Find the coefficients in the ansatz. k=0 iεu t + ε 2 u Vu = E[Φ]u + iεp[a 0 ]e iφ/ε K 2 + a k (t, y)(iε) k. (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 where E and P are the eikonal and transport operators E[Φ] := Φ t + Φ 2 + V Φ, P[a] := a t + 2 φ a + a φ. Solve E[Φ] = 0, P[a 0 ] = 0 and P[a k+1 ] = a k. Then, iεu t + ε 2 u Vu = O(ε K +1 ) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

27 Example: Geometrical optics 1 Use WKB expansion as ansatz (a(t, y) a power series in iε) K 1 u(t, y) = e iφ(t,y)/ε 2 Find the coefficients in the ansatz. k=0 iεu t + ε 2 u Vu = E[Φ]u + iεp[a 0 ]e iφ/ε K 2 + a k (t, y)(iε) k. (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 where E and P are the eikonal and transport operators E[Φ] := Φ t + Φ 2 + V Φ, P[a] := a t + 2 φ a + a φ. Solve E[Φ] = 0, P[a 0 ] = 0 and P[a k+1 ] = a k. Then, iεu t + ε 2 u Vu = O(ε K +1 ) 3 Non-oscillatory problems, easier to solve. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

28 Gaussian approximations Approximate solutions to the wave equations/schrodinger with a Gaussian profile (width ε). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

29 Gaussian approximations Approximate solutions to the wave equations/schrodinger with a Gaussian profile (width ε). For waves studied by e.g. Cerveny, Popov, Babich, Psencik, Ralston, Hörmander, Klimes, Hill, etc. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

30 Gaussian approximations Approximate solutions to the wave equations/schrodinger with a Gaussian profile (width ε). For waves studied by e.g. Cerveny, Popov, Babich, Psencik, Ralston, Hörmander, Klimes, Hill, etc. For Schrödinger, classical (coherent state), Heller, Hagedorn, Herman, Kluk, Kay, etc. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

31 Gaussian approximations Approximate solutions to the wave equations/schrodinger with a Gaussian profile (width ε). For waves studied by e.g. Cerveny, Popov, Babich, Psencik, Ralston, Hörmander, Klimes, Hill, etc. For Schrödinger, classical (coherent state), Heller, Hagedorn, Herman, Kluk, Kay, etc. No break down at caustics. Gives e.g. improved seismic imaging. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

32 Gaussian approximations Approximation of the same form as geometrical optics solutions, v(t, y) = a(t, y)e iφ(t,y)/ε, Φ(t, y) = φ(t, y x(t)), where x(t) is a geometrical optics ray. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

33 Gaussian approximations Approximation of the same form as geometrical optics solutions, v(t, y) = a(t, y)e iφ(t,y)/ε, Φ(t, y) = φ(t, y x(t)), where x(t) is a geometrical optics ray. The phase Φ will now have a positive imaginary part away from the ray x(t). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

34 Gaussian approximations Approximation of the same form as geometrical optics solutions, v(t, y) = a(t, y)e iφ(t,y)/ε, Φ(t, y) = φ(t, y x(t)), where x(t) is a geometrical optics ray. The phase Φ will now have a positive imaginary part away from the ray x(t). Imaginary part of φ y 2 v(t, y) e y x(t) 2 /ε, Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

35 Gaussian approximations Approximation of the same form as geometrical optics solutions, v(t, y) = a(t, y)e iφ(t,y)/ε, Φ(t, y) = φ(t, y x(t)), where x(t) is a geometrical optics ray. The phase Φ will now have a positive imaginary part away from the ray x(t). Imaginary part of φ y 2 v(t, y) e y x(t) 2 /ε, Gaussian with width ε Localized around x(t). (Moves along the space time ray.) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

36 Gaussian approximations Approximation of the same form as geometrical optics solutions, v(t, y) = a(t, y)e iφ(t,y)/ε, Φ(t, y) = φ(t, y x(t)), where x(t) is a geometrical optics ray. The phase Φ will now have a positive imaginary part away from the ray x(t). Imaginary part of φ y 2 v(t, y) e y x(t) 2 /ε, Gaussian with width ε Localized around x(t). (Moves along the space time ray.) Amplitude a(t, y) and the phase Φ(t, y) approximated by polynomials locally around x(t). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

37 Gaussian beams The simplest ("first order") Gaussian beams use the ansatz v(t, y) = a(t)e iφ(t,y)/ε, Φ(t, y) = φ(t, y x(t)), where φ(t, y) = φ 0 (t) + y p(t) y M(t)y. i.e. a(t, y) approximated to 0th order, and Φ(t, y) to 2nd order. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

38 Gaussian beams The simplest ("first order") Gaussian beams use the ansatz v(t, y) = a(t)e iφ(t,y)/ε, Φ(t, y) = φ(t, y x(t)), where φ(t, y) = φ 0 (t) + y p(t) y M(t)y. i.e. a(t, y) approximated to 0th order, and Φ(t, y) to 2nd order. We can require that Φ(t, y) solves eikonal to order O( y x 3 ) and a(t, y) solves transport equation to order O( y x ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

39 First order beams Let us thus require that Φ t + Φ 2 V (y)φ = O( y x(t) 3 ), a t +2 Φ a+a Φ = O( y x(t) ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

40 First order beams Let us thus require that Φ t + Φ 2 V (y)φ = O( y x(t) 3 ), a t +2 Φ a+a Φ = O( y x(t) ). Then we obtain ODEs for φ 0, x, p, M, a 0. ẋ(t) = p, ṗ(t) = V (x), φ 0 (t) = 1 2 p 2 V (x), Ṁ(t) = M 2 x 2 V (x), ȧ 0 (t) = 1 2 a 0Tr(M). Easy to compute numerically! Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

41 Asymptotic order As before, iεv t + ε 2 v Vv = E[Φ]u + iεp[a]e iφ/ε (iε) 2 ae iφ/ε Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

42 Asymptotic order As before, iεv t + ε 2 v Vv = E[Φ]u + iεp[a]e iφ/ε (iε) 2 ae iφ/ε Here we enforce E[Φ] = O( y x 3 ), P[a] = O( y x ), and a = a 0 (t) = 0 by construction. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

43 Asymptotic order As before, iεv t + ε 2 v Vv = E[Φ]u + iεp[a]e iφ/ε (iε) 2 ae iφ/ε Here we enforce E[Φ] = O( y x 3 ), P[a] = O( y x ), and a = a 0 (t) = 0 by construction. Hence, since exp(iφ/ε) exp( y x 2 /ε), ( [ ] iεv t + ε 2 v Vv = O y x 3 + ε y x e y x 2 ε ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

44 Asymptotic order As before, iεv t + ε 2 v Vv = E[Φ]u + iεp[a]e iφ/ε (iε) 2 ae iφ/ε Here we enforce E[Φ] = O( y x 3 ), P[a] = O( y x ), and a = a 0 (t) = 0 by construction. Hence, since exp(iφ/ε) exp( y x 2 /ε), ( [ ] iεv t + ε 2 v Vv = O y x 3 + ε y x e y x 2 ε Using x p e x 2 /ε C p ε p/2 we then get iεv t + ε 2 v Vv = O (ε 3/2) ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

45 Asymptotic order As before, iεv t + ε 2 v Vv = E[Φ]u + iεp[a]e iφ/ε (iε) 2 ae iφ/ε Here we enforce E[Φ] = O( y x 3 ), P[a] = O( y x ), and a = a 0 (t) = 0 by construction. Hence, since exp(iφ/ε) exp( y x 2 /ε), ( [ ] iεv t + ε 2 v Vv = O y x 3 + ε y x e y x 2 ε Using x p e x 2 /ε C p ε p/2 we then get iεv t + ε 2 v Vv = O (ε 3/2) (C.f. GO: iεu t + ε 2 u Vu = O(ε 2 ).) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40 ).

46 Gaussian beams Properties v(t, y) = a 0 (t)e iφ(t,y x(t))/ε, φ(t, y) = φ 0 (t) + y p(t) y M(t)y Φ(t, x(t)) = φ(t, 0) = φ 0 (t) is real valued If M(0) is symmetric and IM(0) is positive definite then this is true for M(t) (which exists) for all t > 0. Second derivatives of φ exist everywhere (no blow-up at caustics) Shape of beam remains Gaussian Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

47 Higher order beams More generally, we can construct higher order beams. Let where, for order K beams, v(t, y) = a(t, y x(t))e iφ(t,y x(t))/ε, Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

48 Higher order beams More generally, we can construct higher order beams. Let where, for order K beams, v(t, y) = a(t, y x(t))e iφ(t,y x(t))/ε, The phase is a Taylor polynomial to order K + 1, φ(t, y) = φ 0 (t) + y p(t) + y 1 K +1 2 M(t)y + β =3 1 β! φ β(t)y β. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

49 Higher order beams More generally, we can construct higher order beams. Let where, for order K beams, v(t, y) = a(t, y x(t))e iφ(t,y x(t))/ε, The phase is a Taylor polynomial to order K + 1, φ(t, y) = φ 0 (t) + y p(t) + y 1 K +1 2 M(t)y + A is now a finite WKB expansion, a(t, y) = K /2 1 j=0 ε j a j (t, y) β =3 1 β! φ β(t)y β. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

50 Higher order beams More generally, we can construct higher order beams. Let where, for order K beams, v(t, y) = a(t, y x(t))e iφ(t,y x(t))/ε, The phase is a Taylor polynomial to order K + 1, φ(t, y) = φ 0 (t) + y p(t) + y 1 K +1 2 M(t)y + A is now a finite WKB expansion, a(t, y) = K /2 1 j=0 ε j a j (t, y) β =3 1 β! φ β(t)y β. Each amplitude term a j is a Taylor polynomial to order K 2j 1 a j (t, y) = K 2j 1 β =0 1 β! a j,β(t)y β Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

51 Higher order beams We now require that φ(t, y) solves eikonal equation to order y x K +2 a j (t, y) solve higher order transport equations to order y x K 2j Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

52 Higher order beams We now require that φ(t, y) solves eikonal equation to order y x K +2 a j (t, y) solve higher order transport equations to order y x K 2j This gives ODEs for all Taylor coefficients, ẋ(t) = p, ṗ(t) = V (x), φ 0 (t) = 1 2 p 2 V (x), Ṁ(t) = M 2 2 x V (x), ȧ j,β (t) =..., φ β (t) =..., Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

53 Higher order beams We now require that φ(t, y) solves eikonal equation to order y x K +2 a j (t, y) solve higher order transport equations to order y x K 2j This gives ODEs for all Taylor coefficients, ẋ(t) = p, ṗ(t) = V (x), φ 0 (t) = 1 2 p 2 V (x), Ṁ(t) = M 2 2 x V (x), ȧ j,β (t) =..., φ β (t) =..., Again, Φ(t, x(t)) is real and IM(t) > 0 no problems at caustcs. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

54 Asymptotic order As before, with K = K /2, iεv t + ε 2 v Vv = E[Φ]v + iεp[a 0 ]e iφ/ε K 2 + (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

55 Asymptotic order As before, with K = K /2, iεv t + ε 2 v Vv = E[Φ]v + iεp[a 0 ]e iφ/ε K 2 + (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 Here we enforce that E[Φ] = O( y x K +2 ), P[a 0 ] = O( y x K ), P[a k+1 ] a k = O( y x K 2k 2 ), k = 0,..., K 2. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

56 Asymptotic order As before, with K = K /2, iεv t + ε 2 v Vv = E[Φ]v + iεp[a 0 ]e iφ/ε K 2 + (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 Here we enforce that E[Φ] = O( y x K +2 ), P[a 0 ] = O( y x K ), P[a k+1 ] a k = O( y x K 2k 2 ), k = 0,..., K 2. This means that ( ) ( ) iεv t + ε 2 v Vv = O y x K +2 e y x 2 ε + O ε y x K e y x 2 ε K 2 + k=1 ( ) ( ) O ε k+2 y x K 2k 2 e y x 2 ε + O ε K +1 e y x 2 ε Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

57 Asymptotic order As before, with K = K /2, iεv t + ε 2 v Vv = E[Φ]v + iεp[a 0 ]e iφ/ε This means that K 2 + (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 iεv t + ε 2 v Vv = O K 2 + k=1 ( ) ( ) y x K +2 e y x 2 ε + O ε y x K e y x 2 ε ( ) ( ) O ε k+2 y x K 2k 2 e y x 2 ε + O ε K +1 e y x 2 ε Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

58 Asymptotic order As before, with K = K /2, iεv t + ε 2 v Vv = E[Φ]v + iεp[a 0 ]e iφ/ε This means that K 2 + (iε) k+2 (P[a k+1 ] a k )e iφ/ε (iε) K +1 a K 1 e iφ/ε k=0 iεv t + ε 2 v Vv = O K 2 + k=1 Recalling that x p e x 2 /ε C p ε p/2, ( ) ( ) y x K +2 e y x 2 ε + O ε y x K e y x 2 ε ( ) ( ) O ε k+2 y x K 2k 2 e y x 2 ε + O ε K +1 e y x 2 ε iεv t + ε 2 v Vv = O (ε K /2+1) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

59 Other gaussian beam like approximations Thawed Gaussian Approximation [Heller, 75] Let φ always be a second order polynomial. Take higher order polynomial in amplitude. (Use more terms in amplitude to correct also for errors in phase.) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

60 Other gaussian beam like approximations Thawed Gaussian Approximation [Heller, 75] Let φ always be a second order polynomial. Take higher order polynomial in amplitude. (Use more terms in amplitude to correct also for errors in phase.) Frozen Gaussian Approximation [Heller, 81], [Herman, Kluk, 84] Let φ always be a second order polynomial, width a fixed second derivative (M(t) =constant). Single frozen Gaussian not an asymptotic solution. Use superposition of frozen Gaussians. ( linear basis expansion) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

61 Other gaussian beam like approximations Thawed Gaussian Approximation [Heller, 75] Let φ always be a second order polynomial. Take higher order polynomial in amplitude. (Use more terms in amplitude to correct also for errors in phase.) Frozen Gaussian Approximation [Heller, 81], [Herman, Kluk, 84] Let φ always be a second order polynomial, width a fixed second derivative (M(t) =constant). Single frozen Gaussian not an asymptotic solution. Use superposition of frozen Gaussians. ( linear basis expansion) Gives ODEs for coefficients. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

62 Approximation Errors Suppose u is exact solution iε t u + ε 2 u Vu = 0. and ũ is the approximate asymptotic solution, iε t ũ + ε 2 ũ V ũ 1. What is the norm error in ũ, i.e. u ũ? Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

63 Approximation Errors Suppose u is exact solution iε t u + ε 2 u Vu = 0. and ũ is the approximate asymptotic solution, iε t ũ + ε 2 ũ V ũ 1. What is the norm error in ũ, i.e. u ũ? Use well-posedness (stability) estimate for PDE: implies that for 0 t T, iε t w + ε 2 w Vw = f (t, x), w(t, ) L 2 w(0, ) L 2 + C(T ) ε sup f (t, ) L 2. t [0,T ] Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

64 Approximation Errors Define P ε as Then P ε [u] = 0 and P ε [w] := iε t w + ε 2 w Vw. P ε [ũ u] = P ε [ũ] P ε [u] = P ε [ũ]. Moreover, the well-posedness estimate can be rewritten w(t, ) L 2 w(0, x) L 2 + C(T ) ε sup P ε [w](0, ) L 2. t [0,T ] Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

65 Approximation Errors Define P ε as Then P ε [u] = 0 and P ε [w] := iε t w + ε 2 w Vw. P ε [ũ u] = P ε [ũ] P ε [u] = P ε [ũ]. Moreover, the well-posedness estimate can be rewritten w(t, ) L 2 w(0, x) L 2 + C(T ) ε Hence, assuming u(0, x) = ũ(0, x), ũ(t, ) u(t, ) L 2 C(T ) ε sup P ε [w](0, ) L 2. t [0,T ] sup P ε [ũ](t, ) L 2, 0 t T. t [0,T ] Error in ũ how well it satisfies equation, minus one order in ε Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

66 Example 1: Geometrical Optics If φ and A satisfy eikonal equation and (high order) transport equations, then for ũ GO = A(t, x)e iφ(t,x)/ε, we have before caustics develop. P ε [ũ GO ] (t, x) = O(ε K +1 ), Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

67 Example 1: Geometrical Optics If φ and A satisfy eikonal equation and (high order) transport equations, then for ũ GO = A(t, x)e iφ(t,x)/ε, we have before caustics develop. P ε [ũ GO ] (t, x) = O(ε K +1 ), Hence, u ũ GO L 2 C ε Pε [ũ GO ] L 2 O(ε K ). (As expected since we cut off WKB expansion at K 1.) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

68 Example 2: Single Gaussian Beam By earlier construction P ε [ũ GB ] (t, x) = O(ε K /2+1 ). Note here that since width of beam ε, ũ GB L 2 ε n/4 and P ε [ũ GB ] L 2 ε K /2+1+n/4. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

69 Example 2: Single Gaussian Beam By earlier construction P ε [ũ GB ] (t, x) = O(ε K /2+1 ). Note here that since width of beam ε, ũ GB L 2 ε n/4 and P ε [ũ GB ] L 2 ε K /2+1+n/4. Therefore, relative error is u ũ GB L 2 ũ GB L 2 C ε 1+n/4 Pε [ũ GO ] L 2 O(ε K /2 ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

70 Example 2: Single Gaussian Beam By earlier construction P ε [ũ GB ] (t, x) = O(ε K /2+1 ). Note here that since width of beam ε, ũ GB L 2 ε n/4 and P ε [ũ GB ] L 2 ε K /2+1+n/4. Therefore, relative error is u ũ GB L 2 ũ GB L 2 C ε 1+n/4 Pε [ũ GO ] L 2 O(ε K /2 ). Proofs of this e.g. by [Lax, 57] for geometrical optics, [Ralston, 82] for Gaussian beams in wave setting and [Hagerdorn, 85] for Thawed Gaussians. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

71 Superpositions of Gaussian beams To approximate more general solutions, use superpositions of beams. Let v(t, y; z) be a beam starting from the point y = z and define u GB (t, y) = ε n 2 v(t, y; z)dz. K 0 (n dimension, K 0 compact set) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

72 Superpositions of Gaussian beams To approximate more general solutions, use superpositions of beams. Let v(t, y; z) be a beam starting from the point y = z and define u GB (t, y) = ε n 2 v(t, y; z)dz. K 0 (n dimension, K 0 compact set) By linearity of the Schrodinger equation a sum of solutions is also a solution. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

73 Superpositions of Gaussian beams To approximate more general solutions, use superpositions of beams. Let v(t, y; z) be a beam starting from the point y = z and define u GB (t, y) = ε n 2 v(t, y; z)dz. K 0 (n dimension, K 0 compact set) By linearity of the Schrodinger equation a sum of solutions is also a solution. u GB (t, y) is an asymptotic solution with initial data u GB (0, y) = ε n 2 v(0, y; z)dz. K 0 Sufficient to describe e.g. WKB data a(y)e iφ(y)/ε. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

74 Superpositions of Gaussian beams To approximate more general solutions, use superpositions of beams. Let v(t, y; z) be a beam starting from the point y = z and define u GB (t, y) = ε n 2 v(t, y; z)dz. K 0 (n dimension, K 0 compact set) By linearity of the Schrodinger equation a sum of solutions is also a solution. u GB (t, y) is an asymptotic solution with initial data u GB (0, y) = ε n 2 v(0, y; z)dz. K 0 Sufficient to describe e.g. WKB data a(y)e iφ(y)/ε. Prefactor scales beams appropriately (u GB = O(1) if v = O(1)). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

75 Superpositions of Gaussian beams More general phase space superposition: Let v(t, y; z, p) be a beam starting from the point y = z with momentum p and define u GB (t, y) = ε n K 0 v(t, y; z, p)dzdp. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

76 Superpositions of Gaussian beams More general phase space superposition: Let v(t, y; z, p) be a beam starting from the point y = z with momentum p and define u GB (t, y) = ε n K 0 v(t, y; z, p)dzdp. u GB (t, y) is an asymptotic solution with initial data u GB (0, y) = ε n K 0 v(0, y; z, p)dzdp. Can describe much more general data. (C.f. FBI transform.) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

77 Numerical methods Approximate superposition integral by sum (trapezoidal rule) u GB (t, y) = ε n 2 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. K 0 j Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

78 Numerical methods Approximate superposition integral by sum (trapezoidal rule) u GB (t, y) = ε n 2 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. K 0 Lagrangian methods Solve ODEs with standard methods. Similar to ray tracing but with all the additional Taylor coefficients computed along the rays (M, a j,β, φ β,... ) [Hill, Klimes,... ] j Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

79 Numerical methods Approximate superposition integral by sum (trapezoidal rule) u GB (t, y) = ε n 2 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. K 0 Lagrangian methods Solve ODEs with standard methods. Similar to ray tracing but with all the additional Taylor coefficients computed along the rays (M, a j,β, φ β,... ) [Hill, Klimes,... ] Eulerian methods obtain parameters from solving PDEs on fixed grids [Leung, Qian, Burridge,07], [Jin, Wu, Yang,08], [Jin, Wu, Yang, Huang, 09], [Leung, Qian,09], [Qian,Ying,10],... j Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

80 Numerical methods Approximate superposition integral by sum (trapezoidal rule) u GB (t, y) = ε n 2 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. K 0 Lagrangian methods Solve ODEs with standard methods. Similar to ray tracing but with all the additional Taylor coefficients computed along the rays (M, a j,β, φ β,... ) [Hill, Klimes,... ] Eulerian methods obtain parameters from solving PDEs on fixed grids [Leung, Qian, Burridge,07], [Jin, Wu, Yang,08], [Jin, Wu, Yang, Huang, 09], [Leung, Qian,09], [Qian,Ying,10],... Wavefront methods solve for parameters on a wave front [Motamed, OR,09] j Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

81 Numerical methods u GB (t, y) = ε n 2 Numerical issues v(t, y; z)dz ε 2 v(t, y; zj ) z n. K 0 Cost number of beams since each beam is O(1). For accuracy need z ε width of beams. cost O(ε n/2 ) C.f. direct solution of wave equations, at least O(ε (n+1) ) n Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

82 Numerical methods u GB (t, y) = ε n 2 Numerical issues v(t, y; z)dz ε 2 v(t, y; zj ) z n. K 0 Cost number of beams since each beam is O(1). For accuracy need z ε width of beams. cost O(ε n/2 ) C.f. direct solution of wave equations, at least O(ε (n+1) ) For phase space superposition would get O(ε n ) but can often be improved (e.g. support in p ε for WKB data) n Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

83 Numerical methods u GB (t, y) = ε n 2 Numerical issues v(t, y; z)dz ε 2 v(t, y; zj ) z n. K 0 Cost number of beams since each beam is O(1). For accuracy need z ε width of beams. cost O(ε n/2 ) C.f. direct solution of wave equations, at least O(ε (n+1) ) For phase space superposition would get O(ε n ) but can often be improved (e.g. support in p ε for WKB data) Spreading of beams Wide beams large Taylor approximation errors n Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

84 Numerical methods u GB (t, y) = ε n 2 Numerical issues v(t, y; z)dz ε 2 v(t, y; zj ) z n. K 0 Cost number of beams since each beam is O(1). For accuracy need z ε width of beams. cost O(ε n/2 ) C.f. direct solution of wave equations, at least O(ε (n+1) ) For phase space superposition would get O(ε n ) but can often be improved (e.g. support in p ε for WKB data) Spreading of beams Wide beams large Taylor approximation errors Initial data approximation Many degrees of freedom. Can have huge impact on accuracy at later times. n Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

85 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre,... ] Need to check how well u GB satisfies equation P ε [u GB ] := iε t u GB +ε 2 u GB Vu GB, u GB (t, y) = ε n 2 K 0 v(t, y; z)dz. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

86 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre,... ] Need to check how well u GB satisfies equation P ε [u GB ] := iε t u GB +ε 2 u GB Vu GB, u GB (t, y) = ε n 2 By linearity, P ε [u GB ] = ε n/2 K 0 P ε [v(t, y; z)]dz. K 0 v(t, y; z)dz. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

87 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre,... ] Need to check how well u GB satisfies equation P ε [u GB ] := iε t u GB +ε 2 u GB Vu GB, u GB (t, y) = ε n 2 By linearity, Coarse estimate gives P ε [u GB ] = ε n/2 K 0 P ε [v(t, y; z)]dz. K 0 v(t, y; z)dz. P ε [u GB ] ε n/2 K 0 P ε [v(t, y; z)] dz Cε n/2 ε K /2+1+n/4. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

88 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre,... ] Need to check how well u GB satisfies equation P ε [u GB ] := iε t u GB +ε 2 u GB Vu GB, u GB (t, y) = ε n 2 By linearity, Coarse estimate gives P ε [u GB ] = ε n/2 K 0 P ε [v(t, y; z)]dz. K 0 v(t, y; z)dz. P ε [u GB ] ε n/2 K 0 P ε [v(t, y; z)] dz Cε n/2 ε K /2+1+n/4. Gives error estimate for u GB u u GB C ε Pε [u GB ] Cε K /2 n/4. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

89 Approximation Errors Superpositions Basic estimate for the Schrödinger equation, [Liu, Ralston, 2010], u(t, ) u GB (t, ) L 2 O(ε K /2 n/4 ). is not sharp. E.g. it does not predict convergence for first order beams in 2D. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

90 Approximation Errors Superpositions Basic estimate for the Schrödinger equation, [Liu, Ralston, 2010], u(t, ) u GB (t, ) L 2 O(ε K /2 n/4 ). is not sharp. E.g. it does not predict convergence for first order beams in 2D. Some improvement for the wave equation [Liu, Ralston, 2009], u(t, ) u GB (t, ) E O(ε K /2+(1 n)/4 ). but still no convergence for first order beams for n large enough Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

91 Approximation Errors Superpositions Basic estimate for the Schrödinger equation, [Liu, Ralston, 2010], u(t, ) u GB (t, ) L 2 O(ε K /2 n/4 ). is not sharp. E.g. it does not predict convergence for first order beams in 2D. Some improvement for the wave equation [Liu, Ralston, 2009], u(t, ) u GB (t, ) E O(ε K /2+(1 n)/4 ). but still no convergence for first order beams for n large enough The dependence on dimensions is introduced when estimating the behavior of the solution at caustics. Artificial? Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

92 Approximation Errors Superpositions Basic estimate for the Schrödinger equation, [Liu, Ralston, 2010], u(t, ) u GB (t, ) L 2 O(ε K /2 n/4 ). is not sharp. E.g. it does not predict convergence for first order beams in 2D. Some improvement for the wave equation [Liu, Ralston, 2009], u(t, ) u GB (t, ) E O(ε K /2+(1 n)/4 ). but still no convergence for first order beams for n large enough The dependence on dimensions is introduced when estimating the behavior of the solution at caustics. Artificial? Problem: The estimate P ε [u GB ] ε n/2 K 0 P ε [v(t, y; z)] dz ignores cancellations between neighbouring beams. Very bad except at caustics where beams interfere constructively. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

93 Approximation Errors Superpositions Basic estimate for the Schrödinger equation, [Liu, Ralston, 2010], u(t, ) u GB (t, ) L 2 O(ε K /2 n/4 ). For phase space superposition, results with no dimensional dependence: Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

94 Approximation Errors Superpositions Basic estimate for the Schrödinger equation, [Liu, Ralston, 2010], u(t, ) u GB (t, ) L 2 O(ε K /2 n/4 ). For phase space superposition, results with no dimensional dependence: For the wave equation with phase space superposition, u(t, ) u GB (t, ) E O(ε K /2 ). [Bougacha, Akian, Alexandre, 2009] Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

95 Approximation Errors Superpositions Basic estimate for the Schrödinger equation, [Liu, Ralston, 2010], u(t, ) u GB (t, ) L 2 O(ε K /2 n/4 ). For phase space superposition, results with no dimensional dependence: For the wave equation with phase space superposition, u(t, ) u GB (t, ) E O(ε K /2 ). [Bougacha, Akian, Alexandre, 2009] For Frozen Gaussian Approximation, [Swart, Rousse, 2009] u(t, ) u GB (t, ) L 2 O(ε K /2 ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

96 Main result Theorem For scalar, strictly hyperbolic m-th order PDEs ( m 1 ε m 1 t l [u(t, ) u GB (t, )] ) H m l 1 l=0 O(ε K /2 ). For the wave equation, u(t, ) u GB (t, ) E O(ε K /2 ). For the Schrödinger equation, u(t, ) u GB (t, ) L 2 O(ε K /2 ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

97 Main result Theorem For scalar, strictly hyperbolic m-th order PDEs ( m 1 ε m 1 t l [u(t, ) u GB (t, )] ) H m l 1 l=0 O(ε K /2 ). For the wave equation, For the Schrödinger equation, u(t, ) u GB (t, ) E O(ε K /2 ). u(t, ) u GB (t, ) L 2 O(ε K /2 ). Superposition in physical space. Initial data approximated on a submanifold of phase space (WKB data). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

98 Main result Theorem For scalar, strictly hyperbolic m-th order PDEs ( m 1 ε m 1 t l [u(t, ) u GB (t, )] ) H m l 1 l=0 O(ε K /2 ). For the wave equation, For the Schrödinger equation, u(t, ) u GB (t, ) E O(ε K /2 ). u(t, ) u GB (t, ) L 2 O(ε K /2 ). Superposition in physical space. Initial data approximated on a submanifold of phase space (WKB data). Convergence of all beams independent of dimension and presence of caustics. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

99 Sketch of proof For all the PDEs P[u] = 0 considered we have an energy estimate of the type t u GB (t, ) u(t, ) S u GB (0, ) u(0, ) S + Cε q P[u GB ](τ, ) L 2dτ, where S is some appropriate norm and q is an integer. (E.g. Schrodinger S = L 2, q = 1, wave equation S = E, q = 1.) 0 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

100 Sketch of proof For all the PDEs P[u] = 0 considered we have an energy estimate of the type t u GB (t, ) u(t, ) S u GB (0, ) u(0, ) S + Cε q P[u GB ](τ, ) L 2dτ, where S is some appropriate norm and q is an integer. (E.g. Schrodinger S = L 2, q = 1, wave equation S = E, q = 1.) For all PDEs considered, we can write J P[u GB ](t, y) = ε K /2 q ε r j Tj ε [f j ](t, y) + O(ε ), where r j 0, J finite and f j L 2 (all independent of ε). : L 2 L 2 belongs to a class of oscillatory integral operators. T ε j j=1 0 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

101 Sketch of proof For all the PDEs P[u] = 0 considered we have an energy estimate of the type t u GB (t, ) u(t, ) S u GB (0, ) u(0, ) S + Cε q P[u GB ](τ, ) L 2dτ, where S is some appropriate norm and q is an integer. (E.g. Schrodinger S = L 2, q = 1, wave equation S = E, q = 1.) For all PDEs considered, we can write J P[u GB ](t, y) = ε K /2 q ε r j Tj ε [f j ](t, y) + O(ε ), where r j 0, J finite and f j L 2 (all independent of ε). : L 2 L 2 belongs to a class of oscillatory integral operators. T ε j j=1 Together we get (if initial data exact) u GB (t, ) u(t, ) S C(T )ε K /2 J j=1 0 ε r j T ε j L 2 f j L 2 + O(ε ) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

102 Sketch of proof, cont. We have u GB (t, ) u(t, ) S C(T )ε K /2 J j=1 T ε j L 2 + O(ε ) where, in its simplest form, T ε [w](t, y) := ε n+ α 2 w(z)(y x(t; z)) α e iφ(t,y x(t;z);z)/ε dz, K 0 for some multi-index α, Gaussian beam phase φ and geometrical optics rays x(t; z) with x(0; z) = z. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

103 Sketch of proof, cont. We have u GB (t, ) u(t, ) S C(T )ε K /2 J j=1 T ε j L 2 + O(ε ) where, in its simplest form, T ε [w](t, y) := ε n+ α 2 w(z)(y x(t; z)) α e iφ(t,y x(t;z);z)/ε dz, K 0 for some multi-index α, Gaussian beam phase φ and geometrical optics rays x(t; z) with x(0; z) = z. Result follows if we prove that T ε is bounded in L 2 independent of ε, T ε L 2 C. This is the key estimate of our proof. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

104 Sketch of proof, cont. Estimate of T ε L 2, where T ε [w](t, y) := ε n+ α 2 w(z)(y x(t; z)) α e iφ(t,y x(t;z);z)/ε dz. K 0 Main difficulty: no globally invertible map x(0; z) = z x(t; z) because of caustics. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

105 Sketch of proof, cont. Estimate of T ε L 2, where T ε [w](t, y) := ε n+ α 2 w(z)(y x(t; z)) α e iφ(t,y x(t;z);z)/ε dz. K 0 Main difficulty: no globally invertible map x(0; z) = z x(t; z) because of caustics. Mapping (x(0; z), p(0; z)) (x(t; z), p(t; z) is however globally invertible and smooth. Gives the "non-squeezing" property, c 1 z z p(t; z) p(t; z ) + x(t; z) x(t; z ) c 2 z z. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

106 Sketch of proof, cont. Estimate of T ε L 2, where T ε [w](t, y) := ε n+ α 2 w(z)(y x(t; z)) α e iφ(t,y x(t;z);z)/ε dz. K 0 Main difficulty: no globally invertible map x(0; z) = z x(t; z) because of caustics. Mapping (x(0; z), p(0; z)) (x(t; z), p(t; z) is however globally invertible and smooth. Gives the "non-squeezing" property, c 1 z z p(t; z) p(t; z ) + x(t; z) x(t; z ) c 2 z z. Allows us to use stationary phase arguments close to caustics, and carefully control cancellations of oscillations there (similar to [Swart,Rousse], [Bougacha, Akian, Alexandre]). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

107 Approximation errors Remarks The estimate u(t, ) u GB (t, ) E O(ε K /2 ) is sharp for individual beams (relative error). But for superpositions? Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

108 Approximation errors Remarks The estimate u(t, ) u GB (t, ) E O(ε K /2 ) is sharp for individual beams (relative error). But for superpositions? Predicts convergence rate of first order beam to be only O( ε). These beams are based on same high frequency approximation as geometrical optics which has O(ε) accuracy. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

109 Approximation errors Remarks The estimate u(t, ) u GB (t, ) E O(ε K /2 ) is sharp for individual beams (relative error). But for superpositions? Predicts convergence rate of first order beam to be only O( ε). These beams are based on same high frequency approximation as geometrical optics which has O(ε) accuracy. For the Helmholtz case we have proved [Motamed, OR] that u(t, ) u GB (t, ) E O(ε K /2 ) for the Taylor expansion part of the error away from caustics. This gives O(ε) for first order beams. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

110 Approximation errors Remarks The estimate u(t, ) u GB (t, ) E O(ε K /2 ) is sharp for individual beams (relative error). But for superpositions? Predicts convergence rate of first order beam to be only O( ε). These beams are based on same high frequency approximation as geometrical optics which has O(ε) accuracy. For the Helmholtz case we have proved [Motamed, OR] that u(t, ) u GB (t, ) E O(ε K /2 ) for the Taylor expansion part of the error away from caustics. This gives O(ε) for first order beams. More error cancellations coming in for odd order beams? ( no gain in using even order beams) Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

111 Approximation errors Remarks The estimate u(t, ) u GB (t, ) E O(ε K /2 ) is sharp for individual beams (relative error). But for superpositions? Predicts convergence rate of first order beam to be only O( ε). These beams are based on same high frequency approximation as geometrical optics which has O(ε) accuracy. For the Helmholtz case we have proved [Motamed, OR] that u(t, ) u GB (t, ) E O(ε K /2 ) for the Taylor expansion part of the error away from caustics. This gives O(ε) for first order beams. More error cancellations coming in for odd order beams? ( no gain in using even order beams) Numerical experiments also in the time-dependent case suggests a better rate for odd order beams Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

112 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

113 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

114 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

115 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

116 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

117 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

118 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

119 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

120 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

121 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

122 Numerical examples Cusp caustic Consider the test case where Φ(0, y) = y 1 + y 2 2, A(0, y) = e 10 y 2. Cusp caustic at t = 0.5 Two fold caustics at t > 0.5 Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

123 Numerical examples Cusp caustic, convergence Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

124 Numerical examples Cusp caustic, convergence Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

125 Partial result (work in progress) For any y away from caustics we can prove that u GB (t, y) u(t, y) C(y)ε K /2. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

126 Partial result (work in progress) For any y away from caustics we can prove that u GB (t, y) u(t, y) C(y)ε K /2. 1 Prove error estimates in higher order Sobolev spaces u GB (t, ) u(t, ) H s = O(ε K /2 s ). Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

127 Partial result (work in progress) For any y away from caustics we can prove that u GB (t, y) u(t, y) C(y)ε K /2. 1 Prove error estimates in higher order Sobolev spaces u GB (t, ) u(t, ) H s = O(ε K /2 s ). 2 Use Sobolev inequalities and K + m order beams u K +m GB to show that u K +m GB (t, ) u(t, ) L ε K /2. for large enough m. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

128 Partial result (work in progress) For any y away from caustics we can prove that u GB (t, y) u(t, y) C(y)ε K /2. 1 Prove error estimates in higher order Sobolev spaces u GB (t, ) u(t, ) H s = O(ε K /2 s ). 2 Use Sobolev inequalities and K + m order beams u K +m GB to show that u K +m GB (t, ) u(t, ) L ε K /2. for large enough m. 3 Consider difference between K order and K + m order beams, u K +m GB u K GB = εk /2 T ε j f j. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40

129 Partial result (work in progress) For any y away from caustics we can prove that u GB (t, y) u(t, y) C(y)ε K /2. 1 Prove error estimates in higher order Sobolev spaces u GB (t, ) u(t, ) H s = O(ε K /2 s ). 2 Use Sobolev inequalities and K + m order beams u K +m GB to show that u K +m GB (t, ) u(t, ) L ε K /2. for large enough m. 3 Consider difference between K order and K + m order beams, 4 Estimate T ε j u K +m GB in max norm. u K GB = εk /2 T ε j f j. Olof Runborg (KTH) Gaussian Beam Approximations München, / 40