Gaussian Beam Approximations of High Frequency Waves

Transcription

1 Gaussian Beam Approximations of High Frequency Waves Olof Runborg Department of Mathematics, KTH Joint with Hailiang Liu, Iowa, James Ralston, UCLA, Nick Tanushev, Z-Terra Corp. ESF Exploratory Workshop IPP Garching, October 2013 Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

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) smooth speed of propagation. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

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) smooth speed of propagation. High frequency short wave length highly oscillatory solutions many gridpoints. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

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) smooth 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 Approximation IPP Garching, / 36

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) smooth 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, quantum mechanics,... Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

6 High frequency material Cauchy problem for scalar wave equation u tt c ε (x) u = 0, (t, x) R + R n, u(0, x) = A(x), u t (0, x) = B(x), where c ε (x) R d d has variations on length scale ε. The functions A(x) and B(x) are smooth (and independent of ε). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

7 High frequency material Cauchy problem for scalar wave equation u tt c ε (x) u = 0, (t, x) R + R n, u(0, x) = A(x), u t (0, x) = B(x), where c ε (x) R d d has variations on length scale ε. The functions A(x) and B(x) are smooth (and independent of ε). Direct numerical solution resolves wavelength: #gridpoints ε n at least cost ε n 1 at least Prohibitively expensive when ε 1, particularly in higher dimensions. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

8 High frequency material Cauchy problem for scalar wave equation u tt c ε (x) u = 0, (t, x) R + R n, u(0, x) = A(x), u t (0, x) = B(x), where c ε (x) R d d has variations on length scale ε. The functions A(x) and B(x) are smooth (and independent of ε). Direct numerical solution resolves wavelength: #gridpoints ε n at least cost ε n 1 at least Prohibitively expensive when ε 1, particularly in higher dimensions. Our approach: Heterogeneous Multiscale Method (HMM) [E,Engquist,2001]. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

9 High frequency material Cauchy problem for scalar wave equation u tt c ε (x) u = 0, (t, x) R + R n, u(0, x) = A(x), u t (0, x) = B(x), where c ε (x) R d d has variations on length scale ε. The functions A(x) and B(x) are smooth (and independent of ε). Direct numerical solution resolves wavelength: #gridpoints ε n at least cost ε n 1 at least Prohibitively expensive when ε 1, particularly in higher dimensions. Our approach: Heterogeneous Multiscale Method (HMM) [E,Engquist,2001]. Solve small micro problems (localized in time and space) to probe effective dynamics, which is approximated on coarse grid ( x ε). Method cost (essentially) independent of ε. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

10 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 Approximation IPP Garching, / 36

11 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 Approximation IPP Garching, / 36

12 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 Approximation IPP Garching, / 36

13 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 Approximation IPP Garching, / 36

14 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 Ray tracing: x(t), p(t) bicharacteristics of the eikonal equation, dx dt = c(x) 2 p, dp dt = c(x) c(x), φ(t, x(t)) = φ(0, x(0)). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

15 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 Ray tracing: x(t), p(t) bicharacteristics of the eikonal equation, dx dt = c(x) 2 p, dp dt = c(x) c(x), φ(t, x(t)) = φ(0, x(0)). Good accuracy for small ε. Computational cost ε-independent. u(t, x) = a(t, x)e iφ(t,x)/ε + O(ε). (#DOF and cost independent of ε) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

16 Geometrical optics The ansatz u(t, x) = a(t, x)e iφ(t,x)/ε, generally breaks down in finite time if valid at t = 0. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

17 Geometrical optics The ansatz u(t, x) = a(t, x)e iφ(t,x)/ε, generally breaks down in finite time if valid at t = 0. Refraction of waves gives rise to multiple crossing waves N u(t, x) = a n (t, x)e iφn(t,x)/ε n=1 Several amplitude and phase functions. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

18 Geometrical optics The ansatz u(t, x) = a(t, x)e iφ(t,x)/ε, generally breaks down in finite time if valid at t = 0. Refraction of waves gives rise to multiple crossing waves N u(t, x) = a n (t, x)e iφn(t,x)/ε n=1 Several amplitude and phase functions. Caustics appear at points of transition = concentration of rays. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

19 Geometrical optics The ansatz u(t, x) = a(t, x)e iφ(t,x)/ε, generally breaks down in finite time if valid at t = 0. Refraction of waves gives rise to multiple crossing waves N u(t, x) = a n (t, x)e iφn(t,x)/ε n=1 Several amplitude and phase functions. Caustics appear at points of transition = concentration of rays. Geometrical optics predicts infinite amplitude at caustics. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

20 Geometrical optics The ansatz u(t, x) = a(t, x)e iφ(t,x)/ε, generally breaks down in finite time if valid at t = 0. Refraction of waves gives rise to multiple crossing waves u(t, x) = N a n (t, x)e iφn(t,x)/ε n=1 Several amplitude and phase functions. Caustics appear at points of transition = concentration of rays. Geometrical optics predicts infinite amplitude at caustics. Handling multiphase solutions tricky for numerical methods with fixed grids. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

21 Caustics Concentration of rays. GO amplitude a(t, y) but should be a(t, y) ε α, 0 < α < 1. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

22 Gaussian beams Approximate, localized, solutions to the wave equation/schrodinger with a Gaussian profile (width ε). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

23 Gaussian beams Approximate, localized, solutions to the wave equation/schrodinger with a Gaussian profile (width ε). Studied in e.g. Geophysics [Cerveny, Popov, Babich, Psencik, Klimes, Kravtsov,... ], Quantum Mechanics,[Heller, Hagedorn, Herman, Kluk, Kay,... ], Plasma Physics, [Pereverzev, Peeters, Maj,... ], Mathematics [Ralston, Hörmander,... ] Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

24 Gaussian beams Approximate, localized, solutions to the wave equation/schrodinger with a Gaussian profile (width ε). Studied in e.g. Geophysics [Cerveny, Popov, Babich, Psencik, Klimes, Kravtsov,... ], Quantum Mechanics,[Heller, Hagedorn, Herman, Kluk, Kay,... ], Plasma Physics, [Pereverzev, Peeters, Maj,... ], Mathematics [Ralston, Hörmander,... ] No breakdown at caustics. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

25 Gaussian beams Gaussian beams are of the same form as geometrical optics solutions, v(t, y) = A(t, y)e iφ(t,y)/ε, centered around a geometrical optics ray x(t), A(t, y) = a(t, y x(t)), Φ(t, y) = φ(t, y x(t)). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

26 Gaussian beams Gaussian beams are of the same form as geometrical optics solutions, v(t, y) = A(t, y)e iφ(t,y)/ε, centered around a geometrical optics ray x(t), A(t, y) = a(t, y x(t)), Φ(t, y) = φ(t, y x(t)). The phase Φ will now have a positive imaginary part away from the ray x(t). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

27 Gaussian beams Gaussian beams are of the same form as geometrical optics solutions, v(t, y) = A(t, y)e iφ(t,y)/ε, centered around a geometrical optics ray x(t), A(t, y) = a(t, y x(t)), Φ(t, y) = φ(t, y x(t)). 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 Approximation IPP Garching, / 36

28 Gaussian beams Gaussian beams are of the same form as geometrical optics solutions, v(t, y) = A(t, y)e iφ(t,y)/ε, centered around a geometrical optics ray x(t), A(t, y) = a(t, y x(t)), Φ(t, y) = φ(t, y x(t)). 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 Approximation IPP Garching, / 36

29 Gaussian beams Gaussian beams are of the same form as geometrical optics solutions, v(t, y) = A(t, y)e iφ(t,y)/ε, centered around a geometrical optics ray x(t), A(t, y) = a(t, y x(t)), Φ(t, y) = φ(t, y x(t)). 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.) Phase Φ(t, y) and amplitude A(t, y) approximated by polynomials locally around x(t) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

30 Gaussian beams Gaussian beams are of the same form as geometrical optics solutions, v(t, y) = A(t, y)e iφ(t,y)/ε, centered around a geometrical optics ray x(t), A(t, y) = a(t, y x(t)), Φ(t, y) = φ(t, y x(t)). 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.) Phase Φ(t, y) and amplitude A(t, y) approximated by polynomials locally around x(t) Φ(t, y) and A(t, y) solve eikonal and transport equation only upto O( y x m ). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

31 First order beams The simplest ("first order") Gaussian beams are of the form v(t, y) = a 0 (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 Approximation IPP Garching, / 36

32 First order beams The simplest ("first order") Gaussian beams are of the form v(t, y) = a 0 (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 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 Approximation IPP Garching, / 36

33 First order beams Let us thus require that Φ A A t + c Φ Φ 2 t c(y) 2 Φ 2 = O( y x(t) 3 ), + c2 Φ Φ tt A = O( y x(t) ), 2c Φ Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

34 First order beams Let us thus require that Φ A A t + c Φ We obtain ODEs for φ 0, x, p, M, a 0. ẋ(t) = c(x) 2 p, φ 0 (t) = 0, Φ 2 t c(y) 2 Φ 2 = O( y x(t) 3 ), + c2 Φ Φ tt A = O( y x(t) ), 2c Φ ṗ(t) = c(x)/c(x), Ṁ(t) = D MB B T M MCM, ȧ 0 (t) = a ( ) 0 c(x)p c(x) c(x) 3 p Mp + c(x) 2 Tr[M], 2 where B, C, D are matrix functions involving x, p and c(x). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

35 First order beams Let us thus require that Φ A A t + c Φ We obtain ODEs for φ 0, x, p, M, a 0. ẋ(t) = c(x) 2 p, φ 0 (t) = 0, Φ 2 t c(y) 2 Φ 2 = O( y x(t) 3 ), + c2 Φ Φ tt A = O( y x(t) ), 2c Φ ṗ(t) = c(x)/c(x), Ṁ(t) = D MB B T M MCM, ȧ 0 (t) = a ( ) 0 c(x)p c(x) c(x) 3 p Mp + c(x) 2 Tr[M], 2 where B, C, D are matrix functions involving x, p and c(x). ODEs easy to solve numerically. Beams easy to evaluate: 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. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

36 First order beams Let us thus require that Φ A A t + c Φ We obtain ODEs for φ 0, x, p, M, a 0. ẋ(t) = c(x) 2 p, φ 0 (t) = 0, Φ 2 t c(y) 2 Φ 2 = O( y x(t) 3 ), + c2 Φ Φ tt A = O( y x(t) ), 2c Φ ṗ(t) = c(x)/c(x), Ṁ(t) = D MB B T M MCM, ȧ 0 (t) = a ( ) 0 c(x)p c(x) c(x) 3 p Mp + c(x) 2 Tr[M], 2 Asymptotic order of accuracy is ( ) 1 v tt c(y) 2 v = O. ε Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

37 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 Approximation IPP Garching, / 36

38 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 of 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 Approximation IPP Garching, / 36

39 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 of 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 Approximation IPP Garching, / 36

40 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 of 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 Approximation IPP Garching, / 36

41 Higher order beams We now require that Φ(t, y) = φ(t, y x)) solves eikonal equation to order y x K +2 a j (t, y x) solve higher order transport equations to order y x K 2j Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

42 Higher order beams We now require that Φ(t, y) = φ(t, y x)) solves eikonal equation to order y x K +2 a j (t, y x) solve higher order transport equations to order y x K 2j Again, this gives ODEs for all Taylor coefficients, ẋ(t) = c(x) 2 p, φ0 (t) = 0, ṗ(t) = c(x)/c(x), Ṁ(t) = D MB B T M MCM, ȧ j,β (t) =..., φ β (t) =..., Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

43 Higher order beams We now require that Φ(t, y) = φ(t, y x)) solves eikonal equation to order y x K +2 a j (t, y x) solve higher order transport equations to order y x K 2j Again, this gives ODEs for all Taylor coefficients, ẋ(t) = c(x) 2 p, φ0 (t) = 0, ṗ(t) = c(x)/c(x), Ṁ(t) = D MB B T M MCM, ȧ j,β (t) =..., φ β (t) =..., Asymptotic order of accuracy is ( v tt c(y) 2 v = O ε K /2 1). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

44 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 Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

45 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 Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

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. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

47 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. a 0 (t) exists everywhere (no blow-up at caustics) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

48 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. a 0 (t) exists everywhere (no blow-up at caustics) Shape of beam remains Gaussian Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

49 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. a 0 (t) exists everywhere (no blow-up at caustics) Shape of beam remains Gaussian For high order need cutoff in a neighborhood of central ray to avoid spurious growth. v(t, y) = a(t, y x(t))e iφ(t,y x(t))/ε ϱ(y x(t)) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

50 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 Approximation IPP Garching, / 36

51 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 a 0 (t; z)e i ε [φ 0(t;z)+(y x(t;z)) p(t;z)+ K 0 (n dimension, K 0 compact set) 1 2 (y x(t;z) M(t;z)(y x(t;z))] Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

52 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 a 0 (t; z)e i ε [φ 0(t;z)+(y x(t;z)) p(t;z)+ K 0 (n dimension, K 0 compact set) 1 2 (y x(t;z) M(t;z)(y x(t;z))] By linearity of the wave equation equation a sum of solutions is also a solution. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

53 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 a 0 (t; z)e i ε [φ 0(t;z)+(y x(t;z)) p(t;z)+ K 0 (n dimension, K 0 compact set) 1 2 (y x(t;z) M(t;z)(y x(t;z))] By linearity of the wave equation equation a sum of solutions is also a solution. u GB (t, y) is an asymptotic solution with initial data u GB (0, y). Sufficient to describe e.g. WKB data: K -th order beams s.t. A(y)e iφ(y)/ε u GB (0, ) = O(ε K /2 ), E [Tanushev, 2007]. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

54 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 a 0 (t; z)e i ε [φ 0(t;z)+(y x(t;z)) p(t;z)+ K 0 (n dimension, K 0 compact set) 1 2 (y x(t;z) M(t;z)(y x(t;z))] By linearity of the wave equation equation a sum of solutions is also a solution. u GB (t, y) is an asymptotic solution with initial data u GB (0, y). Sufficient to describe e.g. WKB data: K -th order beams s.t. A(y)e iφ(y)/ε u GB (0, ) = O(ε K /2 ), E [Tanushev, 2007]. Prefactor normalizes beams appropriately, u GB E = O(1). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

55 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 Approximation IPP Garching, / 36

56 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 more general data. (C.f. FBI transform.) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

57 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 Approximation IPP Garching, / 36

58 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 z j 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 Approximation IPP Garching, / 36

59 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 z j 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 Approximation IPP Garching, / 36

60 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 z j 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 Approximation IPP Garching, / 36

61 Numerical methods u GB (t, y) = ε n 2 Numerical issues K 0 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. 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) ) j Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

62 Numerical methods u GB (t, y) = ε n 2 Numerical issues K 0 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. 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) j Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

63 Numerical methods u GB (t, y) = ε n 2 Numerical issues K 0 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. 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 j Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

64 Numerical methods u GB (t, y) = ε n 2 Numerical issues K 0 v(t, y; z)dz ε n 2 v(t, y; z j ) z n. 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. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36 j

65 Approximation errors Let := tt + c(y) 2. Suppose u is exact solution of wave equation and ũ is the Gaussian beam approximation u = 0, ũ = O(ε K /2 1 ). What is the norm error in ũ, i.e. u ũ? Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

66 Approximation errors Let := tt + c(y) 2. Suppose u is exact solution of wave equation and ũ is the Gaussian beam approximation u = 0, ũ = O(ε K /2 1 ). What is the norm error in ũ, i.e. u ũ? Use ε-scaled energy norm u 2 E := ε2 2 This is O(1) for WKB type initial data, R n u t 2 c(y) 2 + u 2 dy. u(0, x) = A(x)e iφ(x)/ε u(0, ) E = O(1). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

67 Approximation errors Use well-posedness (stability) estimate for wave equation solutions w: w(t, ) E w(0, ) E + εc(t ) sup w(t, ) L 2, 0 t T. t [0,T ] Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

68 Approximation errors Use well-posedness (stability) estimate for wave equation solutions w: w(t, ) E w(0, ) E + εc(t ) sup w(t, ) L 2, 0 t T. t [0,T ] Since, u = 0 and by linearity Hence, assuming u(0, x) = ũ(0, x), [ũ u] = ũ. ũ(t, ) u(t, ) E εc(t ) sup ũ(t, ) L 2, 0 t T. t [0,T ] Error in ũ how well it satisfies equation, plus one order in ε Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

69 Approximation Errors Single Gaussian Beam By earlier construction and ũ GB (t, x) = O(ε K /2 1 ). ũ(t, ) u(t, ) E εc(t ) sup ũ(t, ) L 2, 0 t T. t [0,T ] Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

70 Approximation Errors Single Gaussian Beam By earlier construction and ũ GB (t, x) = O(ε K /2 1 ). ũ(t, ) u(t, ) E εc(t ) sup ũ(t, ) L 2, 0 t T. t [0,T ] Hence, after appropriate normalization (so ũ E = 1) and assuming zero initial data error, [Ralston, 82] Estimate is sharp. sup ũ(t, ) u(t, ) E O(ε K /2 ). t [0,T ] Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

71 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre, Lu, Yang,... ] Need to check how well u GB satisfies equation u GB := tt u GB + c(y) 2 u GB, u GB (t, y) = ε n 2 K 0 v(t, y; z)dz. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

72 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre, Lu, Yang,... ] Need to check how well u GB satisfies equation u GB := tt u GB + c(y) 2 u GB, u GB (t, y) = ε n 2 By linearity, u GB = ε n/2 K 0 v(t, y; z)dz. K 0 v(t, y; z)dz. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

73 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre, Lu, Yang,... ] Need to check how well u GB satisfies equation u GB := tt u GB + c(y) 2 u GB, u GB (t, y) = ε n 2 By linearity and simple estimate, u GB ε n/2 K 0 v(t, y; z) dz. K 0 v(t, y; z)dz. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

74 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre, Lu, Yang,... ] Need to check how well u GB satisfies equation u GB := tt u GB + c(y) 2 u GB, u GB (t, y) = ε n 2 By linearity and simple estimate, u GB ε n/2 K 0 v(t, y; z) dz. After scaling and Cauchy Schwarz, K 0 v(t, y; z)dz. u GB (y) 2 2 Cε n K 0 v(t, y; z) 2 2 dz Cε n ε K 2+n/2. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

75 Approximation Errors Superpositions Norm estimates of u u GB only rather recently derived [Swart, Rousse, Liu, Ralston, Tanushev, Bougacha, Alexandre, Lu, Yang,... ] Need to check how well u GB satisfies equation u GB := tt u GB + c(y) 2 u GB, u GB (t, y) = ε n 2 By linearity and simple estimate, u GB ε n/2 K 0 v(t, y; z) dz. After scaling and Cauchy Schwarz, K 0 v(t, y; z)dz. u GB (y) 2 2 Cε n K 0 v(t, y; z) 2 2 dz Cε n ε K 2+n/2. Gives error estimate for u GB u u GB E εc u GB 2 Cε K /2 n/4. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

76 Approximation errors superpositions Basic estimate u(t, ) u GB (t, ) E 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 Approximation IPP Garching, / 36

77 Approximation errors superpositions Basic estimate u(t, ) u GB (t, ) E O(ε K /2 n/4 ). is not sharp. E.g. it does not predict convergence for first order beams in 2D. Problem: The step u GB ε n/2 K 0 v(t, y; z) dz ignores cancellations between neighbouring beams. Very bad except at caustics where beams interfere constructively. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

78 Approximation errors superpositions Basic estimate u(t, ) u GB (t, ) E O(ε K /2 n/4 ). is not sharp. E.g. it does not predict convergence for first order beams in 2D. Problem: The step u GB ε n/2 K 0 v(t, y; z) dz ignores cancellations between neighbouring beams. Very bad except at caustics where beams interfere constructively. Gives a dependence on dimension n in estimate. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

79 Error estimate Theorem (Liu, Tanushev, O.R.,2010) 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 Approximation IPP Garching, / 36

80 Error estimate Theorem (Liu, Tanushev, O.R.,2010) 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 ). Superposition in physical space. Initial data approximated on a submanifold of phase space (WKB data). Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

81 Error estimate Theorem (Liu, Tanushev, O.R.,2010) 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 ). 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 Approximation IPP Garching, / 36

82 Error estimate Theorem (Liu, Tanushev, O.R.,2010) 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 ). 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. Result also for general scalar, strictly hyperbolic m-th order PDEs. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

83 Error estimate Theorem (Liu, Tanushev, O.R.,2010) 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 ). 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. Result also for general scalar, strictly hyperbolic m-th order PDEs. Cf. [Bougacha, Akian, Alexandre, 2009], [Rousse, Swart, 2009] and [Lu, Yang, 2011] for other settings. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

84 Gaussian beams for Helmholtz For time-harmonic waves consider Helmholtz equation u + (iαε 1 + ε 2 )n 2 u = g, x R d. where n(x) = 1/c(x), α=damping and g supported on a co-dimension one manifold. (Ex. g = g 0 (x 2 )δ(x 1 )/ε.) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

85 Gaussian beams for Helmholtz For time-harmonic waves consider Helmholtz equation u + (iαε 1 + ε 2 )n 2 u = g, x R d. where n(x) = 1/c(x), α=damping and g supported on a co-dimension one manifold. (Ex. g = g 0 (x 2 )δ(x 1 )/ε.) "Blobs" "Fat rays" localized around geometrical optics ray Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

86 Gaussian beams for Helmholtz For time-harmonic waves consider Helmholtz equation u + (iαε 1 + ε 2 )n 2 u = g, x R d. where n(x) = 1/c(x), α=damping and g supported on a co-dimension one manifold. (Ex. g = g 0 (x 2 )δ(x 1 )/ε.) "Blobs" "Fat rays" localized around geometrical optics ray To leading order gaussian transversely to ray Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

87 Gaussian beams for Helmholtz Same ansatz, v = a(s, y x(s))e iφ(s,y x(s))/ε, centered around a geometrical optics ray x(s) but s not time. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

88 Gaussian beams for Helmholtz Same ansatz, v = a(s, y x(s))e iφ(s,y x(s))/ε, centered around a geometrical optics ray x(s) but s not time. First order beams are of the form φ = φ 0 (s) + y p(s) y M(s)y, a = a 0(s), i.e. a approximated to 0th order, and φ to 2nd order. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

89 Gaussian beams for Helmholtz Same ansatz, v = a(s, y x(s))e iφ(s,y x(s))/ε, centered around a geometrical optics ray x(s) but s not time. First order beams are of the form φ = φ 0 (s) + y p(s) y M(s)y, a = a 0(s), i.e. a approximated to 0th order, and φ to 2nd order. Similar ODEs for a 0, x, p, M, φ 0 as in the time-dependent case. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

90 Gaussian beams for Helmholtz Same ansatz, v = a(s, y x(s))e iφ(s,y x(s))/ε, centered around a geometrical optics ray x(s) but s not time. First order beams are of the form φ = φ 0 (s) + y p(s) y M(s)y, a = a 0(s), i.e. a approximated to 0th order, and φ to 2nd order. Similar ODEs for a 0, x, p, M, φ 0 as in the time-dependent case. Similar properties as in time-dependent case: Phase φ evaluated on ray = φ 0 (s) is real valued If M(0) is symmetric and IM(0) is positive definite then this is true for M(s) (which exists) for all s > 0. a 0 (s) exists everywhere (no blow-up at caustics) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

91 Gaussian beams for Helmholtz Extension off ray v(y) = a(s, y x(s))e iφ(s,y x(s))/ε, v(y) y x(s) η x(s*) How to evaluate "(s, y x(s))" in expression for beam? Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

92 Gaussian beams for Helmholtz Extension off ray v(y) = a(s, y x(s ))e iφ(s,y x(s ))/ε, v(y) y x(s) s = s (y) η x(s*) How to evaluate "(s, y x(s))" in expression for beam? No distinguished "time" variable Extend beam by Taylor expansion transversely to ray: Let s = s (y) such that x(s ) is closest point on ray to y. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

93 Gaussian beams for Helmholtz Extension off ray v(y) = a(s, y x(s ))e iφ(s,y x(s ))/ε v(y) y x(s) ϱ(y x(s )), s = s (y) η x(s*) How to evaluate "(s, y x(s))" in expression for beam? No distinguished "time" variable Extend beam by Taylor expansion transversely to ray: Let s = s (y) such that x(s ) is closest point on ray to y. Only well-defined close enough to ray Cutoff ϱ(y) needed also for first order beams (size η) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

94 Gaussian beams for Helmholtz Source Helmholtz with source on Σ = {y : ρ(y) = 0}. u + (iαε 1 + ε 2 )n 2 u = 1 ε g(y)δ(ρ(y)). Σ u (x) x(s) η x 0 u + (x) ρ < 0 ρ > 0 Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

95 Gaussian beams for Helmholtz Source Helmholtz with source on Σ = {y : ρ(y) = 0}. u + (iαε 1 + ε 2 )n 2 u = 1 ε g(y)δ(ρ(y)). Σ u (x) x(s) η x 0 u + (x) ρ < 0 ρ > 0 Beams shoot out orthogonally in each direction from Σ Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

96 Gaussian beams for Helmholtz Source Helmholtz with source on Σ = {y : ρ(y) = 0}. u + (iαε 1 + ε 2 )n 2 u = 1 ε g(y)δ(ρ(y)). Σ u (x) x(s) η x 0 u + (x) ρ < 0 ρ > 0 Beams shoot out orthogonally in each direction from Σ Gives beams v ± (y), with v + (y) = 0 when ρ(y) < 0 etc. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

97 Gaussian beams for Helmholtz Source Helmholtz with source on Σ = {y : ρ(y) = 0}. Lu =: u + (iαε 1 + ε 2 )n 2 u = 1 ε g(y)δ(ρ(y)). Σ u (x) x(s) η x 0 u + (x) ρ < 0 ρ > 0 Beams shoot out orthogonally in each direction from Σ Gives beams v ± (y), with v + (y) = 0 when ρ(y) < 0 etc. Note that v + = v on Σ, but φ + = φ so that L(v + + v ) δ(ρ(y))+ smooth part. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

98 Gaussian beams for Helmholtz Superposition Σ Lu =: u + (iαε 1 + ε 2 )n 2 u u (x) x(s) = 1 ε g(y)δ(ρ(y)). η x 0 u + (x) ρ < 0 ρ > 0 Let v ± (y; z) be the beams starting from z Σ and define superposition u GB (y) = ε n 1 2 [v + (y; z) + v (y; z)]da z (1) Σ Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

99 Gaussian beams for Helmholtz Superposition Σ Lu =: u + (iαε 1 + ε 2 )n 2 u u (x) x(s) = 1 ε g(y)δ(ρ(y)). η x 0 u + (x) ρ < 0 ρ > 0 Let v ± (y; z) be the beams starting from z Σ and define superposition u GB (y) = ε n 1 2 [v + (y; z) + v (y; z)]da z (1) Choose initial data for beam v ± (z; z) such that Σ with g g. Lu GB (y) 1 ε g(y)δ(ρ(y)) + f GB Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

100 Error estimate K th order beams, Helmholtz case Theorem (Liu, Ralston, Tanushev, O.R., 2013) Assume Smooth, compactly supported source g(x) Index of refraction n(x) smooth and constant for x > R No trapped rays: L s.t. x(±l) > 2R if x(0) < R, p(0) = n(x(0)) No initial data error g = g. Then with C independent of ε and α, u u GB L2 ( x <R) Cε K /2, Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

101 Error estimate K th order beams, Helmholtz case Theorem (Liu, Ralston, Tanushev, O.R., 2013) Assume Smooth, compactly supported source g(x) Index of refraction n(x) smooth and constant for x > R No trapped rays: L s.t. x(±l) > 2R if x(0) < R, p(0) = n(x(0)) No initial data error g = g. Then with C independent of ε and α, u u GB L2 ( x <R) Cε K /2, Superposition in physical space. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

102 Error estimate K th order beams, Helmholtz case Theorem (Liu, Ralston, Tanushev, O.R., 2013) Assume Smooth, compactly supported source g(x) Index of refraction n(x) smooth and constant for x > R No trapped rays: L s.t. x(±l) > 2R if x(0) < R, p(0) = n(x(0)) No initial data error g = g. Then with C independent of ε and α, u u GB L2 ( x <R) Cε K /2, Superposition in physical space. Convergence of all beams independent of dimension and presence of caustics. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

103 Sketch of proof, wave equation Use energy estimate u GB (t, ) u(t, ) E u GB (0, ) u(0, ) E + Cε sup u GB (t, ) L 2, t [0,T ] Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

104 Sketch of proof, wave equation Use energy estimate u GB (t, ) u(t, ) E u GB (0, ) u(0, ) E + Cε sup u GB (t, ) L 2, t [0,T ] The residual is of the form u GB (t, y) = ε K /2 q J j=1 ε r j T ε j [f j ](t, y) + O(ε ), where r j 0, J finite and f j L 2 (all independent of ε). T ε j : L 2 L 2 belongs to a class of oscillatory integral operators. Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

105 Sketch of proof, wave equation Use energy estimate u GB (t, ) u(t, ) E u GB (0, ) u(0, ) E + Cε sup u GB (t, ) L 2, t [0,T ] The residual is of the form u GB (t, y) = ε K /2 q J j=1 ε r j T ε j [f j ](t, y) + O(ε ), where r j 0, J finite and f j L 2 (all independent of ε). T ε j : L 2 L 2 belongs to a class of oscillatory integral operators. Together we get (if initial data exact) u GB (t, ) u(t, ) E C(T )ε K /2 J j=1 ε r j T ε j L 2 f j L 2 + O(ε ) Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

106 Sketch of proof, cont. We have u GB (t, ) u(t, ) E 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 Approximation IPP Garching, / 36

107 Sketch of proof, cont. We have u GB (t, ) u(t, ) E 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 Approximation IPP Garching, / 36

108 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 Approximation IPP Garching, / 36

109 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 Approximation IPP Garching, / 36

110 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 Approximation IPP Garching, / 36

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? Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

112 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 Approximation IPP Garching, / 36

113 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. Numerical experiments suggests a better rate for odd order beams Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

114 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. Numerical experiments suggests a better rate for odd order beams For the Helmholtz case we have proved [Motamed, OR] that u(x) u GB(x) 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 Approximation IPP Garching, / 36

115 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. Numerical experiments suggests a better rate for odd order beams For the Helmholtz case we have proved [Motamed, OR] that u(x) u GB(x) 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 Approximation IPP Garching, / 36

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 Approximation IPP Garching, / 36

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 Approximation IPP Garching, / 36

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 Approximation IPP Garching, / 36

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 Approximation IPP Garching, / 36

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 Approximation IPP Garching, / 36

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 Approximation IPP Garching, / 36

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 Approximation IPP Garching, / 36

123 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 Approximation IPP Garching, / 36

124 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 Approximation IPP Garching, / 36

125 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 Approximation IPP Garching, / 36

126 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 Approximation IPP Garching, / 36

127 Numerical examples Cusp caustic, convergence Energy norm Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36

128 Numerical examples Cusp caustic, convergence Max norm Olof Runborg (KTH) Gaussian Beam Approximation IPP Garching, / 36