Yin Huang. Education. Research Interests. Rice University, Houston, TX, USA. Shanghai Jiao Tong University, Shanghai, China

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1 Yin Huang Education Rice University, Houston, TX, USA Ph.D. Candidate, Computational and Applied Mathematics 08/ Present Expected Dissertation Topic: Inverse problems for wave propagation for nonsmooth coefficients Relevant courses: Numerical Analysis; Numerical Linear Algebra; Numerical Differential Equations; Optimization; High Performance Computing Shanghai Jiao Tong University, Shanghai, China M.S., Mathematics and Applied Mathematics 09/ /2009 Dissertation topic: Comparision of many numerical methods for saddle point system arising from the mixed finite element method of elliptic problems with nonsmooth coefficients Research Interests Forward and inverse problems for non-homogeneous medium Seismic waveform tomography Parallel computing

2 Acoustic transparency theorem for the 1D wave equation with a nonsmooth coefficient Yin Huang Advisor: William Symes Rice University March 30, 2012

3 Outline Background One-dimensional acoustic transparency Theorem Plan of the proof Discussion Conclusion References

4 1D constant density acoustic wave equation 1 2 u c 2 t 2 2 u z 2 = 0, u u(z, 0) = u 0 (z), t (z, 0) = u 1(z). simplest system with key features (variable wave velocity, reflected waves,...) z R depth. t R time. c = c(z) wave velocity, nonsmooth u = u(z, t) the acoustic potential at (z, t). u 0, u 1 are initial conditions.

5 Inverse problem Assume that u solves wave equation. Define T c = u t (0, t). Bamberger et al. 1979: if c is piecewise constant min c T c data the layers have equal travel time Neumann boundary condition on z = 0 The solution to the inverse problem: c is unique and depends continuously on data. Relation to acoustic transparency theorem Acoustic transparency theorem is a step of the proof toward this result: For nonsmooth c, c depends continuously on data.

6 Bounded variation function Definition c BV [0, Z] means Var(c) = max P n c(z i ) c(z i 1 ) M < +, i=1 with P the set of ordered points in [0, Z], including 0 and Z. Example: c is piecewise constant with c(z) = c i if z [z i 1, z i ). z i = z i z i 1. Var(c) = i c i c i 1 = i c i c i 1 Z z i = z i 0 dc dz

7 Acoustic energy Density of the material ρ is constant. ρ u t u z is the acoustic pressure, is the acoustic particle velocity, Energy over interval [0, Z] at time t ( ( u E(t) = ρ 2 Z 0 z ) c 2 ( ) ) u 2 (z, t)dz t

8 Acoustic transparency theorem Theorem Suppose u 0 and u 1 are zero outside of [0, Z], T > Z 0 c(z) = c(0) for z < 0. c BV [0, Z] gives 1 c and ke(0) T T ( ) u 2 (0, t) dt KE(0) t with K = sup z [0,Z] c(z) and k = Kexp ( ˆrVar(logc)), where ˆr depends on the supremum and infimum of c. Importance of the lower bound: Otherwise, arbitrarily small part of the energy reaches the surface No possibility of stable solution for the inverse problem.

9 Discussion of existing and relevant results Symes 1983, 1986 provided a proof by flipping the space and time variables Cox and Zuazua 1995 proved energy decay for 1D medium of bounded variation by analyzing the corresponding eigenvalue problem which also gives acoustic transparency.... Why we need a new proof? Existing proofs could not be generalized to multi-dimensional problems. A function space of velocity c that is both necessary and sufficient is needed. A proof to show the necessity and sufficiency of the given function space is needed.

10 Plan of the proof The upper bound is given by integration by parts. By a theorem due to Kahane, if c BV [0, Z], then for every n N there exists piecewise constant c n with n jumps so that max z [0,Z] c(z) c n (z) Var(c) n (see DeVore 1998). Choose piecewise constant approximations c n, then we have u n u in energy, by Theorem of Stolk PhD thesis The lower bound for piecewise constant c is from the analysis of reflections and transmissions at jumps in c. The lower bound we have now depends on the number of layers. Uniform lower bound is the goal.

11 Multi-interfaces with localized initial conditions I Construct the solution by repeating the reflection and transmission. Assume u 0 and u 1 are zero outside of [z i 1, z i ], with velocity c i. Transmission coefficients T k = 2c k c k +c k+1. u iu up-going wave in the i-th interval, Purely transmitted part of the wave u it (0, t) = i 1 k=1 T k u iu (Z(0, t))

12 Multi-interfaces with localized initial conditions II 10 9 t10 8 t (z,t) u T (z,t) u R (z,t) = u(z,t) u T (z,t) (Z(z,t),0) z0 z1 z2 z3 z4 z5 z6 z7 z8 z9 z u 0 and u 1 are zero outside of [z N 1, z N ] u(0, t) = u NT (0, t) for t [t N 1, t N ] N 1 k=1 T k has a possitive lower bound is necessary for acoustic transparency theorem.

13 Current result for purely transmitted wave If c BV [0, Z], product of transmission coefficients N 1 k=1 ) T k exp ( ˆr2 Var(logc), with ˆr depends on the supremum and infimum of c. Let the u T denote the purely transmitted part of the solution with arbitrary initial conditions Z T T «2 ut (0, t) dt t NX i=1! Yi 1 2 c 2 Z zi «T i k ( du0 4c 1 z i 1 dz )2 + u2 1 dz ci 2 k=1

14 Discussion Bounded variation is not necessary for the product of transmission coefficients to be bounded away from zero. c BV [0, Z] N 1 k=1 T k k > 0. k depends on Var(logc). For N layers material, where N is even, let { 1 1 c = N odd layer N even layer, Var(c) + as N +. N 1 k=1 T k exp( 1 2 ) with c BV [0, Z].

15 Discussion What is both necessary and sufficient? BV-2 is both necessary and sufficient (by Demanet) Var 2 (c) = max P ( n ) 1 2 (c(x i ) c(x i 1 )) 2 i=1 M < +, where P is the set of all ordered points of [0, Z]. Approximate BV-2 function c with piecewise constant function Define the transmission coefficient T k c BV-2 N 1 k=1 T k k > 0, k depends on Var 2 (logc).

16 Conclusion Acoustic transparency is necessary for the stable solution of the inverse problem. Lower bound of the acoustic transparency thoerem depends on N 1 k=1 T k and the number of layers N. T ( u T t (0, t)) 2 dt has a lower bound only if N 1 k=1 T k has a positive lower bound. N 1 k=1 T k has a positive lower bound c BV-2.

17 Bamberger A, Chavent G, and Lailly P: About the stability of the inverse problem in 1-d wave equations: application to the interpretation of the seismic profiles, Applied Math and Optim, 5(1):1-47, Cox S and Zuazua E: The rate at which energy decays in a damped string, Comm Partial Differential Equations, 19: , Demanet L and Peyre G: Compressive wave computation, Found Comput Math, 11: ,2011. DeVore R A: Nonlinear approximation, Acta Numerica, 7:51-150, Stolk C C: PhD thesis, On the modeling and inversion of seismic data, PhD thesis, Utrecht University, Symes W W:Impedance profile inversion via the first transport equations, J. Math. Anal. Appl., 84(2): , 1983.

18 Symes W W:On the relation between coefficient and boundary values for solutions of webster s horn equation, SIAM J Math Anal, 17(6): , 1986.