inverse problems for hyperbolic pdes - bcam · inverse problems for hyperbolic pdes arise naturally...

2

Click here to load reader

Upload: tranngoc

Post on 01-Aug-2018

216 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Inverse problems for hyperbolic PDEs - BCAM · Inverse problems for hyperbolic PDEs arise naturally in geophysics, oil prospecting, ... An inverse problem for a layered medium with

Inverse problems for hyperbolic PDEs

Lecturer: Rakesh1

Dates: 17–21 January 2011

Abstract:Inverse problems for hyperbolic PDEs arise naturally in geophysics, oil prospecting, in the design of opti-

cal devices, and in many other areas where the interior of an object is to be imaged using the response of theobject to acoustic waves (satisfying hyperbolic PDEs). There are many unresolved theoretical, algorithmic andcomputational issues and the course will focus mainly on the theoretical issues for such problems.

Many of these problems may be regarded as the study of the inversion of the nonlinear map F from thecoefficients of PDEs to the trace of some special solutions of the PDEs on the boundary of the object. Towardsattempting an inversion of F one also studies the uniqueness question (the injectivity of F) and the stabilityquestion (the continuity of F!1).

We will explore the well understood work on the one dimensional problem (the object is one dimensional)regarding the use of the downward continuation method and then study the results and the many unsolved problemsin the multidimensional case for formally determined data (data and the coefficient depend on the same numberof parameters). There have been attempts to push through the downward continuation argument for the inversionin the multidimensional case, with limited success. Carleman estimates and tools from Harmonic Analysis havebeen used in the study of multidimensional formally determined problems but many problems remain unresolved.We will describe these tools, some of the results, and the many unsolved problems.

The course will be geared towards researchers with a basic knowledge of the study of Partial DifferentialEquations using Sobolev Spaces and Functional Analysis.

Program:

1. Some inverse problems for hyperbolic PDEs

2. Inversion from the Dirichlet to Neumann map data

3. The Gelfand–Levitan–Krein theory for the solution of the one dimensional inverse problem

4. Solution of inverse problems for layered mediums but which do not reduce to one-dimensional problems

5. Romanov’s results for problems with coefficients analytic in some direction

6. Carleman estimates for second order PDEs

7. The Bukhgeim and Klibanov result and its improvements for a formally determined multidimensional prob-lem

8. Unsolved problems

1Department of Mathematical Sciences, University of Delaware. 410 Ewing Hall. Newark, DE 19716, USA. Email [email protected]

Page 2: Inverse problems for hyperbolic PDEs - BCAM · Inverse problems for hyperbolic PDEs arise naturally in geophysics, oil prospecting, ... An inverse problem for a layered medium with

Bibliography:

[1] A. L. Bukhgeim and M. V. Klibanov. Uniqueness in the large of a class of multidimensional inverse problems.(Russian) Dokl. Akad. Nauk SSSR 260 (1981) 269–272.

[2] R. Burridge. The Gel’fand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inversescattering theory, regarded in the context of inverse impulse-response problems. Wave Motion 2 (1980) 305–323.

[3] L. Hormander. Linear Differential Operators. Fourth Printing. Springer-Verlag (1976).

[4] O. Y. Imanuvilov and M. Yamamoto. Global uniqueness and stability in determining coefficients of waveequations. Comm Partial Differential Equations 26 (2001) 1409–1425.

[5] Rakesh. An inverse problem for a layered medium with a point source. Inverse Problems 19 (2003) 497–506.

[6] Rakesh. Inverse problems for the wave equation with a single coincident source-receiver pair. Inverse Prob-lems 24 (2008) 015012, 16 pp.

[7] V. G. Romanov. Investigation methods for inverse problems, VSP (2002).

[8] Z. Sun. On continuous dependence for an inverse initial-boundary value problem for the wave equation. J.Math. Anal. Appl. 150 (1990) 188–204.

[9] J. Sylvester, D. P. Winebrenner and F. Gylys-Colwell. Layer stripping for the Helmholtz equation. SIAM J.Appl. Math 56 (1996) 736–754.

[10] W. W. Symes. Impedance profile inversion via the first transport equation, Journal of Mathematical Analysis94 (1983) 435–453.

[11] D. Tataru. Notes on Carleman estimates. Available at www.math.berkeley.edu/%7Etataru/papers/ucpnotes.ps