Fast methods for three-dimensional inverse obstacle scattering problems. (English) Zbl 1138.65100
Summary: We study the inverse problem to reconstruct the shape of a three dimensional sound-soft obstacle from measurements of scattered acoustic waves. To solve the forward problem we use a wavelet based boundary element method and prove fourth order accuracy both for the evaluation of the forward solution operator and its Fréchet derivative. Moreover, we discuss the characterization and implementation of the adjoint of the Fréchet derivative.
For the solution of the inverse problem we use a regularized Newton method. The boundaries are represented by a class of parametrizations, which include non star-shaped domains and which are not uniquely determined by the obstacle. To prevent degeneration of the parametrizations during the Newton iteration, we introduce an additional penalty term. Numerical examples illustrate the performance of our method.
For the solution of the inverse problem we use a regularized Newton method. The boundaries are represented by a class of parametrizations, which include non star-shaped domains and which are not uniquely determined by the obstacle. To prevent degeneration of the parametrizations during the Newton iteration, we introduce an additional penalty term. Numerical examples illustrate the performance of our method.
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35R30 | Inverse problems for PDEs |
Keywords:
Helmholtz equation; regularization; inverse problem; scattered acoustic waves; wavelet; boundary element method; Newton method; numerical examplesReferences:
| [1] | A.B. Bakushinskii, The problem of the convergence of the iteratively regularized Gauss-Newton method , Comput. Math. Math. Phys. 32 (1992), 1353-1359. · Zbl 0783.65052 |
| [2] | B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method , IMA J. Numerical Anal. 17 (1997), 421-436. · Zbl 0881.65050 · doi:10.1093/imanum/17.3.421 |
| [3] | H. Brackhage and P. Werner, Über das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung , Arch. Math. 16 (1965), 325-329. · Zbl 0132.33601 · doi:10.1007/BF01220037 |
| [4] | D. Colton and R. Kress, Integral equation methods in scattering theory , in Pure and applied mathematics , Chichester, Wiley, New York, 198-3. · Zbl 0522.35001 |
| [5] | ——–, Inverse acoustic and electromagnetic scattering , 2nd Edition, Springer Verlag, Berlin, 199-7. |
| [6] | W. Dahmen, H. Harbrecht and R. Schneider, Compression techniques for boundary integral equations - optimal complexity estimates , SIAM J. Numer. Anal. 43 (2006), 2251-2271. · Zbl 1113.65114 · doi:10.1137/S0036142903428852 |
| [7] | W. Dahmen, A. Kunoth and K. Urban, Biorthogonal spline-wavelets on the interval - stability and moment conditions , Appl. Comp. Harm. Anal. 6 (1999), 259-302. · Zbl 0922.42021 · doi:10.1006/acha.1998.0247 |
| [8] | C. Farhat, R. Tezaur and R. Djellouli, On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method , Inverse Problems 18 (2002), 1229-1246. · Zbl 1009.35088 · doi:10.1088/0266-5611/18/5/302 |
| [9] | K. Giebermann, Schnelle Summationsverfahren zur numerischen Lösung von Integralgleichungen für Streuprobleme im R3, Ph.D. thesis, Universität Karlsruhe, Germany, 199-7. |
| [10] | M. Hanke, F. Hettlich and O. Scherzer, The Landweber iteration for an inverse scattering problem , in Proc. 1995 Design Engineering Technical Conferences Vol. 3 Part C, K.W. Wang et al., eds., 909-915, ASME, New York, 199-5. |
| [11] | M. Hanke, Regularizing properties of a truncated Newton - CG algorithm for nonlinear inverse problems , Numer. Funct. Anal. Optim. 18 (1997), 971-993. · Zbl 0899.65038 · doi:10.1080/01630569708816804 |
| [12] | H. Harbrecht, Wavelet Galerkin schemes for the boundary element method in three dimensions , Ph.D. thesis, Technische Universität Chemnitz, Germany, 200-1. |
| [13] | H. Harbrecht and R. Schneider, Wavelet Galerkin schemes for boundary integral equations - implementation and quadrature , SIAM J. Sci. Comput. 27 (2006), 1347-1370. · Zbl 1117.65162 · doi:10.1137/S1064827503429387 |
| [14] | ——–, Biorthogonal wavelet bases for the boundary element method , Math. Nach. 269 - 270 (2004), 167-188. · Zbl 1055.65136 · doi:10.1002/mana.200310171 |
| [15] | F. Hettlich and W. Rundell, A second degree method for nonlinear ill-posed problems , SIAM J. Numer. Anal. 37 (2000), 587-620. JSTOR: · Zbl 0946.35115 · doi:10.1137/S0036142998341246 |
| [16] | T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem , Inverse Problems 13 (1997), 1279-1299. · Zbl 0895.35109 · doi:10.1088/0266-5611/13/5/012 |
| [17] | ——–, On the numerical solution of a three-dimensional inverse medium scattering problem , Inverse Problems 17 (2001), 1743-1763. · Zbl 0989.35141 · doi:10.1088/0266-5611/17/6/314 |
| [18] | ——–, Iterative methods in inverse obstacle scattering : Regularization theory of linear and nonlinear exponentially ill-posed problems , Ph.D. thesis, Johannes-Kepler-Universität Linz, Austria, 199-9. |
| [19] | D. Huybrechs, J. Simoens and S. Vandewalle, A note on wave number dependence of wavelet matrix compression for integral equations with oscillatory kernel , J. Comput. Appl. Math. 172 (2004), 233-246. · Zbl 1099.78018 · doi:10.1016/j.cam.2004.02.006 |
| [20] | B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems . Kluwer, Dordrecht, 200-5. · Zbl 1145.65037 · doi:10.1515/9783110208276 |
| [21] | H. Kersten, Grenz- und Sprungrelationen für Potentiale mit quadrat-summierbarer Dichte , Resultate d. Math. 3 (1980), 17-24. · Zbl 0429.31007 · doi:10.1007/BF03323345 |
| [22] | A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory , Math. Meth. Appl. Sci. 11 (1989), 789-804. · Zbl 0692.35073 · doi:10.1002/mma.1670110605 |
| [23] | ——–, Properties of far field operators in acoustic scattering , Math. Meth. Appl. Sci. 11 (1989), 773-787. · Zbl 0692.35072 · doi:10.1002/mma.1670110604 |
| [24] | ——–, The domain derivative and two applications in inverse scattering theory , Inverse Problems 9 (1993), 81-96. · Zbl 0773.35085 · doi:10.1088/0266-5611/9/1/005 |
| [25] | R. Kress, Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering , Quart. J. Mech. Appl. Math. 38 (1985), 323-341. · Zbl 0559.73095 · doi:10.1093/qjmam/38.2.323 |
| [26] | ——–, A Newton method in inverse obstacle scattering , in Inverse problems in engineering mechanics , Bui et al., eds., Balkema, Rotterdam, 1994. · Zbl 0823.35172 · doi:10.1088/0266-5611/10/5/011 |
| [27] | R.D. Murch, D.G.H. Tan and D.J.N. Wall, Newton-Kantorovich method applied to two-dimensional inverse scattering for an exterior Helmholtz problem , Inverse Problems 4 (1988), 1117-1128. · Zbl 0659.65127 · doi:10.1088/0266-5611/4/4/012 |
| [28] | R. Schneider, Multiskalen- und Wavelet-Matrixkompression : Analysisbasierte Methoden zur Lösung großer vollbesetzter Gleichungssyteme , B.G. Teubner, Stuttgart, 199-8. |
| [29] | W. Tobocman, Inverse acoustic wave scattering in two dimensions from impenetrable targets , Inverse Problems 5 (1989), 1131-1144. · Zbl 0805.65130 · doi:10.1088/0266-5611/5/6/018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
