## Wavelet moment method for the Cauchy problem for the Helmholtz equation.(English)Zbl 1165.65069

J. Cheng, Y. C. Hon, T. Wei and M. Yamamoto [Z. Angew. Math. Mech. 81, No. 10, 665–674 (2001; Zbl 0999.65100)] investigated a Cauchy problem of the Laplace equation in two space dimensions, where mixed Dirichlet-Neumann conditions are given on an open subset of the boundary. This formulation represents an inverse problem and thus it is ill-posed. The equations are transformed into an equivalent moment problem, i.e., a weak form of a boundary integral equation. Accordingly, Galerkin type methods with polynomials as basis functions can be applied.
In the paper at hand, the authors consider this Cauchy problem of the Helmholtz equation. A corresponding moment problem is formulated in the general case of three space dimensions. The authors focus on the moment approach for a model problem in two space dimensions. The Meyer wavelet generates an orthonormal basis of $$L^2(\mathbb{R})$$. A subset of this basis is applied to construct a numerical approximation of the solution of the boundary integral equation for possibly perturbed data. It follows that a regularisation parameter can be chosen in dependence on a perturbation such that the approximations converge to the exact solution in case of perturbations tending to zero. Accordingly, a regularised formulation of the Helmholtz equation is achieved, where Neumann conditions are given on the whole boundary. Methods for the computation of the defined numerical approximation or simulation results are not within the scope of the article. Instead of that, the authors prove the properties of the numerical approximation using the wavelet projection and corresponding techniques.

### MSC:

 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 44A60 Moment problems 65N38 Boundary element methods for boundary value problems involving PDEs

Zbl 0999.65100
Full Text:

### References:

 [1] Belsky, A.M.; Korneychik, T.M.; Khapalyuk, A.P., Space structure of laser radiation, Izd. BGU, minsk, (1982), (in Russian) [2] Berntsson, F.; Eldén, L., Numerical solution of a Cauchy problem for the Laplace equation, Inverse problems, 17, 4, 839-853, (2001) · Zbl 0993.65119 [3] Cheng, J.; Hon, Y.C.; Wei; Yamamoto, M.T., Numerical computation of a Cauchy problem for laplace’s equation, ZAMM Z. angew. math. mech., 81, 10, 665-674, (2001) · Zbl 0999.65100 [4] Cohen, Albert, Numerical analysis of wavelet methods, (2003), Elsevier · Zbl 1038.65151 [5] Daubechies, Ingrid, Ten lectures on wawelets, (1992), SIAM · Zbl 0776.42018 [6] Dautry, R.; Lions, J.-L., Mathematical analysis and numerical methods for science and technology, (1990), Springer-Verlag [7] DeLillo, T.; Isakov, V.; Valdivia, N.; Wang, L., The detection of surface vibrations from interior acoustical pressure, Inverse problems, 19, 507-524, (2003) · Zbl 1033.76053 [8] Eldén, L.; Berntsson, F., A stability estimate for Cauchy problem for an elliptic partial differential equations, Inverse problems, 21, 5, 1643-1653, (2005) · Zbl 1086.35115 [9] Engl, H.W.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (1996), Kluwer Academic Publishers · Zbl 0859.65054 [10] Hrycak, T.; Isakov, V, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse problems, 20, 697-712, (2004) · Zbl 1086.35080 [11] Isakov, V., Inverse problems for partial differential equations, (1998), Springer New York · Zbl 0908.35134 [12] Kleinman, R.E.; Wendland, W.L., On Neumann method for the exterior Neumann problem for the Helmholtz equation, J. math. anal. appl., 57, 170-202, (1977) · Zbl 0351.35022 [13] Regińska, T.; Regiński, K., Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse problems, 22, 975-989, (2006) · Zbl 1099.35160 [14] Reinhardt, H.-J.; Han, Houde; Hào, Dinh Nho, Stability and regularization of a discrete approximation to the Cauchy problem for laplace’s equation, SIAM J. numer. anal., 36, 3, 890-905, (1999) · Zbl 0928.35184 [15] Siegman, A.E., Lasers, (1986), Oxford University Press [16] Tikhonov, A.N.; Goncharsky, A.V.; Stepanov, V.V.; Yagola, A.G., Numerical methods for the solution of ill-posed problems, (1995), Kluwer Academic Publishers · Zbl 0831.65059 [17] Wojtaszczyk, Przemyslaw, A mathematical introduction to wavelets, (1997), Cambridge University Press · Zbl 0984.42019
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.