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Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation. (English) Zbl 1185.65173

Summary: We propose a discretized Tikhonov regularization for a Cauchy problem for an elliptic equation by a reproducing kernel Hilbert space. We prove the convergence of discretized regularized solutions to an exact solution. Our numerical results demonstrate that our method can stably reconstruct solutions to the Cauchy problems even in severe cases of geometric configurations.

MSC:

65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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