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A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media
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2019
Year
Numerical AnalysisNeumann Boundary ConditionInverse Source ProblemNumerical Method For Partial Differential EquationEngineeringMethod Of Fundamental SolutionPde-constrained OptimizationSemi-implicit MethodHigh-frequency ApproximationInverse Scattering TransformsInverse ProblemsComputational ElectromagneticsConvergent Numerical MethodInhomogenous MediaApproximation TheoryBoundary Element MethodNew Numerical Method
A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in which the unknown source function is not involved. The Dirichlet boundary condition is given on the entire boundary of the domain of interest and the Neumann boundary condition is given on a part of this boundary. To solve this problem, the quasi-reversibility method is applied. Uniqueness and existence of the minimizer are proven. A new Carleman estimate is established. Next, the convergence of those minimizers to the exact solution is proven using that Carleman estimate. Results of numerical tests are presented.