Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment
-
Pauline Achieng
,
Fredrik Berntsson
Abstract
We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.
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Articles in the same Issue
- Frontmatter
- Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment
- Numerical Approximation of Gaussian Random Fields on Closed Surfaces
- Multivariate Analysis-Suitable T-Splines of Arbitrary Degree
- Symmetrized Two-Scale Finite Element Discretizations for Partial Differential Equations with Symmetric Solutions
- Relaxation Quadratic Approximation Greedy Pursuit Method Based on Sparse Learning
- Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation
- Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations
- An Optimal Method for High-Order Mixed Derivatives of Bivariate Functions
- A Convenient Inclusion of Inhomogeneous Boundary Conditions in Minimal Residual Methods
- A 𝐶1-𝑃7 Bell Finite Element on Triangle
- A Conforming Virtual Element Method for Parabolic Integro-Differential Equations
- Three Low Order H-Curl-Curl Finite Elements on Triangular Meshes
Articles in the same Issue
- Frontmatter
- Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment
- Numerical Approximation of Gaussian Random Fields on Closed Surfaces
- Multivariate Analysis-Suitable T-Splines of Arbitrary Degree
- Symmetrized Two-Scale Finite Element Discretizations for Partial Differential Equations with Symmetric Solutions
- Relaxation Quadratic Approximation Greedy Pursuit Method Based on Sparse Learning
- Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation
- Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations
- An Optimal Method for High-Order Mixed Derivatives of Bivariate Functions
- A Convenient Inclusion of Inhomogeneous Boundary Conditions in Minimal Residual Methods
- A 𝐶1-𝑃7 Bell Finite Element on Triangle
- A Conforming Virtual Element Method for Parabolic Integro-Differential Equations
- Three Low Order H-Curl-Curl Finite Elements on Triangular Meshes
