TY  - JOUR
A1  - Clausnitzer, Julian
A1  - Kleefeld, Andreas
T1  - A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a C²  boundary
JF  - Discrete and Continuous Dynamical Systems - Series B
N2  - We consider the numerical approximation of second-order semi-linear parabolic stochastic partial differential equations interpreted in the mild sense which we solve on general two-dimensional domains with a C² boundary with homogeneous Dirichlet boundary conditions. The equations are driven by Gaussian additive noise, and several Lipschitz-like conditions are imposed on the nonlinear function. We discretize in space with a spectral Galerkin method and in time using an explicit Euler-like scheme. For irregular shapes, the necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary integral equation method. This yields a nonlinear eigenvalue problem, which is discretized using a boundary element collocation method and is solved with the Beyn contour integral algorithm. We present an error analysis as well as numerical results on an exemplary asymmetric shape, and point out limitations of the approach.
KW  - Nonlinear eigenvalue problems
KW  - Boundary integral equations,
KW  - Exponential Euler scheme,
KW  - Parabolic SPDEs
Y1  - 2024
U6  - https://doi.org/10.3934/dcdsb.2023148
SN  - 1531-3492
SN  - 1553-524X (eISSN)
VL  - 29
IS  - 4
SP  - 1624
EP  - 1651
PB  - AIMS
CY  - Springfield
ER  - 
TY  - JOUR
A1  - Ayala, Rafael Ceja
A1  - Harris, Isaac
A1  - Kleefeld, Andreas
T1  - Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary
JF  - Inverse Problems and Imaging
N2  - In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.
Y1  - 2024
U6  - https://doi.org/10.3934/ipi.2023051
SN  - 1930-8337
SN  - 1930-8345 (eISSN)
VL  - 18
IS  - 3
SP  - 708
EP  - 729
PB  - AIMS
CY  - Springfield
ER  -