110 research outputs found
Analysis of new direct sampling indicators for far-field measurements
This article focuses on the analysis of three direct sampling indicators
which can be used for recovering scatterers from the far-field pattern of
time-harmonic acoustic measurements. These methods fall under the category of
sampling methods where an indicator function is constructed using the far-field
operator. Motivated by some recent work, we study the standard indicator using
the far-field operator and two indicators derived from the factorization
method. We show the equivalence of two indicators previously studied as well as
propose a new indicator based on the Tikhonov regularization applied to the
far-field equation for the factorization method. Finally, we give some
numerical examples to show how the reconstructions compare to other direct
sampling methods
Comparing intrusive and non-intrusive polynomial chaos for a class of exponential time differencing schemes
We consider the numerical approximation of different ordinary differential
equations (ODEs) and partial differential equations (PDEs) with periodic
boundary conditions involving a one-dimensional random parameter, comparing the
intrusive and non-intrusive polynomial chaos expansion (PCE) method. We
demonstrate how to modify two schemes for intrusive PCE (iPCE) which are highly
efficient in solving nonlinear reaction-diffusion equations: A second-order
exponential time differencing scheme (ETD-RDP-IF) as well as a spectral
exponential time differencing fourth-order Runge-Kutta scheme (ETDRK4). In
numerical experiments, we show that these schemes show superior accuracy to
simpler schemes such as the EE scheme for a range of model equations and we
investigate whether they are competitive with non-intrusive PCE (niPCE)
methods. We observe that the iPCE schemes are competitive with niPCE for some
model equations, but that iPCE breaks down for complex pattern formation models
such as the Gray-Scott system.Comment: 25 pages, 13 figures, 3 table
On trajectories of complex-valued interior transmission eigenvalues
This paper investigates properties of complex-valued eigenvalue trajectories for the interior transmission problem parametrized by the index of refraction for homogeneous media. Our theoretical analysis for the unit disk shows that the only intersection points with the real axis, as well as the unique trajectorial limit points as the refractive index tends to infinity, are Dirichlet eigenvalues of the Laplacian. Complementing numerical experiments even give rise to an underlying one-to-one correspondence between Dirichlet eigenvalues of the Laplacian and complex-valued interior transmission eigenvalue trajectories. We also examine other scatterers than the disk for which similar numerical observations can be made. We summarize our results in a conjecture for general simply-connected scatterers
A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a C2-boundary
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 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.Comment: 23 pages, 7 figure
Detecting and approximating decision boundaries in low dimensional spaces
A method for detecting and approximating fault lines or surfaces,
respectively, or decision curves in two and three dimensions with guaranteed
accuracy is presented. Reformulated as a classification problem, our method
starts from a set of scattered points along with the corresponding
classification algorithm to construct a representation of a decision curve by
points with prescribed maximal distance to the true decision curve. Hereby, our
algorithm ensures that the representing point set covers the decision curve in
its entire extent and features local refinement based on the geometric
properties of the decision curve. We demonstrate applications of our method to
problems related to the detection of faults, to Multi-Criteria Decision Aid
and, in combination with Kirsch's factorization method, to solving an inverse
acoustic scattering problem. In all applications we considered in this work,
our method requires significantly less pointwise classifications than
previously employed algorithms
Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary
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.Comment: arXiv admin note: substantial text overlap with arXiv:2301.1002
Theoretical Foundation of the Weighted Laplace Inpainting Problem
Laplace interpolation is a popular approach in image inpainting using partial
differential equations. The classic approach considers the Laplace equation
with mixed boundary conditions. Recently a more general formulation has been
proposed where the differential operator consists of a point-wise convex
combination of the Laplacian and the known image data. We provide the first
detailed analysis on existence and uniqueness of solutions for the arising
mixed boundary value problem. Our approach considers the corresponding weak
formulation and aims at using the Theorem of Lax-Milgram to assert the
existence of a solution. To this end we have to resort to weighted Sobolev
spaces. Our analysis shows that solutions do not exist unconditionally. The
weights need some regularity and fulfil certain growth conditions. The results
from this work complement findings which were previously only available for a
discrete setup.Comment: 16 pages, 2 Figure
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