Related papers: Method of Fundamental Solutions with Optimal Regul…
We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable…
We present a new approach to convexification of the Tikhonov regularization using a continuation method strategy. We embed the original minimization problem into a one-parameter family of minimization problems. Both the penalty term and the…
We propose and analyze a perturbative regularization method to approximate quadratic optimization problems with finite-dimensional degeneracy. The original problem is first approximated by a regularized problem depending on a small positive…
For the first time, a globally convergent numerical method is presented for ill-posed Cauchy problems for quasilinear PDEs. The key idea is to use Carleman Weight Functions to construct globally strictly convex Tikhonov-like cost…
The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here we present new scalable MFS…
This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific…
We study multi-parameter regularization (multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters…
We study multi-parameter Tikhonov regularization, i.e., with multiple penalties. Such models are useful when the sought-for solution exhibits several distinct features simultaneously. Two choice rules, i.e., discrepancy principle and…
We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account…
The solution, $x$, of the linear system of equations $A x\approx b$ arising from the discretization of an ill-posed integral equation with a square integrable kernel $H(s,t)$ is considered. The Tikhonov regularized solution $ x(\lambda)$ is…
This work is concerned with linear inverse problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex…
We consider the identification of scattering and absorption rates in the stationary radiative transfer equation. For a stable solution of this parameter identification problem, we consider Tikhonov regularization within Banach spaces. A…
In the paper we offer a functional-discrete method for solving the Cauchy problem for the first order ordinary differential equations (ODEs). This method (FD-method) is in some sense similar to the Adomian Decomposition Method. But it is…
We consider the inverse problem for the wave equation which consists of determining an unknown space-dependent force function acting on a vibrating structure from Cauchy boundary data. Since only boundary data are used as measurements, the…
The method of fundamental solutions (MFS), also known as the method of auxiliary sources (MAS), is a well-known computational method for the solution of boundary-value problems. The final solution ("MAS solution") is obtained once we have…
The Tikhonov-Phillips method is widely used for regularizing ill-posed inverse problems mainly due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the…
Computations of incompressible flows with velocity boundary conditions require solution of a Poisson equation for pressure with all Neumann boundary conditions. Discretization of such a Poisson equation results in a rank-deficient matrix of…
The determination of an unknown spacewice dependent force function acting on a vibrating string from over-specified Cauchy boundary data is investigated numerically using the boundary element method (BEM) combined with a regularized method…
The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the…
We consider the Cauchy problem for the Zakharov-Kuznetsov equation in the cylinder. We improve the local wellposedness to spaces of regularity $s > 1/2$. The result is optimal in terms of the corresponding bilinear estimate or Picard…