On a new notion of the solution to an ill-posed problem

A new understanding of the notion of the stable solution to ill-posed problems is proposed. The new notion is more realistic than the old one and better fits the practical computational needs. A method for constructing stable solutions in the new sense is proposed and justified. The basic point is: in the traditional definition of the stable solution to an ill-posed problem $Au=f$, where $A$ is a linear or nonlinear operator in a Hilbert space $H$, it is assumed that the noisy data $\{f_\delta, \delta\}$ are given, $||f-f_\delta||\leq \delta$, and a stable solution u_\d:=R_\d f_\d is defined by the relation \lim_{\d\to 0}||R_\d f_\d-y||=0, where $y$ solves the equation $Au=f$, i.e., $Ay=f$. In this definition $y$ and $f$ are unknown. Any f\in B(f_\d,\d) can be the exact data, where B(f_\d,\d):=\{f: ||f-f_\delta||\leq \delta\}.The new notion of the stable solution excludes the unknown $y$ and $f$ from the definition of the solution

On Compact Finite Difference Schemes With Applications To Moving Boundary Problems

Compact finite differences are introduced with the purpose of developing compact methods of higher order for the numerical solution of ordinary and elliptic partial differential equations.;The notion of poisedness of a compact finite difference is introduced. It is shown that if the incidence matrix of the underlying interpolation problem contains no odd unsupported sequences then the Polya conditions are necessary and sufficient for poisedness.;A Pade Operator method is used to construct compact formulae valid for uniform three point grids. A second Function-Theoretic method extends compact formulae to variably-spaced three point grids with no deterioration in the order of the truncation error.;A new fourth order compact method (CI4) leading to matrix systems with block tridiagonal structure, is applied to boundary value problems associated with second order ordinary differential equations. Numerical experiments with both linear and nonlinear problems and on uniform and nonuniform grids indicate rates of convergence of four.;An application is considered to the time-dependent one-dimensional nonlinear Burgers\u27 equation in which an initial sinusoidal disturbance develops a very sharp boundary layer. It is found that the CI4 method, with a small number of points placed on a highly stretched grid, is capable of accurately resolving the boundary layer.;A new method (LCM) based on local polynomial collocation and Gauss-type quadrature and leading to matrix systems with block tridiagonal structure, is used to generate high order compact methods for ordinary differential equations. A tenth order method is shown to be considerably more efficient than the CI4 method.;A new fourth order compact method, based on the CI4 method, is developed for the solution, on variable grids, of two-dimensional, time independent elliptic partial differential equations. The method is applied to the ill-posed problem of calculating the interface in receding Hele-Shaw flow. Comparisons with exact solutions indicate that the numerical method behaves as expected for early times.;Finally, in an application to the simulation of contaminant transport within a porous medium under an evolving free surface, new fourth order explicit compact expressions for mixed derivatives are developed

Necessary conditions for variational regularization schemes

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization method, we start with an investigation of the converse question: How could necessary conditions for a variational method to provide a regularization method look like? To this end, we formalize the notion of a variational scheme and start with comparison of three different instances of variational methods. Then we focus on the data space model and investigate the role and interplay of the topological structure, the convergence notion and the discrepancy functional. Especially, we deduce necessary conditions for the discrepancy functional to fulfill usual continuity assumptions. The results are applied to discrepancy functionals given by Bregman distances and especially to the Kullback-Leibler divergence.Comment: To appear in Inverse Problem

The Residual Method for Regularizing Ill-Posed Problems

Although the \emph{residual method}, or \emph{constrained regularization}, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals. We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on $L^p$-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.Comment: 29 pages, one figur

Solving ill-posed bilevel programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem

Numerical modeling of black holes as sources of gravitational waves in a nutshell

These notes summarize basic concepts underlying numerical relativity and in particular the numerical modeling of black hole dynamics as a source of gravitational waves. Main topics are the 3+1 decomposition of general relativity, the concept of a well-posed initial value problem, the construction of initial data for general relativity, trapped surfaces and gravitational waves. Also, a brief summary is given of recent progress regarding the numerical evolution of black hole binary systems.Comment: 28 pages, lectures given at winter school 'Conceptual and Numerical Challenges in Femto- and Peta-Scale Physics' in Schladming, Austria, 200

Evolution PDEs and augmented eigenfunctions. I finite interval

The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, which will be referred to as problems of type~I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated initial-boundary value problems, which will be referred to as problems of type~II, there does \emph{not} exist a classical transform pair and the solution \emph{cannot} be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type~II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions introduced in the sixties by Gel'fand and his co-authors