Singular limits for stochastic equations

We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and increasing roughness of the noise, so that the noise in the limit would be too rough to define a solution to the limiting equations. Simultaneously, the limit is singular in the sense that the leading order differential operator may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. We give an abstract framework for the main error estimates, that first reduce to bounds on a residual and in a second step to bounds on the stochastic convolution. Moreover, we apply it to a singularly regularized Allen-Cahn equation and the Cahn-Hilliard/Allen-Cahn homotopy.Comment: 23 pages, 36 reference

Optimal trajectory tracking

This thesis investigates optimal trajectory tracking of nonlinear dynamical systems with affine controls. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible. The concept of so-called exactly realizable trajectories is proposed. For exactly realizable desired trajectories exists a control signal which enforces the state to exactly follow the desired trajectory. For a given affine control system, these trajectories are characterized by the so-called constraint equation. This approach does not only yield an explicit expression for the control signal in terms of the desired trajectory, but also identifies a particularly simple class of nonlinear control systems. Based on that insight, the regularization parameter is used as the small parameter for a perturbation expansion. This results in a reinterpretation of affine optimal control problems with small regularization term as singularly perturbed differential equations. The small parameter originates from the formulation of the control problem and does not involve simplifying assumptions about the system dynamics. Combining this approach with the linearizing assumption, approximate and partly linear equations for the optimal trajectory tracking of arbitrary desired trajectories are derived. For vanishing regularization parameter, the state trajectory becomes discontinuous and the control signal diverges. On the other hand, the analytical treatment becomes exact and the solutions are exclusively governed by linear differential equations. Thus, the possibility of linear structures underlying nonlinear optimal control is revealed. This fact enables the derivation of exact analytical solutions to an entire class of nonlinear trajectory tracking problems with affine controls. This class comprises mechanical control systems in one spatial dimension and the FitzHugh-Nagumo model.Comment: 240 pages, 36 figures, PhD thesi

Dynamics of Patterns

This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Adaptive Wavelet Methods for Inverse Problems: Acceleration Strategies, Adaptive Rothe Method and Generalized Tensor Wavelets

In general, inverse problems can be described as the task of inferring conclusions about the cause u from given observations y of its effect. This can be described as the inversion of an operator equation K(u) = y, which is assumed to be ill-posed or ill-conditioned. To arrive at a meaningful solution in this setting, regularization schemes need to be applied. One of the most important regularization methods is the so called Tikhonov regularization. As an approximation to the unknown truth u it is possible to consider the minimizer v of the sum of the data error K(v)-y (in a certain norm) and a weighted penalty term F(v). The development of efficient schemes for the computation of the minimizers is a field of ongoing research and a central Task in this thesis. Most computation schemes for v are based on some generalized gradient descent approach. For problems with weighted lp-norm penalty terms this typically leads to iterated soft shrinkage methods. Without additional assumptions the convergence of these iterations is only guaranteed for subsequences, and even then only to stationary points. In general, stationary points of the minimization problem do not have any regularization properties. Also, the basic iterated soft shrinkage algorithm is known to converge very poorly in practice. This is critical as each iteration step includes the application of the nonlinear operator K and the adjoint of its derivative. This in itself may already be numerically demanding. This thesis is concerned with the development of strategies for the fast computation of the solution of inverse problems with provable convergence rates. In particular, the application and generalization of efficient numerical schemes for the treatment of the arising nonlinear operator equations is considered. The first result of this thesis is a general acceleration strategy for the iterated soft thresholding iteration to compute the solution of the inverse problem. It is based on a decreasing strategy for the weights of the penalty term. The new method converges with linear rate to a global minimizer. A very important class of inverse problems are parameter identification problems for partial differential equations. As a prototype for this class of problems the identification of parameters in a specific parabolic partial differential equation is investigated. The arising operators are analyzed, the applicability of Tikhonov Regularization is proven and the parameters in a simplified test equation are reconstructed. The parabolic differential equations are solved by means of the so called horizontal method of lines, also known as Rothes method. Here the parabolic problem is interpreted as an abstract Cauchy problem. It is discretized in time by means of an implicit scheme. This is combined with a discretization of the resulting system of spatial problems. In this thesis the application of adaptive discretization schemes to solve the spatial subproblems is investigated. Such methods realize highly nonuniform discretizations. Therefore, they tend to require much less degrees of freedom than classical discretization schemes. To ensure the convergence of the resulting inexact Rothe method, a rigorous convergence proof is given. In particular, the application of implementable asymptotically optimal adaptive methods, based on wavelet bases, is considered. An upper bound for the degrees of freedom of the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance is derived. As an important case study, the complexity of the approximate solution of the heat equation is investigated. To this end a regularity result for the spatial equations that arise in the Rothe method is proven. The rate of convergence of asymptotically optimal adaptive methods deteriorates with the spatial dimension of the problem. This is often called the curse of dimensionality. One way to avoid this problem is to consider tensor wavelet discretizations. Such discretizations lead to dimension independent convergence rates. However, the classical tensor wavelet construction is limited to domains with simple product geometry. Therefor, in this thesis, a generalized tensor wavelet basis is constructed. It spans a range of Sobolev spaces over a domain with a fairly general geometry. The construction is based on the application of extension operators to appropriate local bases on subdomains that form a non-overlapping domain decomposition. The best m-term approximation of functions with the new generalized tensor product basis converges with a rate that is independent of the spatial dimension of the domain. For two- and three-dimensional polytopes it is shown that the solution of Poisson type problems satisfies the required regularity condition. Numerical tests show that the dimension independent rate is indeed realized in practice

The 2nd International Conference on Mathematical Modelling in Applied Sciences, ICMMAS’19, Belgorod, Russia, August 20-24, 2019 : book of abstracts

The proposed Scientific Program of the conference is including plenary lectures, contributed oral talks, poster sessions and listeners. Five suggested special sessions / mini-symposium are also considered by the scientific committe

Some results on anisotropic mean curvature and other phase transition models

The present thesis is divided into three parts. In the first part, we analyze a suitable regularization \u2014 which we call nonlinear multidomain model \u2014 of the motion of a hypersurface under smooth anisotropic mean curvature flow. The second part of the thesis deals with crystalline mean curvature of facets of a solid set of R^3 . Finally, in the third part we study a phase-transition model for Plateau\u2019s type problems based on the theory of coverings and of BV functions