Othmar Koch

Dynamical Low-Rank Approximation

For the low-rank approximation of time-dependent data matrices and of solutions to matrix differential equations, an increment-based computational approach is proposed and analyzed. In this method, the derivative is projected onto the tangent space of the manifold of rank-

Dynamical Tensor Approximation

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Efficient Collocation Schemes for Singular Boundary Value Problems

matrices at the current approximation. With an appropriate decomposition of rank-

Iterated Defect Correction for the Solution of Singular Initial Value Problems

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Analysis of a New Error Estimate for Collocation Methods Applied to Singular Boundary Value Problems

matrices and their tangent matrices, this yields nonlinear differential equations that are well suited for numerical integration. The error analysis compares the result with the pointwise best approximation in the Frobenius norm. It is shown that the approach gives locally quasi-optimal low-rank approximations. Numerical experiments illustrate the theoretical results.

Asymptotically correct error estimation for collocation methods applied to singular boundary value problems

For the approximation of time-dependent data tensors and of solutions to tensor differential equations by tensors of low Tucker rank, we study a computational approach that can be viewed as a continuous-time updating procedure. This approach works with the increments rather than the full tensor and avoids the computation of decompositions of large matrices. In this method, the derivative is projected onto the tangent space of the manifold of tensors of Tucker rank

A Collocation Code for Singular Boundary Value Problems in Ordinary Differential Equations

(r_1,\dots,r_N)

Computation of Self-similar Solution Profiles for the Nonlinear Schrödinger Equation

at the current approximation. This yields nonlinear differential equations for the factors in a Tucker decomposition, suitable for numerical integration. Approximation properties of this approach are analyzed.

Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.

Analytical and numerical treatment of a singular initial value problem in avalanche modeling

We investigate the convergence properties of the iterated defect correction (IDeC) method based on the implicit Euler rule for the solution of singular initial value problems with a singularity of the first kind. We show that the method retains its classical order of convergence, which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure.