arXiv subjects

Thorsten Raasch

Publications and source records attributed to Thorsten Raasch.

9 recordsLinked to original sources

arXiv2026

A penalty-type method for relaxed inverse optimal control problems

This paper is devoted to the introduction and analysis of a penalty-type method for the numerical treatment of a class of bilevel optimization problems arising from inverse optimal control. The algorithm is designed to compute stationary points of the associated relaxed value function reformulation. This is achieved by determining a sequence of stationary points associated with a sequence of surrogate problems where the relaxed value function constraint is penalized, where the updates of upper- and lower-level decision variables are decoupled, and where the penalty parameter is enlarged only in those iterations which do not come along with a sufficient improvement of some feasibility measure. The resulting method does not comprise any linesearch, the lower-level problem has to be evaluated just once per iteration, and the penalty parameter does not need to be driven to infinity. Nevertheless, subsequential convergence results are obtained under reasonable assumptions. Numerical experiments, where the relaxation parameter is also driven to zero, visualize effectiveness of the approach.

math.OC↗

arXiv2023

Adaptive Quarklet Tree Approximation

This paper is concerned with near-optimal approximation of a given function $f \in L_2([0,1])$ with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by $hp$-approximation techniques of Binev, we use the underlying tree structure of the frame elements to derive an adaptive algorithm that, under standard assumptions concerning the local errors, can be used to create approximations with an error close to the best tree approximation error for a given cardinality. We support our findings by numerical experiments demonstrating that this approach can be used to achieve inverse-exponential convergence rates.

math.NA↗

arXiv2022

B-Spline Quarklets and Biorthogonal Multiwavelets

We show that B-spline quarks and the associated quarklets fit into the theory of biorthogonal multiwavelets. Quark vectors are used to define sequences of subspaces $ V_{p,j} $ of $ L_{2}(\mathbb{R}) $ which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters they even satisfy all those properties. Moreover we prove that quarks and quarklets possess modulation matrices which fulfill the perfect reconstruction condition. Furthermore we show the existence of generalized dual quarks and quarklets which are known to be at least compactly supported tempered distributions from $\mathcal{S}'(\mathbb{R})$. Finally we also verify that quarks and quarklets can be used to define sequences of subspaces $ W_{p,j} $ of $ L_{2}(\mathbb{R}) $ that yield non-orthogonal decompositions of $ L_{2}(\mathbb{R}) $.

math.FA↗

arXiv2022

Real eigenstructure of regular simplex tensors

We are concerned with the eigenstructure of supersymmetric tensors. Like in the matrix case, normalized tensor eigenvectors are fixed points of the tensor power iteration map. However, unless the given tensor is orthogonally decomposable, some of these fixed points may be repelling and therefore be undetectable by any numerical scheme. In this paper, we consider the case of regular simplex tensors whose symmetric decomposition is induced by an overcomplete, equiangular set of $n+1$ vectors from $\mathbb R^n$. We discuss the full real eigenstructure of such tensors, including the robustness analysis of all normalized eigenvectors. As it turns out, regular simplex tensors exhibit robust as well as non-robust eigenvectors which, moreover, only partly coincide with the generators from the symmetric tensor decomposition.

math.NA↗

arXiv2020

Newton-type method for bilevel programs with linear lower level problem and application to toll optimization

We consider a bilevel program involving a linear lower level problem with left-hand-side perturbation. We then consider the Karush-Kuhn-Tucker reformulation of the problem and subsequently build a tractable optimization problem with linear constraints by means of a partial exact penalization. A semismooth system of equations is then generated from the later problem and a Newton-type method is developed to solve it. Finally, we illustrate the convergence and practical implementation of the algorithm on the optimal toll-setting problem in transportation networks.

math.OC↗

arXiv2016

A hybrid particle-continuum resolution method and its application to a homopolymer solution

We discuss in detail a recently proposed hybrid particle-continuum scheme for complex fluids and evaluate it at the example of a confined homopolymer solution in slit geometry. The hybrid scheme treats polymer chains near the impenetrable walls as particles keeping the configuration details, and chains in the bulk region as continuous density fields. Polymers can switch resolutions on the fly, controlled by an inhomogeneous tuning function. By properly choosing the tuning function, the representation of the system can be adjusted to reach an optimal balance between physical accuracy and computational efficiency. The hybrid simulation reproduces the results of a reference particle simulation and is significantly faster (about a factor of 3.5 in our application example).

cond-mat.soft↗

arXiv2015

A globally convergent and locally quadratically convergent modified B-semismooth Newton method for $\ell_1$-penalized minimization

We consider the efficient minimization of a nonlinear, strictly convex functional with $\ell_1$-penalty term. Such minimization problems appear in a wide range of applications like Tikhonov regularization of (non)linear inverse problems with sparsity constraints. In (2015 Inverse Problems (31) 025005), a globalized Bouligand-semismooth Newton method was presented for $\ell_1$-Tikhonov regularization of linear inverse problems. Nevertheless, a technical assumption on the accumulation point of the sequence of iterates was necessary to prove global convergence. Here, we generalize this method to general nonlinear problems and present a modified semismooth Newton method for which global convergence is proven without any additional requirements. Moreover, under a technical assumption, full Newton steps are eventually accepted and locally quadratic convergence is achieved. Numerical examples from image deblurring and robust regression demonstrate the performance of the method.

math.OC↗

arXiv2015

On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.

math.PR↗

arXiv2010

Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

math.PR↗