Thorsten Raasch

Universität Siegen · Department of Mathematics

This paper is concerned with near-optimal approximation of a given univariate function 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 conce...

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 regularized Newton system of equations is then generat...

In this paper, 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 [Formula: see text] of [Formula: see text] which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters, they even s...

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 assumpt...

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 prope...

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 regularized Newton system of equations is then generat...

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 pape...

This paper is concerned with new discretization methods for the numerical treatment of elliptic partial differential equations. We derive an adaptive approximation scheme that is based on frames of quarkonial type, which can be interpreted as a wavelet version of $hp$ finite element dictionaries. These new frames are constructed from a finite set o...

This paper is concerned with approximation properties of polynomially enriched wavelet systems, so-called quarklet frames. We show that certain model singularities that arise in elliptic boundary value problems on polygonal domains can be approximated from the span of such quarklet systems at inverse-exponential rates. In order to realize these, we...

In the spirit of subatomic or quarkonial decomposition of function spaces (Triebel in Fractals and spectra related to fourier analysis and function spaces. Birkhäuser, Boston, 1997), we construct compactly supported, piecewise polynomial functions whose properly weighted dilates and translates generate frames for Sobolev spaces on the real line. Al...

We report on the first application of model reduction techniques to the numerical simulation of self-consistent field (SCF) models for polymer melts at equilibrium. As a complement to the classical SCF iteration, the parametric reaction-diffusion equations for the chain propagators are treated by a reduced basis (RB) approximation from spaces of ba...

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 field...

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...

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 metho...

This paper is concerned with the development of numerical schemes for the minimization of functionals involving sparsity constraints and nonconvex fidelity terms. These functionals appear in a natural way in the context of Tikhonov regularization of nonlinear inverse problems with ℓ1 penalty terms. Our method of minimization is based on a generaliz...

We are concerned with Tikhonov regularization of linear ill-posed problems with l1 coefficient penalties. Griesse and Lorenz (2008 Inverse Problems 24 035007) proposed a semismooth Newton method for the efficient minimization of the corresponding Tikhonov functionals. In the class of high-precision solvers for such problems, semismooth Newton metho...

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...

This paper is concerned with the convergence analysis of the horizontal method of lines for evolution equations of the parabolic type. Following a semidiscretization in time by \(S\)-stage one-step methods, the resulting elliptic stage equations per time step are solved with adaptive space discretization schemes. We investigate how the tolerances i...

We review a series of results that have been obtained in the context of the DFG-SPP 1324 project “Adaptive wavelet methods for SPDEs”. This project has been concerned with the construction and analysis of adaptive wavelet methods for second order parabolic stochastic partial differential equations on bounded, possibly nonsmooth domains O ⊂ ℝd. A de...

We discuss the connection between the theory of quarkonial decompositions for function spaces developed by Hans Triebel, and the multilevel partition of unity method. The central result is an alternative approach to the stability of quarkonial decompositions in Besov spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty...

We extend the qualitative reconstruction method for inverse source problems for time-harmonic acoustic and electromagnetic waves in free space, recently developed in the first part [R. Griesmaier, M. Hanke and T. Raasch, SIAM J. Sci. Comput. 34, No. 3, A1544–A1562 (2012; Zbl 1251.35187)], to a relevant three-dimensional setting. The reconstruction...

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ℝd . The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an...

A new approach to determining the unit-cell vectors from single-crystal diffraction data based on clustering analysis is proposed. The method uses the density-based clustering algorithm DBSCAN. Unit-cell determination through the clustering procedure is particularly useful for limited tilt sequences and noisy data, and therefore is optimal for sing...

In this paper, we combine concepts from two different mathematical research topics: adaptive wavelet techniques for well-posed problems and regularization theory for nonlinear inverse problems with sparsity constraints. We are concerned with identifying certain parameters in a parabolic reaction-diffusion equation from measured data. Analytical pro...

We are concerned with the efficient numerical solution of minimization problems in Hilbert spaces involving sparsity constraints. These optimizations arise, e.g., in the context of inverse problems. In this work we analyze an efficient variant of the well-known iterative soft-shrinkage algorithm for large or even infinite dimensional problems. This...

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 approxi...

This paper is concerned with the numerical treatment of inverse heat conduction problems. In particular, we combine recent results on the regularization of ill-posed problems by iterated soft shrinkage with adaptive wavelet algorithms for the forward problem. The analysis is applied to an inverse parabolic problem that stems from the industrial pro...

This article is concerned with adaptive numerical frame methods for elliptic operator equations. We show how specific noncanonical frame expansions on domains can be constructed. Moreover, we study the approximation order of best n-term frame approximation, which serves as the benchmark for the performance of adaptive schemes. We also discuss numer...

We are concerned with the efficient numerical solution of minimization problems in Hilbert spaces involving sparsity constraints. These optimizations arise, e.g., in the context of inverse problems. In this work we analyze a very efficient variant of the well-known iterative soft-shrinkage algorithm for large or even infinite dimensional problems....

We are concerned with the numerical solution of linear parameter identification problems for parabolic PDE, written as an operator equation $Ku=f$. The target object $u$ is assumed to have a sparse expansion with respect to a wavelet system $Psi={psi_lambda}$ in space-time, being equivalent to a priori information on the regularity of $u=mathbf u^...

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfa...

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The sch...

This thesis is concerned with the application of wavelet methods to the adaptive numerical solution of elliptic and parabolic operator equations over a polygonal domain. Driven by the insight that the construction of wavelet bases on more general domains is complicated and may pose stability problems, we analyze the option to replace the concept of...

We discuss the use of Gelfand frames for the adaptive numerical solution of linear elliptic operator equations. After a transformation into an equivalent infinite–dimensional system Lu = f in frame coordinates, the operator equation can be solved within a prescribed target accuracy by approximate Richardson or gradient iteration schemes. (© 2005 WI...