A Robust Multigrid Solver for Isogeometric Analysis Based on Multiplicative Schwarz Smoothers

The design of fast solvers for isogeometric analysis is receiving a lot of attention due to the challenge that offers to find an algorithm with a robust convergence with respect to the spline degree. Here, we analyze the application of geometric multigrid methods to this type of discretization, and we propose a multigrid approach based on overlapping multiplicative Schwarz methods as smoothers. The size of the blocks considered within these relaxation procedures is adapted to the spline degree. A simple multigrid V-cycle with only one step of presmoothing results in a very efficient algorithm, whose convergence is independent on the spline degree and the spatial discretization parameter. Local Fourier analysis is shown to be very useful for the understanding of the problems encountered in the design of a robust multigrid method for IGA, and it is performed to support the good convergence properties of the proposed solver. In fact, an analysis for any spline degree and an arbitrary size of the blocks within the Schwarz smoother is presented for the one-dimensional case. The efficiency of the solver is also demonstrated through several numerical experiments, including a two-dimensional problem on a nontrivial computational domain

Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration $(X_h)$ via a sort of repeated squaring. The equations we consider are Stein equations $X - A^*XA=Q$, Lyapunov equations $A^*X+XA+Q=0$, discrete-time algebraic Riccati equations $X=Q+A^*X(I+GX)^{-1}A$, continuous-time algebraic Riccati equations $Q+A^*X+XA-XGX=0$, palindromic quadratic matrix equations $A+QY+A^*Y^2=0$, and nonlinear matrix equations $X+A^*X^{-1}A=Q$. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.Comment: Review article for GAMM Mitteilunge

Nonlocal Myriad Filters for Cauchy Noise Removal

The contribution of this paper is two-fold. First, we introduce a generalized myriad filter, which is a method to compute the joint maximum likelihood estimator of the location and the scale parameter of the Cauchy distribution. Estimating only the location parameter is known as myriad filter. We propose an efficient algorithm to compute the generalized myriad filter and prove its convergence. Special cases of this algorithm result in the classical myriad filtering, respective an algorithm for estimating only the scale parameter. Based on an asymptotic analysis, we develop a second, even faster generalized myriad filtering technique. Second, we use our new approaches within a nonlocal, fully unsupervised method to denoise images corrupted by Cauchy noise. Special attention is paid to the determination of similar patches in noisy images. Numerical examples demonstrate the excellent performance of our algorithms which have moreover the advantage to be robust with respect to the parameter choice

Computational Inverse Problems

Inverse problem typically deal with the identification of unknown quantities from indirect measurements and appear in many areas in technology, medicine, biology, finance, and econometrics. The computational solution of such problems is a very active, interdisciplinary field with close connections to optimization, control theory, differential equations, asymptotic analysis, statistics, and probability. The focus of this workshop was on hybrid methods, model reduction, regularization in Banach spaces, and statistical approaches

Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective

The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of problems, e.g., nonnegativity or sparsity-constrained factorization, we take a {\it top-down} approach: we start with general optimization theory (e.g., inexact and accelerated block coordinate descent, stochastic optimization, and Gauss-Newton methods) that covers a wide range of factorization problems with diverse constraints and regularization terms of engineering interest. Then, we go `under the hood' to showcase specific algorithm design under these introduced principles. We pay a particular attention to recent algorithmic developments in structured tensor and matrix factorization (e.g., random sketching and adaptive step size based stochastic optimization and structure-exploiting second-order algorithms), which are the state of the art---yet much less touched upon in the literature compared to {\it block coordinate descent} (BCD)-based methods. We expect that the article to have an educational values in the field of structured factorization and hope to stimulate more research in this important and exciting direction.Comment: Final Version; to appear in IEEE Signal Processing Magazine; title revised to comply with the journal's rul

Dichtematrix-Renormierung, angewandt auf nichtlineare dynamische Systeme

Bogner T. Density matrix renormalisation applied to nonlinear dynamical systems. Bielefeld (Germany): Bielefeld University; 2007.In dieser Dissertation wird die effektive numerische Beschreibung nichtlinearer dynamischer Systeme untersucht. Systeme dieser Art tauchen praktisch überall auf, wo zeitabhängige Größen quantitativ untersucht werden, d.h. in fast allen Bereichen der Physik, aber auch in der Biologie, Ökonomie oder Mathematik. Ziel ist die Bestimmung reduzierter Modelle, deren Phasenraum eine signifikant reduzierte Dimensionalität aufweist. Dies wird erreicht durch Benutzung von Konzepten aus der Dichtematrix-Renormierung. In dieser Arbeit werden drei neue Anwendungen vorgeschlagen. Zuerst wird eine Dichtematrix-Renormierungsmethode zur Berechnung einer Schur-Zerlegung vorgestellt. Verglichen mit bereits existierenden Arbeiten liegt der Vorteil dieses Ansatzes in der Möglichkeit, auch für nicht-normale Operatoren orthonormale Basen von sukzessive invarianten Unterräumen zu bestimmen. Der Algorithmus wird dann angewandt auf Gittermodelle stochastischer Systeme, wobei als Beispiele ein Reaktions-Diffusions- und ein Oberflächenablagerungs-Modell dienen. Als Nächstes wird ein Dichtematrix-Renormierungsansatz für die orthogonale Zerlegung (proper orthogonal decomposition) entwickelt. Diese Zerlegung erlaubt die Bestimmung relevanter linearer Unterräume auch für nichtlineare Systeme. Durch die Verwendung der Dichtematrix-Renormierung werden alle Berechnungen nur für kleine Untersysteme durchgeführt. Dabei werden diskretisierte partielle Differentialgleichungen, d.h. die Diffusionsgleichung, die Burgers-Gleichung und eine nichtlineare Diffusionsgleichung als numerische Beispiele betrachtet. Schließlich wird das vorige Konzept auf höherdimensionale Probleme in Form eines Variationsverfahrens erweitert. Dies Verfahren wird dann an den zweidimensionalen Navier-Stokes-Gleichungen erprobt.In this work the effective numerical description of nonlinear dynamical systems is investigated. Such systems arise in most fields of physics, as well as in mathematics, biology, economy and essentially in all problems for which a quantitative description of a time evolution is considered. The aim is to find reduced models with a phase space of significantly reduced dimensionality. This is achieved by the use of concepts from density matrix renormalisation. Three new applications are proposed in this work. First, a density matrix renormalisation method for calculating a Schur decomposition is introduced. The advantage of this approach, compared to existing work, is the possibility to obtain orthonormal bases for successively invariant subspaces even if the generator of evolution is not normal. The algorithm is applied to lattice models for stochastic systems, namely a reaction diffusion and a surface deposition model. Next, a density matrix renormalisation approach to the proper orthogonal decomposition is developed. This allows the determination of relevant linear subspaces even for nonlinear systems. Due to the use of density matrix renormalisation concepts, all calculations are done on small subsystems. Here discretised partial differential equations, i.e. the diffusion equation, the Burgers equation and a nonlinear diffusion equation are considered as numerical examples. Finally, the previous concept is extended to higher dimensional problems in a variational form. This method is then applied to the two-dimensional, incompressible Navier-Stokes equations as testing ground