A mixed ℓ1\ell_1 regularization approach for sparse simultaneous approximation of parameterized PDEs

We present and analyze a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a jointly sparse reconstruction problem through the reformulation of the standard basis pursuit denoising, where the set of jointly sparse vectors is infinite. To achieve global reconstruction of sparse solutions to parameterized elliptic PDEs over both physical and parametric domains, we combine the standard measurement scheme developed for compressed sensing in the context of bounded orthonormal systems with a novel mixed-norm based $\ell_1$ regularization method that exploits both energy and sparsity. In addition, we are able to prove that, with minimal sample complexity, error estimates comparable to the best $s$-term and quasi-optimal approximations are achievable, while requiring only a priori bounds on polynomial truncation error with respect to the energy norm. Finally, we perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure

Higher order numerical methods for singular perturbation problems

Philosophiae Doctor - PhDIn recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis.South Afric

Rosenbrock-Typ-Methoden für semilineare parabolische Gleichungen

In this work we examine the viability of Rosenbrock-type time-stepping methods - specifically Rosenbrock-Wanner (ROW) methods and W-methods - for the temporal discretization of certain parabolic partial differential equations (PDEs) that have a dominating linear term (usually the Laplacian) and lower order nonlinearities, such as convective or reactive terms.In dieser Arbeit untersuchen wir die Eignung von Rosenbrock-Typ-Methoden (RTM) - speziell von Rosenbrock-Wanner- (ROW) und W-Methoden - für die zeitliche Diskretisierung von bestimmten parabolischen partiellen Differentialgleichungen (PDG), die einen dominierenden linearen Term haben (z. B. den Laplace-Operator) und Nichtlinearitäten niedrigerer Ordnung, wie z. B. konvektive oder reaktive Terme

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

Virtual Element Methods for Magnetohydrodynamics on General Polygonal and Polyhedral Meshes

The aim of this dissertation is to construct a virtual element method (VEM) for models in magneto-hydrodynamics (MHD), an area that studies the behavior and properties of electrically conducting fluids such as a plasma. MHD models are a coupling of the Maxwell’s equations for electromagnetics and models for fluid flow. First we consider a simplified resistive MHD sub-model where we assume that the fluid flow is prescribed, along with a resistive term in Ohm’s law. This approach is called Kinematics of MHD, and we use it to predict the evolution of the electric and magnetic fields. Then we consider the full coupled MHD system in two spatial dimensions (2D) where the flow is not prescribed and design another novel VEM for the discretization of Maxwell’s and Stokes’ equations. We present variational formulations for each of these models. These formulations reveal two chains of spaces where the exact solutions lie. Our study focuses on developing discrete versions of these chains in both two (2D) and three (3D) spatial dimensions for MHD Kinematics and in 2D for the full MHD system. By defining a series of computable projectors, each of the terms in the continuous problem are approximated. In all our studies we present analysis of the stability of the VEM method by exploiting well-known techniques from the theory of saddle-point problems. The VEMs developed can be implemented on a very general class of polygonal/polyhedral meshes. Moreover, these methods are guaranteed to preserve the divergence of the magnetic field at the discrete level. In the last chapter, we present a study of opinion dynamics applied specifically to debates between legislators, which forms the topic for an interdisciplinary chapter requirement for the NRT program in ”Risk and Uncertainty Quantification in the Marine Sciences”. The context of the study is the preservation of cultural keystone species (CKS) that are part of the core of indigeneous peoples culture. In this chapter, we explore how we can use mathematical modeling to design strategies to influence legislation that supports the protection of CKS