Übungen zur Vorlesung Theoretische Physik III: Elektrodynamik/Computergestützte Elektrodynamik im Wintersemester 2003/04 für den Studiengang Physik und den Bakkalaureusstudiengang Computational Science

Übungen zur Vorlesung Theoretische Physik III: Elektrodynamik/Computergestützte Elektrodynamik im Wintersemester 2004/05 für den Studiengang Physik und den Bakkalaureusstudiengang Computational Science

The rank revealing QR decomposition is a special form of the well known QR decomposition of a matrix. It uses specialized pivoting strategies and allows for an easy and efficient numerical rank decision for arbitrary matrices. It is especially valuable when column compression of rectangular matrices needs to be performed. Here we provide documentation and compilation instructions for a MATLAB MEX implementation of the RRQR allowing the easy usage of this decomposition inside the MATLAB environment.

We address the important field of large scale matrix based algorithms in control and model order reduction. Many important tools from theory and applications in systems theory have been widely ignored during the recent decades in the context of PDE constraint optimal control problems and simulation of electric circuits. Often this is due to the fact that large scale matrices are suspected to be unsolvable in large scale applications. Since around 2000 efficient low rank theory for matrix equation solvers exists for sparse and also data sparse systems. Unfortunately upto now only incomplete or experimental Matlab implementations of most of these solvers have existed. Here we aim on the implementation of these algorithms in a higher programming language (in our case C) that allows for a high performance solver for many matrix equations arising in the context of large scale standard and generalized state space systems. We especially focus on efficient memory saving data structures and implementation techniques as well as the shared memory parallelization of the underlying algorithms.

In this thesis, we study several nonlocal models to obtain their numerical solutions accurately and efficiently. In contrast to the classical (local) partial differential equation models, these nonlocal models are integro-differential equations that do not contain spatial derivatives. As a result, these nonlocal models allow their solutions to have discontinuities. Hence, they can be widely used for fracture problems and anisotropic problems. This thesis mainly includes two parts. The first part focuses on presenting accurate and efficient numerical methods. In this part, we first introduce three meshless methods including two global schemes, namely the radial basis functions collocation method (RBFCM) and the radial ba- sis functions-based pseudo-spectral method (RBF-PSM) and a localized scheme, namely the localized radial basis functions-based pseudo-spectral method (LRBF-PSM), which also gives the development process of the RBF methods from global to local. The comparison of these methods shows that LRBF-PSM not only avoids the Runge phenomenon but also has similar accuracy to the global scheme. Since the LRBF-PSM uses only a small subset of points, the calculation consumes less CPU time. Afterwards, we improve this scheme by adding enrichment functions so that it can be effectively applied to discontinuity problems. This thesis abbreviates this enriched method as LERBF-PSM (Localized enriched radial basis functions-based pseudo-spectral method). In the second part, we focus on applying the derived methods from the first part to nonlocal topics of current research, including nonlocal diffusion models, linear peridynamic models, parabolic/hyperbolic nonlocal phase field models, and nonlocal nonlinear Schrödinger equations arising in quantum mechanics. The first point worth noting is that in order to verify the meshless nature of LRBF-PSM, we apply this method to solve a two-dimensional steady-state continuous peridynamic model in regular, irregular (L-shaped and Y-shaped) domains with uniform and non-uniform discretizations and even extend this method to three dimensions. It is also worth noting that before solving nonlinear nonlocal Schrödinger equations, according to the property of the convolution, these partial integro-differential equations are transformed into equivalent or approximate partial differential equations (PDEs) in the whole space and then the LRBF-PSM is used for the spatial discretization in a finite domain with suitable boundary conditions. Therefore, the solutions can be quickly approximated.

This thesis considers classical methods to solve stochastic control problems and valuation problems from financial mathematics numerically. To this end, (linear) partial differential equations (PDEs) in non-divergence form or the optimality conditions known as the (nonlinear) Hamilton-Jacobi-Bellman (HJB) equations are solved by means of finite differences, volumes and elements. We consider all of these three approaches in detail after a thorough introduction to stochastic control problems and discuss various solution terms including classical solutions, strong solutions, weak solutions and viscosity solutions. A particular role in this thesis play degenerate problems. Here, a new model for the optimal control of an energy storage facility is developed which extends the model introduced in [Chen, Forsyth (2007)]. This four-dimensional HJB equation is solved by the classical finite difference Kushner-Dupuis scheme [Kushner, Dupuis (2001)] and a semi-Lagrangian variant which are both discussed in detail. Additionally, a convergence proof of the standard scheme in the setting of parabolic HJB equations is given. Finite volume schemes are another classical method to solve partial differential equations numerically. Sharing similarities to both finite difference and finite element schemes we develop a vertex-centered dual finite volume scheme. We discuss convergence properties and apply the scheme to the solution of HJB equations, which has not been done in such a broad context, to the best of our knowledge. Astonishingly, this is one of the first times the finite volume approach is systematically discussed for the solution of HJB equations. Furthermore, we give many examples which show advantages and disadvantages of the approach. Finally, we investigate novel tailored non-conforming finite element approximations of second-order PDEs in non-divergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study approximations with both continuous and discontinuous trial functions. Of particular interest are a-priori and a-posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature. We discuss implementations of all three approaches in MATLAB (finite differences and volumes) and FEniCS (finite elements) publicly available in GitHub repositories under https://github.com/janblechschmidt. Many numerical experiments show convergence properties as well as pros and cons of the respective approach. Additionally, a new postprocessing procedure for policies obtained from numerical solutions of HJB equations is developed which improves the accuracy of control laws and their incurred values.

Diese Arbeit behandelt die effiziente Vorkonditionierung von Finite-Elemente-Matrizen unter Verwendung hierarchischer Matrizen.

Optimierungsaufgaben unter Partiellen Differentialgleichungen (PDGLs) tauchen in verschiedensten Anwendungen der Wissenschaft und Technik auf. Wenn wir ein PDGL Problem formulieren, kann es aufgrund seiner Größe unmöglich werden, das Problem mit konventionellen Methoden zu lösen. Zusätzlich noch eine Optimierung auszuführen birgt zusätzliche Schwierigkeiten. In vielen Fällen können wir das PDGL Problem in einem kompakteren Format formulieren indem wir der zugrundeliegenden Kronecker-Produkt Struktur zwischen Raum- und Zeitdimension Aufmerksamkeit schenken. Wenn die PDGL zusätzlich mit Isogeometrischer Analysis diskretisiert wurde, können wir zusätlich eine Niedrig-Rang Approximation zwischen den einzelnen Raumdimensionen erzeugen. Diese Niedrig-Rang Approximation lässt uns die Systemmatrizen schnell und speicherschonend aufstellen. Das folgende PDGL-Problem lässt sich als Summe aus Kronecker-Produkten beschreiben, welche als eine Niedrig-Rang Tensortrain Formulierung interpretiert werden kann. Diese kann effizient im Niedrig-Rang Format gelöst werden. Wir illustrieren dies mit unterschiedlichen, anspruchsvollen Beispielproblemen.:Introduction Tensor Train Format Isogeometric Analysis PDE-constrained Optimization Bayesian Inverse Problems A low-rank tensor method for PDE-constrained optimization with Isogeometric Analysis A low-rank matrix equation method for solving PDE-constrained optimization problems A low-rank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis Theses and Summary Bibilography / Optimization problems governed by Partial Differential Equations (PDEs) arise in various applications of science and engineering. If we formulate a discretization of a PDE problem, it may become infeasible to treat the problem with conventional methods due to its size. Solving an optimization problem on top of the forward problem poses additional difficulties. Often, we can formulate the PDE problem in a more compact format by paying attention to the underlying Kronecker product structure between the space and time dimension of the discretization. When the PDE is discretized with Isogeometric Analysis we can additionally formulate a low-rank representation with Kronecker products between its individual spatial dimensions. This low-rank formulation gives rise to a fast and memory efficient assembly for the system matrices. The PDE problem represented as a sum of Kronecker products can then be interpreted as a low-rank tensor train formulation, which can be efficiently solved in a low-rank format. We illustrate this for several challenging PDE-constrained problems.:Introduction Tensor Train Format Isogeometric Analysis PDE-constrained Optimization Bayesian Inverse Problems A low-rank tensor method for PDE-constrained optimization with Isogeometric Analysis A low-rank matrix equation method for solving PDE-constrained optimization problems A low-rank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis Theses and Summary Bibilography

Dissertation ueber die effektive Vorkonditionierung linearer Gleichungssysteme resultierend aus der Diskretisierung eines elliptischen Randwertproblems 2. Ordnung mittels der Methode der Finiten Elementen. Als Vorkonditionierer werden multi-level artige Vorkonditionierer (BPX, Multi-grid, Wavelets) benutzt.

In this paper, finite element discretizations of the degenerated operator -ω²(y) u_xx-ω²(x)u_yy=g in the unit square are investigated, where the weight function satisfies ω(ξ)=ξ^α with α ≥ 0. We propose two multi-level methods in order to solve the resulting system of linear algebraic equations. The first method is a multi-grid algorithm with line-smoother. A proof of the smoothing property is given. The second method is a BPX-like preconditioner which we call MTS-BPX preconditioner. We show that the upper eigenvalue bound of the MTS-BPX preconditioned system matrix grows proportionally to the level number.