Michael Lülfesmann

Interactively exploring elimination orderings in symbolic sparse Cholesky factorization

When large sparse symmetric systems of linear equations are solved by the Cholesky factorization, nonzero elements can be generated at positions where the original matrix contains zero elements. This phenomenon is called fill-in and it is often crucial in large-scale problems. The symbolic Cholesky factorization solely takes into account the nonzero structure of a sparse matrix to determine the nonzero structure of its Cholesky factor. Sequences of elimination graphs are typically used to model this combinatorial problem. We propose an interactive educational module to visualize and explore the symbolic Cholesky factorization in terms of both elimination graphs and matrix representation. We describe the design and implementation of this interactive module that is intended to be used in a face-to-face learning environment.

On the Influence of Constitutive Models on Shape Optimization for Artificial Blood Pumps

We report on a shape optimization framework that couples a highlyparallel finite element solver with a geometric kernel and different optimization algorithms. The entire optimization framework is transformed with automatic differentiation techniques, and the derivative code is employed to compute derivatives of the optimal shapes with respect to viscosity. This methodology provides a powerful tool to investigate the necessity of intricate constitutive models by taking derivatives with respect to model parameters

Partial Jacobian computation in the domain-specific program transformation system ADiCape

Sensitivities of functions given in the form of computer models are crucial in various areas of computational science and engineering. We consider computer models written in CapeML, a domain-specific XML-based language used in process engineering. Rather than computing all nonzero entries of a sparse Jacobian matrix, we are interested in obtaining only a subset of these entries. For the solution of this problem called partial Jacobian computation, we transform a CapeML model of an industrial distillation column using the automatic differentiation system ADiCape.

Graphfärbung zur Berechnung benötigter Matrixelemente

ZusammenfassungProbleme im Bereich der Informatik können oft als Graphprobleme modelliert werden. Dieses gilt auch für viele kombinatorische Probleme im wissenschaftlichen Rechnen. In der numerischen linearen Algebra sind Graphen insbesondere dann allgegenwärtig, wenn die zugrunde liegenden Matrizen dünnbesetzt sind. Im Folgenden soll nun ein spezielles Problem als Graphfärbungsproblem betrachtet werden, nämlich das partielle (teilweise) Berechnen einer Jacobi-Matrix.

Sub-exponential graph coloring algorithm for stencil-based Jacobian computations

Abstract Partial differential equations can be discretized using a regular Cartesian grid and a stencil-based method to approximate the partial derivatives. The computational effort for determining the associated Jacobian matrix can be reduced. This reduction can be modeled as a (grid) coloring problem. Currently, this problem is solved by using a heuristic approach for general graphs or by developing a formula for every single stencil. We introduce a sub-exponential algorithm using the Lipton–Tarjan separator in a divide-and-conquer approach to compute an optimal coloring. The practical relevance of the algorithm is evaluated when compared with an exponential algorithm and a greedy heuristic.