Person# Daniel Kressner

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MATH-101(de): Analysis I (German)

Es werden die Grundlagen der Analysis sowie der Differential- und Integralrechnung von Funktionen einer reellen Veränderlichen erarbeitet.

MATH-250: Numerical analysis

Construction and analysis of numerical methods for the solution of problems from linear algebra, integration, approximation, and differentiation.

MATH-403: Low-rank approximation techniques

Low-rank approximation techniques have become a key tool in scientific computing to deal with large-scale problems and high-dimensional data. This course covers state-of-the-art algorithms and current research in this area.

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Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

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By the addition of entropic regularization, multimarginal optimal transport problems can be trans-formed into tensor scaling problems, which can be solved numerically using the multimarginal Sinkhorn algorithm. The main computational bottleneck of this algorithm is the repeated eval-uation of marginals. Recently, it has been suggested that this evaluation can be accelerated when the application features an underlying graphical model. In this work, we accelerate the computation further by combining the tensor network dual of the graphical model with additional low-rank ap-proximations. We provide an example for the color transfer between several images, in which these additional low-rank approximations save more than 96% of the computation time.

The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positive definite matrix A. In this work, we propose a mixed precision variant of LOBPCG that uses a (sparse) Cholesky factorization of A computed in lower precision as the preconditioner. To further enhance performance, a mixed precision orthogonalization strategy is proposed. To analyze the impact of reducing precision in the preconditioner on performance, we carry out a rounding error and convergence analysis of PINVIT, a simplified variant of LOBPCG. Our theoretical results predict and our numerical experiments confirm that the impact on convergence remains marginal. In practice, our mixed precision LOBPCG algorithm typically reduces the computation time by a factor of 1.4-2.0 on both CPUs and GPUs.

Daniel Kressner, Ivana Sain Glibic

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of 8-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases 8-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.