10:40
Numerical Linear Algebra (Part 3)
Chair: Chandrasekhar Venkataraman
10:40
25 mins
|
High Performance Solution of Skew-symmetric Eigenvalue Problems with Applications in Solving the Bethe-Salpeter Eigenvalue Problem
Peter Benner, Andreas Marek, Carolin Penke
Abstract: A matrix $A\in\mathbb{R}^{n\times n}$ is called skew-symmetric when $A=-A^T$. We present a direct solver for computing the eigenvalues and eigenvectors of $A$, that is highly efficient and scalable on modern compute architectures.
The symmetric eigenvalue problem, i.e. the case $A=A^T$, has been studied in depth for many years. It lies at the core of many applications in different areas such as electronic structure computations. As the involved matrices easily become extremely large when more complex systems are investigated, parallel algorithms running on supercomputers are necessary. One optimized library that provides the necessary tools is the ELPA library. It contains highly competitive direct solvers for symmetric (and Hermitian) eigenvalue problems running on distributed memory machines such as compute clusters.
The skew-symmetric case lacks the ubiquitous presence of its symmetric counterpart and has not received the same extensive treatment. We close this gap by extending the ELPA methodology to the skew-symmetric case. The motivation is to accelerate the structure-preserving solution of the Bethe-Salpeter eigenvalue problem. The solution of this problem allows a more accurate prediction of optical properties in quantum chemical systems. The resulting matrices are large and dense and call for a parallelizable and scalable algorithm. One particular method relies on efficiently solving a skew-symmetric eigenvalue problem and can be accelerated using the newly implemented method.
ELPA employs a two-step band reduction to transform a matrix to tridiagonal form and subsequently solves the tridiagonal eigenvalue problem. If the eigenvectors are required, a two-step back transformation is performed. This method is carried over to the skew-symmetric case. The reduction via Householder transformations is adapted to work on skew-symmetric matrices. An equivalent symmetric tridiagonal eigenvalue problem is solved with available methods from the ELPA package. This induces an additional step in the back transformation. The remaining back transformation of eigenvectors is performed as in the symmetric case.
|
11:05
25 mins
|
Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement
Xiujie Shan, Martin van Gijzen
Abstract: Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.
|
11:30
25 mins
|
A New Algebraically Stabilized Method for Convection-Diffusion-Reaction Equations
Petr Knobloch
Abstract: Most of the methods developed for the numerical solution of convection-dominated problems either do not suppress spurious oscillations in layer regions sufficiently, or introduce too much artificial diffusion and lead to a pronounced smearing of layers. Nevertheless, some of the algebraically stabilized methods seem not to suffer from these two deficiencies. These schemes are designed to satisfy the discrete maximum principle by construction (so that spurious oscillations cannot appear) and provide sharp approximations of layers.
Based on our previous research on algebraic flux correction schemes, we propose a new class of algebraically stabilized methods for convection-diffusion problems. We shall present a general existence and convergence theory and discuss the choice of limiters leading to the validity of the discrete maximum principle on arbitrary meshes. The theoretical findings will be illustrated by numerical results.
|
11:55
25 mins
|
Reusing Model Order Reduction Information for Accelerating Topology Optimization Iterations
Arnoud Delissen, Reinaldo Astudillo, Fred van Keulen, Martin van Gijzen, Matthijs Langelaar
Abstract: Topology optimization is a popular subfield of structural optimization, that
has become widely used in the design of continuum structures. For the design
of structures which are subjected to dynamic loads, topology optimization
requires the solution of large-scale eigenvalue problems every iteration. Hence,
it is necessary to apply model order reduction (MOR) techniques to manage the
computational complexity.
This work is focused on accelerating MOR algorithms applied to topology
optimization problems, by reusing the information obtained in previous design
iterations. We compare the performance and accuracy of two well-known MOR
algorithms: moment matching and modal truncation. In moment matching
and modal truncation, a large part of the computation is devoted to create
the so-called reduction bases by respectively solving several systems of linear
equations and calculating eigenvectors of large matrices. We propose variants
of the forementioned MOR algorithms based on the reuse of reduction bases
from previous topology optimization iterations to accelerate the computations.
Our discussion and proposed algorithms are supported by an assortment of
numerical examples.
|
|
