The calculation of the distance to a nearby defective matrix

In this paper a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam & Bora (Linear Algebra Appl., 396 (2005), pp.~273--301) and reduces to finding when a parameter-dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the Implicit Determinant Method introduced by Spence & Poulton (J. Comput. Phys., 204 (2005), pp.~65--81). Numerical results for several examples illustrate the performance of the algorithm.Comment: 12 page

Low-rank solutions to the stochastic Helmholtz equation

In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtain an advantage in terms of storage and CPU time compared to computations in full rank. We also propose a general approach to implement a preconditioner using the low-rank format efficiently

Balanced truncation and singular perturbation approximation model order reduction for stochastically controlled linear systems

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive L'evy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory

Time-limited Balanced Truncation for Data Assimilation Problems

Balanced truncation is a well-established model order reduction method which has been applied to a variety of problems. Recently, a connection between linear Gaussian Bayesian inference problems and the system-theoretic concept of balanced truncation has been drawn. Although this connection is new, the application of balanced truncation to data assimilation is not a novel idea: it has already been used in four-dimensional variational data assimilation (4D-Var). This paper discusses the application of balanced truncation to linear Gaussian Bayesian inference, and, in particular, the 4D-Var method, thereby strengthening the link between systems theory and data assimilation further. Similarities between both types of data assimilation problems enable a generalisation of the state-of-the-art approach to the use of arbitrary prior covariances as reachability Gramians. Furthermore, we propose an enhanced approach using time-limited balanced truncation that allows to balance Bayesian inference for unstable systems and in addition improves the numerical results for short observation periods.Comment: 24 pages, 5 figure

Balanced truncation and singular perturbation approximation model order reduction for stochastically controlled linear systems

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory

Sparse grid based Chebyshev HOPGD for parameterized linear systems

We consider approximating solutions to parameterized linear systems of the form $A(\mu_1,\mu_2) x(\mu_1,\mu_2) = b$, where $(\mu_1, \mu_2) \in \mathbb{R}^2$. Here the matrix $A(\mu_1,\mu_2) \in \mathbb{R}^{n \times n}$ is nonsingular, large, and sparse and depends nonlinearly on the parameters $\mu_1$ and $\mu_2$. Specifically, the system arises from a discretization of a partial differential equation and $x(\mu_1,\mu_2) \in \mathbb{R}^n$, $b \in \mathbb{R}^n$. This work combines companion linearization with the Krylov subspace method preconditioned bi-conjugate gradient (BiCG) and a decomposition of a tensor matrix of precomputed solutions, called snapshots. As a result, a reduced order model of $x(\mu_1,\mu_2)$ is constructed, and this model can be evaluated in a cheap way for many values of the parameters. The decomposition is performed efficiently using the sparse grid based higher-order proper generalized decomposition (HOPGD), and the snapshots are generated as one variable functions of $\mu_1$ or of $\mu_2$. Tensor decompositions performed on a set of snapshots can fail to reach a certain level of accuracy, and it is not possible to know a priori if the decomposition will be successful. This method offers a way to generate a new set of solutions on the same parameter space at little additional cost. An interpolation of the model is used to produce approximations on the entire parameter space, and this method can be used to solve a parameter estimation problem. Numerical examples of a parameterized Helmholtz equation show the competitiveness of our approach. The simulations are reproducible, and the software is available online