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Person# Petar Sirkovic

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For studying spectral properties of a non-normal matrix A ∈ Cn×n, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε-pseudospectra, that is the level-sets of the resolvent norm function g(z) = ‖(zI − A)−1‖2. The computation of ε-pseudospectra requires determining the smallest singular values σmin(zI − A) on for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation that provides highly accurate estimates of pseudospectra in the region of interest. It incorporates the sampled singular vectors of zI − A for different values of z and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. We also present a comparison of our approach to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.

For studying spectral properties of a nonnormal matrix A is an element of Cnxn, information about its spectrum sigma(A) alone is usually not enough. Effects of perturbations on sigma(A) can be studied by computing epsilon-pseudospectra, i.e. the level sets of the resolvent norm function g(z)=||(zI-A)-1||2. The computation of epsilon-pseudospectra requires determining the smallest singular values sigma min(zI-A) for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A. It incorporates the sampled singular vectors of zI - A for different values of z, and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.

The focus of this thesis is on developing efficient algorithms for two important problems arising in model reduction, estimation of the smallest eigenvalue for a parameter-dependent Hermitian matrix and solving large-scale linear matrix equations, by extracting and exploiting underlying low-rank properties. Availability of reliable and efficient algorithms for estimating the smallest eigenvalue of a parameter-dependent Hermitian matrix $A(\mu)$ for many parameter values $\mu$ is important in a variety of applications. Most notably, it plays a crucial role in \textit{a posteriori} estimation of reduced basis methods for parametrized partial differential equations. We propose a novel subspace approach, which builds upon the current state-of-the-art approach, the Successive Constraint Method (SCM), and improves it by additionally incorporating the sampled smallest eigenvectors and implicitly exploiting their smoothness properties. Like SCM, our approach also provides rigorous lower and upper bounds for the smallest eigenvalues on the parameter domain $D$. We present theoretical and experimental evidence to demonstrate that our approach represents a significant improvement over SCM in the sense that the bounds are often much tighter, at a negligible additional cost. We have successfully applied the approach to computation of the coercivity and the inf-sup constants, as well as computation of $\varepsilon$-pseudospectra. Solving an $m\times n$ linear matrix equation $A_1 X B_1^T + \cdots + A_K X B_K^T = C$ as an $m n \times m n$ linear system, typically limits the feasible values of $m,n$ to a few hundreds at most. We propose a new approach, which exploits the fact that the solution $X$ can often be well approximated by a low-rank matrix, and computes it by combining greedy low-rank techniques with Galerkin projection as well as preconditioned gradients. This can be implemented in a way where only linear systems of size $m \times m$ and $n \times n$ need to be solved. Moreover, these linear systems inherit the sparsity of the coefficient matrices, which allows to address linear matrix equations as large as $m = n = O(10^5)$. Numerical experiments demonstrate that the proposed methods perform well for generalized Lyapunov equations, as well as for the standard Lyapunov equations. Finally, we combine the ideas used for addressing matrix equations and parameter-dependent eigenvalue problems, and propose a low-rank reduced basis approach for solving parameter-dependent Lyapunov equations.

Related research domains (8)

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

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

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is