Subspace acceleration for large-scale parameter-dependent Hermitian eigenproblems *

Abstract This work is concerned with approximating the smallest eigenvalue of a parameterdependent Hermitian matrix A(µ) for many parameter values µ in a domain D ⊂ R P . The design of reliable and efficient algorithms for addressing this task is of importance in a variety of applications. Most notably, it plays a crucial role in estimating the error of reduced basis methods for parametrized partial differential equations. The current state-of-the-art approach, the so called Successive Constraint Method (SCM), addresses affine linear parameter dependencies by combining sampled Rayleigh quotients with linear programming techniques. In this work, we propose a subspace approach that additionally incorporates the sampled eigenvectors of A(µ) and implicitly exploits their smoothness properties. Like SCM, our approach results in rigorous lower and upper bounds for the smallest eigenvalues on D. Theoretical and experimental evidence is given to demonstrate that our approach represents a significant improvement over SCM in the sense that the bounds are often much tighter, at negligible additional cost

MATHICSE Technical Report : Multi-index stochastic collocation for random PDEs

In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most eective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more eective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi- Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods

MATHICSE Technical Report : Subspace acceleration for large-scale parameter-dependent Hermitian eigenproblems

This work is concerned with approximating the smallest eigenvalue of a parameter-dependent Hermitian matrix A(μ) for many parameter values μ ∈ RP. The design of reliable and efficient algorithms for addressing this task is of importance in a variety of applications. Most notably, it plays a crucial role in estimating the error of reduced basis methods for parametrized partial differential equations. The current state-of-the-art approach, the so called Successive Constraint Method (SCM), addresses affine linear parameter dependencies by combining sampled Rayleigh quotients with linear programming techniques. In this work, we propose a subspace approach that additionally incorporates the sampled eigenvectors of A(μ) and implicitly exploits their smoothness properties. Like SCM, our approach results in rigorous lower and upper bounds for the smallest eigenvalues on D. Theoretical and experimental evidence is given to demonstrate that our approach represents a significant improvement over SCM in the sense that the bounds are often much tighter, at negligible additional cost

Low-rank methods for parameter-dependent eigenvalue problems and matrix equations

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