Matrix differential equation | EPFL Graph Search

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For example, a first-order matrix ordinary differential equation is : \mathbf{\dot{x}}(t) = \mathbf{A}(t)\mathbf{x}(t) where \mathbf{x}(t) is an n \times 1 vector of functions of an underlying variable t, \mathbf{\dot{x}}(t) is the vector of first derivatives of these functions, and \mathbf{A}(t) is an n \times n matrix of coefficients. In the case where \mathbf{A} is constant and has n linearly independent eigenvectors, this differential equation has the following general solution, : \mathbf{x}(t) = c_1 e^{\lambda_1 t} \mathbf{u}_1 + c_2 e^{\l

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

Petar Sirkovic

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.

EPFL2016

On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes

Stefano Massei

Matrix equations of the kind A(1)X(2)+A(0)X+A(-1)=X, where both the matrix coefficients and the unknown are semi-infinite matrices belonging to a Banach algebra, are considered. These equations, where coefficients are quasi-Toeplitz matrices, are encountered in certain quasi-birth-death processes as the tandem Jackson queue or in any other processes that can be modeled as a reflecting random walk in the quarter plane. We provide a numerical framework for approximating the minimal nonnegative solution of these equations that relies on semi-infinite quasi-Toeplitz matrix arithmetic. In particular, we show that the algorithm of cyclic reduction can be effectively applied and can approximate the infinite-dimensional solutions with quadratic convergence at a cost that is comparable to that of the finite case. This way, we may compute a finite approximation of the sought solution and of the invariant probability measure of the associated quasi-birth-death process, within a given accuracy. Numerical experiments, performed on a collection of benchmarks, confirm the theoretical analysis.

WILEY2018

MATHICSE Technical Report : Fast solvers for 2D fractional differential equations using rank structured matrices

Stefano Massei, Marco Mazza

We consider the discretization of time-space diffusion equations with fractional derivatives in space and either 1D or 2D spatial domains. The use of implicit Euler scheme in time and finite differences or finite elements in space, leads to a sequence of dense large scale linear systems describing the behavior of the solution over a time interval. We prove that the coefficient matrices arising in the 1D context are rank structured and can be efficiently represented using hierarchical formats (H-matrices, HODLR). Quantitative estimates for the rank of the off-diagonal blocks of these matrices are presented. We analyze the use of HODLR arithmetic for solving the 1D case and we compare this strategy with existing methods that exploit the Toeplitz-like structure to precondition the GMRES iteration. The numerical tests demonstrate the convenience of the HODLR format when at least a reasonably low number of time steps is needed. Finally, we explain how these properties can be leveraged to design fast solvers for problems with 2D spatial domains that can be reformulated as matrix equations. The experiments show that the approach based on the

MATHICSE2018