Matrix-equation-based strategies for convection-diffusion equations

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on the matrix equation formulation of the problem are proposed, which naturally approximate the original discretized problem. For certain types of convection coefficients, we show that the explicit solution of the matrix equation can effectively replace the linear system solution. Numerical experiments with data stemming from two and three dimensional problems are reported, illustrating the potential of the proposed methodology

Numerical methods for large-scale Lyapunov equations with symmetric banded data

The numerical solution of large-scale Lyapunov matrix equations with symmetric banded data has so far received little attention in the rich literature on Lyapunov equations. We aim to contribute to this open problem by introducing two efficient solution methods, which respectively address the cases of well conditioned and ill conditioned coefficient matrices. The proposed approaches conveniently exploit the possibly hidden structure of the solution matrix so as to deliver memory and computation saving approximate solutions. Numerical experiments are reported to illustrate the potential of the described methods

Solving rank structured Sylvester and Lyapunov equations

We consider the problem of efficiently solving Sylvester and Lyapunov equations of medium and large scale, in case of rank-structured data, i.e., when the coefficient matrices and the right-hand side have low-rank off-diagonal blocks. This comprises problems with banded data, recently studied by Haber and Verhaegen in "Sparse solution of the Lyapunov equation for large-scale interconnected systems", Automatica, 2016, and by Palitta and Simoncini in "Numerical methods for large-scale Lyapunov equations with symmetric banded data", SISC, 2018, which often arise in the discretization of elliptic PDEs. We show that, under suitable assumptions, the quasiseparable structure is guaranteed to be numerically present in the solution, and explicit novel estimates of the numerical rank of the off-diagonal blocks are provided. Efficient solution schemes that rely on the technology of hierarchical matrices are described, and several numerical experiments confirm the applicability and efficiency of the approaches. We develop a MATLAB toolbox that allows easy replication of the experiments and a ready-to-use interface for the solvers. The performances of the different approaches are compared, and we show that the new methods described are efficient on several classes of relevant problems

Matrix Equation Techniques for Certain Evolutionary Partial Differential Equations

We show that the discrete operator stemming from time-space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples

Optimality Properties of Galerkin and Petrov-Galerkin Methods for Linear Matrix Equations

none2siGalerkin and Petrov–Galerkin methods are some of the most successful solution procedures in numerical analysis. Their popularity is mainly due to the optimality properties of their approximate solution. We show that these features carry over to the (Petrov-) Galerkin methods applied for the solution of linear matrix equations. Some novel considerations about the use of Galerkin and Petrov–Galerkin schemes in the numerical treatment of general linear matrix equations are expounded and the use of constrained minimization techniques in the Petrov–Galerkin framework is proposed.nonePalitta D.; Simoncini V.Palitta D.; Simoncini V

A new ParaDiag time-parallel time integration method

Time-parallel time integration has received a lot of attention in the high performance computing community over the past two decades. Indeed, it has been shown that parallel-in-time techniques have the potential to remedy one of the main computational drawbacks of parallel-in-space solvers. In particular, it is well-known that for large-scale evolution problems space parallelization saturates long before all processing cores are effectively used on today's large scale parallel computers. Among the many approaches for time-parallel time integration, ParaDiag schemes have proved themselves to be a very effective approach. In this framework, the time stepping matrix or an approximation thereof is diagonalized by Fourier techniques, so that computations taking place at different time steps can be indeed carried out in parallel. We propose here a new ParaDiag algorithm combining the Sherman-Morrison-Woodbury formula and Krylov techniques. A panel of diverse numerical examples illustrates the potential of our new solver. In particular, we show that it performs very well compared to different ParaDiag algorithms recently proposed in the literature

An Efficient, Memory-Saving Approach for the Loewner Framework

The Loewner framework is one of the most successful data-driven model order reduction techniques. If N is the cardinality of a given data set, the so-called Loewner and shifted Loewner matrices [Formula: see text] and [Formula: see text] can be defined by solely relying on information encoded in the considered data set and they play a crucial role in the computation of the sought rational model approximation.In particular, the singular value decomposition of a linear combination of [Formula: see text] and [Formula: see text] provides the tools needed to construct accurate models which fulfill important approximation properties with respect to the original data set. However, for highly-sampled data sets, the dense nature of [Formula: see text] and [Formula: see text] leads to numerical difficulties, namely the failure to allocate these matrices in certain memory-limited environments or excessive computational costs. Even though they do not possess any sparsity pattern, the Loewner and shifted Loewner matrices are extremely structured and, in this paper, we show how to fully exploit their Cauchy-like structure to reduce the cost of computing accurate rational models while avoiding the explicit allocation of [Formula: see text] and [Formula: see text] . In particular, the use of the hierarchically semiseparable format allows us to remarkably lower both the computational cost and the memory requirements of the Loewner framework obtaining a novel scheme whose costs scale with [Formula: see text]