Davide Palitta

Davide Palitta, Valeria Simoncini

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.

Davide Palitta, Valeria Simoncini

In the numerical treatment of large-scale Sylvester and Lyapunov equations, projection methods require solving a reduced problem to check convergence. As the approximation space expands, this solution takes an increasing portion of the overall computational effort. When data are symmetric, we show that the Frobenius norm of the residual matrix can be computed at significantly lower cost than with available methods, without explicitly solving the reduced problem. For certain classes of problems, the new residual norm expression combined with a memory-reducing device make classical Krylov strategies competitive with respect to more recent projection methods. Numerical experiments illustrate the effectiveness of the new implementation for standard and extended Krylov subspace methods.

Davide Palitta, Valeria Simoncini

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.

Elias Jarlebring, Giampaolo Mele, Davide Palitta et al.

We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator $Π$ with a particular structure. More precisely, the commutator of the matrix coefficients of the operator $Π$ and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low-rank approximability of this problem, i.e., the solution to this matrix equation can be approximated with a low-rank matrix. Projection methods have successfully been used to solve other matrix equations with low-rank approximability. We propose a new projection method for this class of matrix equations. The choice of subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low-rank commutators. We illustrate the effectiveness of our method by solving large-scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated.

Stefano Massei, Davide Palitta, Leonardo Robol

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.

Davide Palitta, Catherine E. Powell, Valeria Simoncini

We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. These iterative methods combine local subspace projection to speed up convergence with rank truncation strategies and randomization procedures to limit memory consumption. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.

Vincenzo Citro, Davide Palitta

BoostConv has been introduced in earlier works as an effective acceleration technique for nonlinear iterative processes and has been successfully employed in a variety of applications to enhance convergence rates or to compute unstable fixed points that are otherwise inaccessible through standard approaches. Despite its demonstrated practical effectiveness, the theoretical properties of the method have not yet been fully characterized. In this work, we present a more robust formulation of the BoostConv algorithm and, for the first time, provide a rigorous proof of its convergence. The proposed analysis places BoostConv within a precise mathematical framework, clarifying its interpretation as a nonlinear convergence accelerator and establishing sufficient conditions under which convergence to a fixed point is guaranteed. The theoretical findings are illustrated through several numerical examples, spanning from a linear problem to a low-dimensional benchmark and a large-scale incompressible Navier-Stokes simulation. These results demonstrate the robustness and practical relevance of the proposed method and bridge the gap between empirical performance and rigorous analysis, paving the way for further developments and applications to complex nonlinear problems.

Martina Iannacito, Sascha Portaro, Davide Palitta et al.

In the last decades, tensors have emerged as the right tool to represent multidimensional data in a compact yet informative manner. Moreover, it is well-known that by performing low-rank factorizations of such tensors one is often able to effectively unveil possible hidden structure in data, mainly due to unexpected dependencies among the different variables encoded in the given tensor. However, computing these factorizations is extremely energy-consuming and memory-demanding, especially for high-dimensional tensors, namely those with a large number of modes. In this paper we focus on two state-of-the-art tensor decompositions: the Tucker and H-Tucker decompositions. We propose novel numerical strategies able to perform these factorizations in a \emph{mode-parallel} fashion, that is the operations required by the algorithm along all modes are performed in parallel. This is in contrast to what is achieved by many procedures available in the literature that parallelize some of the operations along each mode, e.g., tensor-times-matrix steps, while still visiting one mode at the time in a sequential manner. Our strategies make use of cutting-edge randomization techniques comprising fiber sampling and randomized range-finding steps. We provide upper bounds on the expected value of the error provided by our factorizations while a panel of numerical results showcases the potential of our approach in reducing both the running time and the storage demand of the whole procedure. Moreover, experiments carried out in HPC environments illustrate the good scaling of our mode-parallel approach.