Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Development of scalable linear solvers for engineering applications

The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, particularly those with diffusive character, sparse linear systems with symmetric positive definite (SPD) matrices need to be solved, and multilevel methods represent common choices for the role of iterative solvers or preconditioners. The weak scalability showed by those techniques is one of the main reasons for their popularity, since it allows the solution of linear systems with growing size without requiring a substantial increase in the computational time and number of iterations. On the other hand, single-level preconditioners such as the adaptive Factorized Sparse Approximate Inverse (aFSAI) might be attractive for reaching strong scalability due to their simpler setup. In this thesis, we propose four multilevel preconditioners based on aFSAI targeting the efficient solution of ill-conditioned SPD systems through parallel computing. The first two novel methods, namely Block Tridiagonal FSAI (BTFSAI) and Domain Decomposition FSAI (DDFSAI), rely on graph reordering techniques and approximate block factorizations carried out by aFSAI. Then, we introduce an extension of the previous techniques called the Multilevel Factorization with Low-Rank corrections (MFLR) that ensures positive definiteness of the Schur complements as well as improves their approximation with the aid of tall-and-skinny correction matrices. Lastly, we present the adaptive Smoothing and Prolongation Algebraic MultiGrid (aSPAMG) preconditioner belonging to the adaptive AMG family that introduces the use of aFSAI as a flexible smoother; three strategies for uncovering the near-null space of the system matrix and two new approaches to dynamically compute the prolongation operator. We assess the performance of the proposed preconditioners through the solution of a set of model problems along with real-world engineering test cases. Moreover, we perform comparisons to other approaches such as aFSAI, ILU (ILUPACK), and BoomerAMG (HYPRE), showing that our new methods prove comparable, if not superior, in many test cases

Data-driven deep-learning methods for the accelerated simulation of Eulerian fluid dynamics

Deep-learning (DL) methods for the fast inference of the temporal evolution of fluid-dynamics systems, based on the previous recognition of features underlying large sets of fluid-dynamics data, have been studied. Specifically, models based on convolution neural networks (CNNs) and graph neural networks (GNNs) were proposed and discussed. A U-Net, a popular fully-convolutional architecture, was trained to infer wave dynamics on liquid surfaces surrounded by walls, given as input the system state at previous time-points. A term for penalising the error of the spatial derivatives was added to the loss function, which resulted in a suppression of spurious oscillations and a more accurate location and length of the predicted wavefronts. This model proved to accurately generalise to complex wall geometries not seen during training. As opposed to the image data-structures processed by CNNs, graphs offer higher freedom on how data is organised and processed. This motivated the use of graphs to represent the state of fluid-dynamic systems discretised by unstructured sets of nodes, and GNNs to process such graphs. Graphs have enabled more accurate representations of curvilinear geometries and higher resolution placement exclusively in areas where physics is more challenging to resolve. Two novel GNN architectures were designed for fluid-dynamics inference: the MuS-GNN, a multi-scale GNN, and the REMuS-GNN, a rotation-equivariant multi-scale GNN. Both architectures work by repeatedly passing messages from each node to its nearest nodes in the graph. Additionally, lower-resolutions graphs, with a reduced number of nodes, are defined from the original graph, and messages are also passed from finer to coarser graphs and vice-versa. The low-resolution graphs allowed for efficiently capturing physics encompassing a range of lengthscales. Advection and fluid flow, modelled by the incompressible Navier-Stokes equations, were the two types of problems used to assess the proposed GNNs. Whereas a single-scale GNN was sufficient to achieve high generalisation accuracy in advection simulations, flow simulation highly benefited from an increasing number of low-resolution graphs. The generalisation and long-term accuracy of these simulations were further improved by the REMuS-GNN architecture, which processes the system state independently of the orientation of the coordinate system thanks to a rotation-invariant representation and carefully designed components. To the best of the author’s knowledge, the REMuS-GNN architecture was the first rotation-equivariant and multi-scale GNN. The simulations were accelerated between one (in a CPU) and three (in a GPU) orders of magnitude with respect to a CPU-based numerical solver. Additionally, the parallelisation of multi-scale GNNs resulted in a close-to-linear speedup with the number of CPU cores or GPUs.Open Acces

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Parallel algorithms and efficient implementation techniques for finite element approximations

In this thesis we study the efficient implementation of the finite element method for the numerical solution of partial differential equations (PDE) on modern parallel computer archi- tectures, such as Cray and IBM supercomputers. The domain-decomposition (DD) method represents the basis of parallel finite element software and is generally implemented such that the number of subdomains is equal to the number of MPI processes. We are interested in breaking this paradigm by introducing a second level of parallelism. Each subdomain is assigned to more than one processor and either MPI processes or multiple threads are used to implement the parallelism on the second level. The thesis is devoted to the study of this second level of parallelism and includes the stages described below. The algebraic additive Schwarz (AAS) domain-decomposition preconditioner is an integral part of the solution process. We seek to understand its performance on the parallel computers which we target and we introduce an improved construction approach for the parallel precon- ditioner. We examine a novel strategy for solving the AAS subdomain problems, using multiple MPI processes. At the subdomain level, this is represented by the ShyLU preconditioner. We bring improvements to its algorithm in the form of a novel inexact solver based on an incomplete QR (IQR) factorization. The performance of the new preconditioner framework is studied for Laplacian and advection-diffusion-reaction (ADR) problems and for Navier-Stokes problems, as a component within a larger framework of specialized preconditioners. The partitioning of the computational mesh comes with considerable memory limitations, when done at runtime on parallel computers, due to the low amount of available memory per processor. We describe and implement a solution to this problem, based on offloading the partitioning process to a preliminary offline stage of the simulation process. We also present the efficient implementation, based on parallel MPI collective instructions, of the routines which load the mesh parts during the simulation. We discuss an alternative parallel implementation of the finite element system assembly based on multi-threading. This new approach is used to supplement the existing one based on MPI parallelism, in situations where MPI alone can not make use of all the available parallel hardware resources. The work presented in the thesis has been done in the framework of two software projects: the Trilinos project and the LifeV parallel finite element modeling library. All the new develop- ments have been contributed back to the respective projects, to be used freely in subsequent public releases of the software

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described