BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

In the present paper, we introduce a new extension of the conjugate residual (CR) for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A-orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8

CoLA: Exploiting Compositional Structure for Automatic and Efficient Numerical Linear Algebra

Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker, convolutional, block diagonal, sum, or product structure. In this paper, we propose a simple but general framework for large-scale linear algebra problems in machine learning, named CoLA (Compositional Linear Algebra). By combining a linear operator abstraction with compositional dispatch rules, CoLA automatically constructs memory and runtime efficient numerical algorithms. Moreover, CoLA provides memory efficient automatic differentiation, low precision computation, and GPU acceleration in both JAX and PyTorch, while also accommodating new objects, operations, and rules in downstream packages via multiple dispatch. CoLA can accelerate many algebraic operations, while making it easy to prototype matrix structures and algorithms, providing an appealing drop-in tool for virtually any computational effort that requires linear algebra. We showcase its efficacy across a broad range of applications, including partial differential equations, Gaussian processes, equivariant model construction, and unsupervised learning.Comment: Code available at https://github.com/wilson-labs/col

Accelerating advanced preconditioning methods on hybrid architectures

Un gran número de problemas, en diversas áreas de la ciencia y la ingeniería, involucran la solución de sistemas dispersos de ecuaciones lineales de gran escala. En muchos de estos escenarios, son además un cuello de botella desde el punto de vista computacional, y por esa razón, su implementación eficiente ha motivado una cantidad enorme de trabajos científicos. Por muchos años, los métodos directos basados en el proceso de la Eliminación Gaussiana han sido la herramienta de referencia para resolver dichos sistemas, pero la dimensión de los problemas abordados actualmente impone serios desafíos a la mayoría de estos algoritmos, considerando sus requerimientos de memoria, su tiempo de cómputo y la complejidad de su implementación. Propulsados por los avances en las técnicas de precondicionado, los métodos iterativos se han vuelto más confiables, y por lo tanto emergen como alternativas a los métodos directos, ofreciendo soluciones de alta calidad a un menor costo computacional. Sin embargo, estos avances muchas veces son relativos a un problema específico, o dotan a los precondicionadores de una complejidad tal, que su aplicación en diversos problemas se vuelve poco práctica en términos de tiempo de ejecución y consumo de memoria. Como respuesta a esta situación, es común la utilización de estrategias de Computación de Alto Desempeño, ya que el desarrollo sostenido de las plataformas de hardware permite la ejecución simultánea de cada vez más operaciones. Un claro ejemplo de esta evolución son las plataformas compuestas por procesadores multi-núcleo y aceleradoras de hardware como las Unidades de Procesamiento Gráfico (GPU). Particularmente, las GPU se han convertido en poderosos procesadores paralelos, capaces de integrar miles de núcleos a precios y consumo energético razonables.Por estas razones, las GPU son ahora una plataforma de hardware de gran importancia para la ciencia y la ingeniería, y su uso eficiente es crucial para alcanzar un buen desempeño en la mayoría de las aplicaciones. Esta tesis se centra en el uso de GPUs para acelerar la solución de sistemas dispersos de ecuaciones lineales usando métodos iterativos precondicionados con técnicas modernas. En particular, se trabaja sobre ILUPACK, que ofrece implementaciones de los métodos iterativos más importantes, y presenta un interesante y moderno precondicionador de tipo ILU multinivel. En este trabajo, se desarrollan versiones del precondicionador y de los métodos incluidos en el paquete, capaces de explotar el paralelismo de datos mediante el uso de GPUs sin afectar las propiedades numéricas del precondicionador. Además, se habilita y analiza el uso de las GPU en versiones paralelas existentes, basadas en paralelismo de tareas para plataformas de memoria compartida y distribuida. Los resultados obtenidos muestran una sensible mejora en el tiempo de ejecución de los métodos abordados, así como la posibilidad de resolver problemas de gran escala de forma eficiente

Accurate and efficient solutions of electromagnetic problems with the multilevel fast multipole algorithm

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 2009.Thesis (Ph.D.) -- Bilkent University, 2009.Includes bibliographical references leaves 434-226.The multilevel fast multipole algorithm (MLFMA) is a powerful method for the fast and efficient solution of electromagnetics problems discretized with large numbers of unknowns. This method reduces the complexity of matrix-vector multiplications required by iterative solvers and enables the solution of largescale problems that cannot be investigated by using traditional methods. On the other hand, efficiency and accuracy of solutions via MLFMA depend on many parameters, such as the integral-equation formulation, discretization, iterative solver, preconditioning, computing platform, parallelization, and many other details of the numerical implementation. This dissertation is based on our efforts to develop sophisticated implementations of MLFMA for the solution of real-life scattering and radiation problems involving three-dimensional complicated objects with arbitrary geometries.Ergül, Özgür SalihPh.D