An odyssey into local refinement and multilevel preconditioning III: Implementation and numerical experiments

In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems, which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on Bramble, Pasciak, and Xu (BPX)-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis (HB) method, and their use in the local mesh refinement setting. In parts I and II of this trilogy, it was shown that both BPX and wavelet stabilizations of HB have uniformly bounded condition numbers on several classes of locally refined two- and three-dimensional meshes based on fairly standard (and easily implementable) red and red-green mesh refinement algorithms. In this third part of the trilogy, we describe in detail the implementation of these types of algorithms, including detailed discussions of the data structures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard data types, available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. We have successfully used these data structure ideas for both MATLAB and C implementations using the FEtk, an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement

Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs

We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a K-cycle iteration solve phase with a $\ell^1$-Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness.Comment: 18 pages, 3 figure

Multilevel Numerical Algorithms for Systems of Nonlinear Parabolic Partial Differential Equations

This thesis is concerned with the development of efficient and reliable numerical algorithms for the solution of nonlinear systems of partial differential equations (PDEs) of elliptic and parabolic type. The main focus is on the implementation and performance of three different nonlinear multilevel algorithms, following discretisation of the PDEs: the Full Approximation Scheme (FAS), Newton-Multigrid (Newton-MG) and a Newton-Krylov solver with a novel pre- conditioner that we have developed based on the use of Algebraic Multigrid (AMG). In recent years these algorithms have been commonly used to solve nonlinear systems that arise from the discretisation of PDEs due to the fact that their execution time can scale linearly (or close to linearly) with the number of degrees of freedom used in the discretisation. We consider two mathematical models: a thin film flow and the Cahn-Hilliard-Hele-Shaw model. These mathematical models consist of nonlinear, time-dependent and coupled PDEs systems. Using a Finite Difference Method (FDM) in space and Backward Differentiation For- mulae (BDF) in time, we discrete the two models, to produce nonlinear algebraic systems. We are able to solve these nonlinear systems implicitly in computationally demanding 2D situa- tions. We present numerical results, for both steady-state and time-dependent problems, that demonstrate the optimality of the three numerical algorithms for the thin film flow model. We show optimality of the FAS and Newton-Krylov approaches for the time-dependent Cahn- Hilliard-Hele-Shaw (CHHS) problem. The main contribution is to address the question of which of these three nonlinear solvers is likely to be the best (i.e. computationally most effective) in practice. In order to asses this, we discuss the careful implementation and timing of these algorithms in order to permit a fair direct comparison of their computational cost. We then present extensive numerical results in order to make this comparison between these nonlinear multilevel methods. The conclusion emerging from this investigation is that it does not appear that there is a single superior approach, but rather that the best approach is problem dependent. Specifically, we find that our optimally preconditioned Newton-Krylov approach is best for the thin film flow model in the steady-state and time-dependent form, whilst the FAS solver appears best for the time-dependent CHHS model

On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes

On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes