Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two-by-two structure, with one of the submatrices block diagonal. Each one of the diagonal blocks in this submatrix is closely related to the deterministic mean-value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix-vector multiplications for the off-diagonal blocks. Neither the global matrix, nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good preconditioned for the mean-value deterministic problem. We provide a condition number bound for a model elliptic problem and the performance of the method is illustrated by numerical experiments.Comment: 15 pages, 2 figures, 9 tables, (updated numerical experiments

Scalable partitioning for parallel position based dynamics

We introduce a practical partitioning technique designed for parallelizing Position Based Dynamics, and exploiting the ubiquitous multi-core processors present in current commodity GPUs. The input is a set of particles whose dynamics is influenced by spatial constraints. In the initialization phase, we build a graph in which each node corresponds to a constraint and two constraints are connected by an edge if they influence at least one common particle. We introduce a novel greedy algorithm for inserting additional constraints (phantoms) in the graph such that the resulting topology is q-colourable, where ˆ qˆ ≥ 2 is an arbitrary number. We color the graph, and the constraints with the same color are assigned to the same partition. Then, the set of constraints belonging to each partition is solved in parallel during the animation phase. We demonstrate this by using our partitioning technique; the performance hit caused by the GPU kernel calls is significantly decreased, leaving unaffected the visual quality, robustness and speed of serial position based dynamics

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

Problem-orientable numerical algorithm for modelling multi-dimensional radiative MHD flows in astrophysics -- the hierarchical solution scenario

We present a hierarchical approach for enhancing the robustness of numerical solvers for modelling radiative MHD flows in multi-dimensions. This approach is based on clustering the entries of the global Jacobian in a hierarchical manner that enables employing a variety of solution procedures ranging from a purely explicit time-stepping up to fully implicit schemes. A gradual coupling of the radiative MHD equation with the radiative transfer equation in higher dimensions is possible. Using this approach, it is possible to follow the evolution of strongly time-dependent flows with low/high accuracies and with efficiency comparable to explicit methods, as well as searching quasi-stationary solutions for highly viscous flows. In particular, it is shown that the hierarchical approach is capable of modelling the formation of jets in active galactic nuclei and reproduce the corresponding spectral energy distribution with a reasonable accuracy.Comment: 28 pages, 9 figure