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The Golub-Welsch algorithm [Math. Comp., 23: 221-230 (1969)] for computing Gaussian quadrature rules is of importance in estimating quadratic forms. Quadrature rules based on this algorithm have long been assumed to be symmetric. Recent research indicates that the presence of asymmetric quadrature nodes may be more often. Such a divergence has led to varying error analyses of the Lanczos quadrature method. Since symmetry often implies simplicity, it is of great interest to ask when do Lanczos iterations generate symmetric quadrature rules. This paper derives a sufficient condition that ensures symmetric quadrature nodes which partially answers the question that when the Ritz values of a symmetric matrix are symmetrically distributed. Additionally, we establish both lower and upper bounds on the disparity between the minimum Lanczos iterations required for symmetric and asymmetric quadrature., Comment: 17 pages, 2 figures

Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional MPI processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time-to-solution (when applicable). However, the development of fast, scalable all-at-once methods has largely been limited to time-dependent (advection-)diffusion problems. This paper introduces the concept of space-time block preconditioning for the all-at-once solution of incompressible flow. By extending well-known concepts of spatial block preconditioning to the space-time setting, we develop a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and a space-time matrix-vector multiplication. The new method is tested on four classical models in incompressible flow. Results indicate perfect scalability in refinement of spatial and temporal mesh spacing, perfect scalability in nonlinear Picard iterations count when applied to a nonlinear Navier-Stokes problem, and minimal overhead in terms of number of preconditioner applications compared with sequential time-stepping., Comment: 28 pages, 7 figures, 4 tables

Algorithms for Gaussian process, marginal likelihood methods or restricted maximum likelihood methods often require derivatives of log determinant terms. These log determinants are usually parametric with variance parameters of the underlying statistical models. This paper demonstrates that, when the underlying matrix is sparse, how to take the advantage of sparse inversion---selected inversion which share the same sparsity as the original matrix---to accelerate evaluating the derivative of log determinant., Comment: 15

For both isothermal and thermal petroleum reservoir simulation, the Constrained Pressure Residual (CPR) method is the industry-standard preconditioner. This method is a two-stage process involving the solution of a restricted pressure system. While initially designed for the isothermal case, CPR is also the standard for thermal cases. However, its treatment of the energy conservation equation does not incorporate heat diffusion, which is often dominant in thermal cases. In this paper, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted pressure-temperature system is solved in the first stage. In previous work, we introduced a block preconditioner with an efficient Schur complement approximation for a pressure-temperature system. Here, we extend this method for multiphase flow as the first stage of CPTR. The algorithmic performance of different two-stage preconditioners is evaluated for reservoir simulation test cases., Comment: 28 pages, 2 figures. Sources/sinks description in arXiv:1902.00095

In petroleum reservoir simulation, the industry standard preconditioner, the constrained pressure residual method (CPR), is a two-stage process which involves solving a restricted pressure system with Algebraic Multigrid (AMG). Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not have a specific treatment of the additional energy conservation equation and temperature variable. We seek to develop preconditioners which better capture thermal effects such as heat diffusion. In order to study the effects of both pressure and temperature on fluid and heat flow, we consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Both the pressure block and the approximate Schur complement are approximately inverted using an AMG V-cycle. The resulting solver is scalable with respect to problem size and parallelization., Comment: 35 pages, 3 figures

The restricted maximum likelihood method enhances popularity of maximum likelihood methods for variance component analysis on large scale unbalanced data. As the high throughput biological data sets and the emerged science on uncertainty quantification, such a method receives increasing attention. Estimating the unknown variance parameters with restricted maximum likelihood method usually requires an nonlinear iterative method. Therefore proper formulae for the log-likelihood function and its derivatives play an essential role in practical algorithm design. It is our aim to provide a mathematical introduction to this method, and supply a self-contained derivation on some available formulae used in practical algorithms. Some new proof are supplied., Comment: arXiv admin note: substantial text overlap with arXiv:1608.07207

For a prescribed porosity, the coupled magma/mantle flow equations can be formulated as a two-field system of equations with velocity and pressure as unknowns. Previous work has shown that while optimal preconditioners for the two-field formulation can be obtained, the construction of preconditioners that are uniform with respect to model parameters is difficult. This limits the applicability of two-field preconditioners in certain regimes of practical interest. We address this issue by reformulating the governing equations as a three-field problem, which removes a term that was problematic in the two-field formulation in favour of an additional equation for a pressure-like field. For the three-field problem, we develop and analyse new preconditioners and we show numerically that they are optimal in terms of problem size and less sensitive to model parameters, compared to the two-field preconditioner. This extends the applicability of optimal preconditioners for coupled mantle/magma dynamics into parameter regimes of physical interest., Comment: To appear in SIAM Journal on Scientific Computing

This article considers the iterative solution of a finite element discretisation of the magma dynamics equations. In simplified form, the magma dynamics equations share some features of the Stokes equations. We therefore formulate, analyse and numerically test a Elman, Silvester and Wathen-type block preconditioner for magma dynamics. We prove analytically and demonstrate numerically the optimality of the preconditioner. The presented analysis highlights the dependence of the preconditioner on parameters in the magma dynamics equations that can affect convergence of iterative linear solvers. The analysis is verified through a range of two- and three-dimensional numerical examples on unstructured grids, from simple illustrative problems through to large problems on subduction zone-like geometries. The computer code to reproduce all numerical examples is freely available as supporting material.

This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. Making use of a recently derived inf-sup condition [13] and the Brezzi stability and convergence theorem for this approximation scheme, we show that the linear system can be optimally preconditioned with a suitable block-diagonal preconditioner. Numerical experiments with a non-uniform distribution of data points support the theoretical conclusions., Comment: 14 pages, 1 figure, submitted