Current iterative methods for solving linear equations assume reliability of data (no “bit flips”) and arithmetic (correct up to rounding error). If faults occur, the solver usually either aborts, or computes the wrong answer without indication. System reliability guarantees consume energy or reduces performance. As processor counts continue to grow, these costs will become unbearable. Instead,...

This paper presents our work on designing scalable linear solvers for large-scale reservoir simulations. The main objective is to support implementation of parallel reservoir simulators on distributed-memory parallel systems, where MPI (Message Passing Interface) is employed for communications among computation nodes. Distributed matrix and vector modules are designed, which are the base of our...

The Laplace-Beltrami system of nonlinear, elliptic, partial differential equations has utility in the generation of computational grids on complex and highly curved geometry. Discretization of this system using the finite element method accommodates unstructured grids, but generates a large, sparse, ill-conditioned system of nonlinear discrete equations. The use of the Laplace-Beltrami approach...

Computational uid dynamics and meteorology in particular are among the major consumers of high performance computer technology. The next generation of atmospheric models will be capable of representing uid ow phenomena at very small scales in the atmosphere. The Mesoscale Compressible Community (MC2) model represents one of the rst successful applications of a semi-implicit, semi-Lagrangian sch...

Globally convergent homotopy methods are used to solve difficult nonlinear systems of equations by tracking the zero curve of a homotopy map. Homotopy curve tracking involves solving a sequence of linear systems, which often vary greatly in difficulty. In this research, a popular iterative solution tool, GMRES(k), is adapted to deal with the sequence of such systems. The proposed adaptive strat...

We introduce an iterative method named Gpmr (general partitioned minimum residual) for solving block unsymmetric linear systems. is based on a new process that simultaneously reduces two rectangular matrices to upper Hessenberg form and closely related the block-Arnoldi process. tantamount Block-Gmres with right-hand sides in which approximate solutions are summed at each iteration, but its sto...

We investigate the convergence of GMRES for an n by n Jordan block J . For each k that divides n we derive the exact form of the kth ideal GMRES polynomial and prove the equality max ‖v‖=1 min p∈πk ‖p(J)v‖ = min p∈πk max ‖v‖=1 ‖p(J)v‖, where πk denotes the set of polynomials of degree at most k and with value one at the origin, and ‖ · ‖ denotes the Euclidean norm. In other words, we show that ...

We present a domain decomposition solver for the 2D Helmholtz equation, with a special choice of integral transmission condition that involves polarizing the waves into oneway components. This refinement of the transmission condition is the key to combining local direct solves into an efficient iterative scheme, which can then be deployed in a highperformance computing environment. The method i...

For Stokes equations with a discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, it is known that the pressure is discontinuous and the velocity is non-smooth. It has been shown that these discontinuities are coupled together, which makes it difficult to obtain accurate numerical solutions. In this paper, a new numerical method that decouples the jum...

We study an iterative low-rank approximation method for the solution of the steadystate stochastic Navier–Stokes equations with uncertain viscosity. The method is based on linearization schemes using Picard and Newton iterations and stochastic finite element discretizations of the linearized problems. For computing the low-rank approximate solution, we adapt the nonlinear iterations to an inexa...