Daniel Osei-Kuffuor

Modification and Compensation Strategies for Threshold-based Incomplete Factorizations

Standard (single-level) incomplete factorization preconditioners are known to successfully accelerate Krylov subspace iterations for many linear systems. The classical modified incomplete LU (MILU) factorization approach improves the acceleration given by (standard) ILU approaches, by modifying the nonunit diagonal in the factorization to match the action of the system matrix on a given vector, typically the constant vector. Here, we examine the role of similar modifications within the dual-threshold ILUT algorithm. We introduce column and row variants of the modified ILUT algorithm and discuss optimal ways of modifying the columns or rows of the computed factors to improve their accuracy and stability. Modifications are considered for both the diagonal and off-diagonal entries of the factors, based on one or many vectors, chosen a priori or through an Arnoldi iteration. Numerical results are presented to support our findings.

A Scalable

Traditional algorithms for first-principles molecular dynamics (FPMD) simulations only gain a modest capability increase from current petascale computers, due to their

O(N)

O(N^3)

Algorithm for Large-Scale Parallel First-Principles Molecular Dynamics Simulations

complexity and their heavy use of global communications. To address this issue, we are developing a truly scalable

Modeling dilute solutions using first-principles molecular dynamics: computing more than a million atoms with over a million cores

O(N)

The basic matrix library (BML) for quantum chemistry

complexity FPMD algorithm, based on density functional theory (DFT), which avoids global communications. The computational model uses a general nonorthogonal orbital formulation for the DFT energy functional, which requires knowledge of selected elements of the inverse of the associated overlap matrix. We present a scalable algorithm for approximately computing selected entries of the inverse of the overlap matrix, based on an approximate inverse technique, by inverting local blocks corresponding to principal submatrices of the global overlap matrix. The new FPMD algorithm exploits sparsity and uses nearest neighbor communication to provide a computational scheme capable of extreme scalability. Accuracy is contr...

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions

First-Principles Molecular Dynamics (FPMD) methods, although powerful, are notoriously expensive computationally due to the quantum modeling of electrons. Traditional FPMD approaches have typically been limited to a few thousand atoms at most, due to O(N3) or worse solver complexity and the large amount of communication required for highly parallel implementations. Attempts to lower the complexity have often introduced uncontrolled approximations or systematic errors. Using a robust new algorithm, we have developed an O(N) complexity solver for electronic structure problems with fully controllable numerical error. Its minimal use of global communications yields excellent scalability, allowing for very accurate FPMD simulations of more than a million atoms on over a million cores. At these scales, this approach provides multiple orders of magnitude speedup compared to the standard plane-wave approach typically used in condensed matter applications, without sacrificing accuracy. This will open up entire new classes of FPMD simulations, e.g. dilute aqueous solutions.

Preconditioning Helmholtz linear systems

The basic matrix library package (BML) provides a common application programming interface (API) for linear algebra and matrix functions in C and Fortran for quantum chemistry codes. The BML API is matrix format independent. Currently the dense, compressed sparse row, and ELLPACK-R sparse matrix data types are available, each with different implementations. We show how the second-order spectral projection (SP2) algorithm used to compute the electronic structure of a molecular system represented with a tight-binding Hamiltonian can be successfully implemented with the aid of this library.

Improved numerical solvers for implicit coupling of subsurface and overland flow

Abstract Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switching technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. We also show that the strategy is efficient and scales optimally with problem size.