A two-level ILU preconditioner for electromagnetic applications

[EN] Computational electromagnetics based on the solution of the integral form of Maxwell s equations with boundary element methods require the solution of large and dense linear systems. For large-scale problems the solution is obtained by using iterative Krylov-type methods provided that a fast method for performing matrix vector products is available. In addition, for ill-conditioned problems some kind of preconditioning technique must be applied to the linear system in order to accelerate the convergence of the iterative method and improve its performance. For many applications it has been reported that incomplete factorizations often suffer from numerical instability due to the indefiniteness of the coefficient matrix. In this context, approximate inverse preconditioners based on Frobenius-norm minimization have emerged as a robust and highly parallel alternative. In this work we propose a two-level ILU preconditioner for the preconditioned GMRES method. The computation and application of the preconditioner is based on graph partitioning techniques. Numerical experiments are presented for different problems and show that with this technique it is possible to obtain robust ILU preconditioners that perform competitively compared with Frobenius-norm minimization preconditioners.This work was supported by the Spanish Ministerio de Economía y Competitividad under grant MTM2014-58159-P and MTM2015-68805-REDT.Cerdán Soriano, JM.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2017). A two-level ILU preconditioner for electromagnetic applications. Journal of Computational and Applied Mathematics. 309:371-382. https://doi.org/10.1016/j.cam.2016.03.012S37138230

Fast Direct Solvers for Elliptic Partial Differential Equations

The dissertation describes fast, robust, and highly accurate numerical methods for solving boundary value problems associated with elliptic PDEs such as Laplace\u27s and Helmholtz\u27 equations, the equations of elasticity, and time-harmonic Maxwell\u27s equation. In many areas of science and engineering, the cost of solving such problems determines what can and cannot be modeled computationally. Elliptic boundary value problems may be solved either via discretization of the PDE (e.g., finite element methods) or by first reformulating the equation as an integral equation, and then discretizing the integral equation. In either case, one is left with the task of solving a system of linear algebraic equations that could be very large. There exist a broad range of schemes with linear complexity for solving these equations (multigrid, preconditioned Krylov methods, etc). Most of these schemes are based on ``iterative\u27\u27 techniques that build a sequence of approximate solutions that converges to the exact solution. In contrast, the methods described here are ``direct\u27\u27 in the sense that they construct an approximation to the inverse (or LU/Cholesky factorization) of the coefficient matrix. Such direct solvers tend to be more robust, versatile, and stable than iterative methods, but have until recently been considered prohibitively expensive for large scale problems. The objective of the dissertation is to demonstrate that in important environments it is possible to construct an approximate inverse with linear computational cost. The methods are for a single solve competitive with the best iterative methods, and can be far faster than any previously available methods in situations where the same coefficient matrix is used in a sequence of problems. In addition, a new discretization technique for elliptic boundary value problems is proposed. The idea is to first compute the solution operator of a large collection of small domains. The small domains are chosen such that the operator is easily computed to high accuracy. A global equilibrium equation is then built by equating the fluxes through all internal domain boundaries. The resulting linear system is well-suited to the newly developed fast direct solvers

Research and Education in Computational Science and Engineering

Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.Comment: Major revision, to appear in SIAM Revie

Large-scale Reservoir Simulations on IBM Blue Gene/Q

This paper presents our work on simulation of large-scale reservoir models on IBM Blue Gene/Q and studying the scalability of our parallel reservoir simulators. An in-house black oil simulator has been implemented. It uses MPI for communication and is capable of simulating reservoir models with hundreds of millions of grid cells. Benchmarks show that our parallel simulator are thousands of times faster than sequential simulators that designed for workstations and personal computers, and the simulator has excellent scalability

Hydrodynamic approach to the evolution of cosmological structures

A hydrodynamic formulation of the evolution of large-scale structure in the Universe is presented. It relies on the spatially coarse-grained description of the dynamical evolution of a many-body gravitating system. Because of the assumed irrelevance of short-range (``collisional'') interactions, the way to tackle the hydrodynamic equations is essentially different from the usual case. The main assumption is that the influence of the small scales over the large-scale evolution is weak: this idea is implemented in the form of a large-scale expansion for the coarse-grained equations. This expansion builds a framework in which to derive in a controlled manner the popular ``dust'' model (as the lowest-order term) and the ``adhesion'' model (as the first-order correction). It provides a clear physical interpretation of the assumptions involved in these models and also the possibility to improve over them.Comment: 14 pages, 3 figures. Version to appear in Phys. Rev.

A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3,700 are used to verify the accuracy and physical fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational Physic

A mass action model of a fibroblast growth factor signaling pathway and its simplification

We consider a kinetic law of mass action model for Fibroblast Growth Factor (FGF) signaling, focusing on the induction of the RAS-MAP kinase pathway via GRB2 binding. Our biologically simple model suffers a combinatorial explosion in the number of differential equations required to simulate the system. In addition to numerically solving the full model, we show that it can be accurately simplified. This requires combining matched asymptotics, the quasi-steady state hypothesis, and the fact subsets of the equations decouple asymptotically. Both the full and simplified models reproduce the qualitative dynamics observed experimentally and in previous stochastic models. The simplified model also elucidates both the qualitative features of GRB2 binding and the complex relationship between SHP2 levels, the rate SHP2 induces dephosphorylation and levels of bound GRB2. In addition to providing insight into the important and redundant features of FGF signaling, such work further highlights the usefulness of numerous simplification techniques in the study of mass action models of signal transduction, as also illustrated recently by Borisov and co-workers (Borisov et al. in Biophys. J. 89, 951–66, 2005, Biosystems 83, 152–66, 2006; Kiyatkin et al. in J. Biol. Chem. 281, 19925–9938, 2006). These developments will facilitate the construction of tractable models of FGF signaling, incorporating further biological realism, such as spatial effects or realistic binding stoichiometries, despite a more severe combinatorial explosion associated with the latter