A parallel Heap-Cell Method for Eikonal equations

Numerous applications of Eikonal equations prompted the development of many efficient numerical algorithms. The Heap-Cell Method (HCM) is a recent serial two-scale technique that has been shown to have advantages over other serial state-of-the-art solvers for a wide range of problems. This paper presents a parallelization of HCM for a shared memory architecture. The numerical experiments in $R^3$ show that the parallel HCM exhibits good algorithmic behavior and scales well, resulting in a very fast and practical solver. We further explore the influence on performance and scaling of data precision, early termination criteria, and the hardware architecture. A shorter version of this manuscript (omitting these more detailed tests) has been submitted to SIAM Journal on Scientific Computing in 2012.Comment: (a minor update to address the reviewers' comments) 31 pages; 15 figures; this is an expanded version of a paper accepted by SIAM Journal on Scientific Computin

Stabilised finite element methods for ill-posed problems with conditional stability

In this paper we discuss the adjoint stabilised finite element method introduced in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on Scientific Computing, and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equation

Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver

In the numerical solution of large-scale eigenvalue problems, Davidson-type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson-type methods, the factorization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi-Davidson is one of the most effective variants. However, parallel implementations of this method are not widely available, particularly for non-symmetric problems. We present a parallel implementation that has been included in SLEPc, the Scalable Library for Eigenvalue Problem Computations, and test it in the context of a highly scalable plasma turbulence simulation code. We analyze its parallel efficiency and compare it with a Krylov-Schur eigensolver. © 2011 John Wiley and Sons, Ltd..The authors are indebted to Florian Merz for providing us with the test cases and for his useful suggestions. The authors acknowledge the computer resources provided by the Barcelona Supercomputing Center (BSC). This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519.Romero Alcalde, E.; Román Moltó, JE. (2011). Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver. Concurrency and Computation: Practice and Experience. 23:2179-2191. https://doi.org/10.1002/cpe.1740S2179219123Hochstenbach, M. E., & Notay, Y. (2009). Controlling Inner Iterations in the Jacobi–Davidson Method. SIAM Journal on Matrix Analysis and Applications, 31(2), 460-477. doi:10.1137/080732110Heuveline, V., Philippe, B., & Sadkane, M. (1997). 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ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Romero, E., Cruz, M. B., Roman, J. E., & Vasconcelos, P. B. (2011). A Parallel Implementation of the Jacobi-Davidson Eigensolver for Unsymmetric Matrices. High Performance Computing for Computational Science – VECPAR 2010, 380-393. doi:10.1007/978-3-642-19328-6_35Romero, E., & Roman, J. E. (2010). A Parallel Implementation of the Jacobi-Davidson Eigensolver and Its Application in a Plasma Turbulence Code. Lecture Notes in Computer Science, 101-112. doi:10.1007/978-3-642-15291-7_11Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen*). (1846). Journal für die reine und angewandte Mathematik (Crelles Journal), 1846(30), 51-94. doi:10.1515/crll.1846.30.51G. Sleijpen, G. L., & Van der Vorst, H. A. (1996). A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems. 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Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numerical Mathematics, 36(3), 595-633. doi:10.1007/bf01731936Balay S Buschelman K Eijkhout V Gropp W Kaushik D Knepley M McInnes LC Smith B Zhang H PETSc users manual 2010Hernandez, V., Roman, J. E., & Tomas, A. (2007). Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Computing, 33(7-8), 521-540. doi:10.1016/j.parco.2007.04.004Dannert, T., & Jenko, F. (2005). Gyrokinetic simulation of collisionless trapped-electron mode turbulence. Physics of Plasmas, 12(7), 072309. doi:10.1063/1.1947447Roman, J. E., Kammerer, M., Merz, F., & Jenko, F. (2010). Fast eigenvalue calculations in a massively parallel plasma turbulence code. Parallel Computing, 36(5-6), 339-358. doi:10.1016/j.parco.2009.12.001Merz, F., & Jenko, F. (2010). Nonlinear interplay of TEM and ITG turbulence and its effect on transport. Nuclear Fusion, 50(5), 054005. doi:10.1088/0029-5515/50/5/054005Simoncini, V., & Szyld, D. B. (2002). Flexible Inner-Outer Krylov Subspace Methods. SIAM Journal on Numerical Analysis, 40(6), 2219-2239. doi:10.1137/s0036142902401074Morgan, R. B. (2002). GMRES with Deflated Restarting. SIAM Journal on Scientific Computing, 24(1), 20-37. doi:10.1137/s106482759936465