A Parallel Implementation of the Jacobi-Davidson Eigensolver and its Application in a Plasma Turbulence Code ? Eloy Romero and Jose E. Roman Instituto ITACA, Universidad Polit´ ecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain {eromero,jroman}@itaca.upv.es Abstract. 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 factor- ization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi-Davidson is one of the most effective vari- ants. However, parallel implementations of this method are not widely available, particularly for non-symmetric problems. We present a par- allel implementation to be released 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 effi- ciency and compare it with Krylov-type eigensolvers. Keywords: Message-passing parallelization, eigenvalue computations, Jacobi-Davidson, plasma simulation. 1 Introduction We are concerned with the partial solution of the standard eigenvalue problem defined by a large, sparse matrix A of order n, Ax = λx, where the scalar λ is called the eigenvalue, and the n-vector x is called the eigenvector. Many iter- ative methods are available for the partial solution of the above problem, that is, for computing a subset of the eigenvalues. The most popular ones are Krylov projection methods such as Lanczos, Arnoldi or Krylov-Schur, and Davidson- type methods such as Generalized Davidson or Jacobi-Davidson. Details of these methods can be found in [1]. Krylov methods achieve good performance when computing extreme eigenvalues, but usually fail to compute interior eigenvalues. In that case, the convergence can be improved by combining the method with a spectral transformation technique, i.e., to solve (A - σ) -1 x = θx instead of ? This work was partially supported by the Spanish Ministerio de Ciencia e Innovaci´ on under project TIN2009-07519.