Jacobi-Davidson methods for generalized MHD-eigenvalue problems

A Jacobi-Davidson algorithm for computing selected eigenvalues and associated eigenvectors of the generalized eigenvalue problem $Ax = lambda Bx$ is presented. In this paper the emphasis is put on the case where one of the matrices, say the B-matrix, is Hermitian positive definite. The method is an inner-outer iterative scheme, in which the inner iteration process consists of solving linear systems to some accuracy. The factorization of either matrix is avoided. Numerical experiments are presented for problems arising in magnetohydrodynamics (MHD)

Accelerating Inexact Newton Schemes for Large Systems of Nonlinear Equations

Classical iteration methods for linear systems, such as Jacobi iteration, can be accelerated considerably by Krylov subspace methods like GMRES. In this paper, we describe how inexact Newton methods for nonlinear problems can be accelerated in a similar way and how this leads to a general framework that includes many well-known techniques for solving linear and nonlinear systems, as well as new ones. Inexact Newton methods are frequently used in practice to avoid the expensive exact solution of the large linear system arising in the (possibly also inexact) linearization step of Newton’s process. Our framework includes acceleration techniques for the “linear steps” as well as for the “nonlinear steps” in Newton’s process. The described class of methods, the accelerated inexact Newton (AIN) methods, contains methods like GMRES and GMRESR for linear systems, Arnoldi and Jacobi–Davidson for linear eigenproblems, and many variants of Newton’s method, like damped Newton, for general nonlinear problems. As numerical experiments suggest, the AIN approach may be useful for the construction of efficient schemes for solving nonlinear problems

Jacobi-Davidson methods for generalized MHD-eigenvalue problems

A Jacobi-Davidson algorithm for computing selected eigenvalues and associated eigenvectors of the generalized eigenvalue problem $Ax = lambda Bx$ is presented. In this paper the emphasis is put on the case where one of the matrices, say the B-matrix, is Hermitian positive definite. The method is an inner-outer iterative scheme, in which the inner iteration process consists of solving linear systems to some accuracy. The factorization of either matrix is avoided. Numerical experiments are presented for problems arising in magnetohydrodynamics (MHD)

Phase II study of oral platinum drug JM216 as first-line treatment in patients with small-cell lung cancer.

PURPOSE: This multicenter phase II trial was performed to determine tumor efficacy and tolerance of the oral platinum drug JM216 in patients with small-cell lung cancer (SCLC). PATIENTS AND METHODS: Patients with SCLC limited disease unfit for intensive chemotherapy or those with extensive disease received JM216 120 mg/m(2)/d for 5 consecutive days every 3 weeks. Individual dose escalation to 140 mg/m(2)/d was allowed if toxicity was </= grade 2 according to the National Cancer Institute Common Toxicity Criteria. Tumor response was evaluated according to World Health Organization criteria. RESULTS: Twenty-seven patients were assessable for toxicity and 26 for tumor response. Eighty-eight cycles were administered. Common Toxicity Criteria grade 3 and 4 hematologic toxicities were neutropenia in 15.9% and 3.7%, lymphocytopenia in 47.6% and 17.1%, and thrombocytopenia in 19.5% and 10.3% of cycles, respectively. One patient suffered from neutropenic fever. Nausea, vomiting, and diarrhea were the most common nonhematologic toxicities. Except for grade 4 diarrhea in one patient, no grade 4 nonhematologic toxicity was observed. No severe neurotoxicity or nephrotoxicity was observed. Tumor response rate was 10 of 26 (38%; 95% confidence interval, 19% to 58%), excluding five unconfirmed partial responses. No complete responses were observed. Median overall time to progression was 110 days (range, 5 to 624 days). Median overall survival time was 210 days (range, 5 to 624 days). CONCLUSION: Oral JM216 is active in previously untreated patients with SCLC and shows mild toxicities

A Parallel Implementation of the Jacobi-Davidson Eigensolver and its Application in a Plasma Turbulence Code ⋆

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 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 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 efficiency and compare it with Krylov-type eigensolvers. Keywords: Message-passing parallelization, eigenvalue computations, Jacobi-Davidson, plasma simulation.