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We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic using three different techniques. We derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. In particular, we apply both a symmetric variant and Forsgren's variant of the simplified Newton method. This work has implications for the implementation of quasi-Newton methods regardless of the scale of the calculation or the machine.

In molecular dynamics simulations we can often increase the time step by imposing constraints on bond lengths and bond angles. This allows us to extend the length of the time interval and therefore the range of physical phenomena that we can afford to simulate. We examine the existing algorithms and software for solving nonlinear constraint equations in parallel and we explain why it is necessary to advance the state-of-the-art. We present ILVES-PC, a new algorithm for imposing bond constraints on proteins accurately and efficiently. It solves the same system of differential algebraic equations as the celebrated SHAKE algorithm, but ILVES-PC solves the nonlinear constraint equations using Newton’s method rather than the nonlinear Gauss-Seidel method. Moreover, ILVES-PC solves the necessary linear systems using a specialized linear solver that exploits the structure of the protein. ILVES-PC can rapidly solve constraint equations as accurately as the hardware will allow. The run-time of ILVES-PC is proportional to the number of constraints. We have integrated ILVES-PC into GROMACS and simulated proteins of different sizes. Compared with SHAKE, we have achieved speedups of up to 4.9× in single-threaded executions and up to 76× in shared-memory multi-threaded executions. Moreover, ILVES-PC is more accurate than P-LINCS algorithm. Our work is a proof-of-concept of the utility of software designed specifically for the simulation of polymers.

We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic. In particular, we derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. We briefly discuss implications for parallel solvers.

In this paper, we present the StarNEig library for solving dense nonsymmetric standard and generalized eigenvalue problems. The library is built on top of the StarPU runtime system and targets both shared and distributed memory machines. Some components of the library have support for GPU acceleration. The library currently applies to real matrices with real and complex eigenvalues and all calculations are done using real arithmetic. Support for complex matrices is planned for a future release. This paper is aimed at potential users of the library. We describe the design choices and capabilities of the library, and contrast them to existing software such as LAPACK and ScaLAPACK. StarNEig implements a ScaLAPACK compatibility layer which should assist new users in the transition to StarNEig. We demonstrate the performance of the library with a sample of computational experiments.

Abstract. In this paper, we present the StarNEig library for solvingdense nonsymmetric (generalized) eigenvalue problems. The library isbuilt on top of the StarPU runtime system and targets both shared anddistributed memory machines. Some components of the library supportGPUs. The library is currently in an early beta state and only real arith-metic is supported. Support for complex data types is planned for afuture release. This paper is aimed at potential users of the library. Wedescribe the design choices and capabilities of the library, and contrastthem to existing software such as ScaLAPACK. StarNEig implements aScaLAPACK compatibility layer that should make it easy for new usersto transition to StarNEig. We demonstrate the performance of the librarywith a small set of computational experiments.

In this paper we consider the problem of computing generalized eigenvectors of a matrix pencil in real Schur form. In exact arithmetic, this problem can be solved using substitution. In practice, substitution is vulnerable to floating-point overflow. The robust solvers xtgevc in LAPACK prevent overflow by dynamically scaling the eigenvectors.These subroutines are scalar and sequential codes which compute theeigenvectors one by one. In this paper, we discuss how to derive robust algorithms which are blocked and parallel. The new StarNEig librarycontains a robust task-parallel solver Zazamoukh which runs on top of StarPU. Our numerical experiments show that Zazamoukh achieves a super-linear speedup compared with dtgevc for sufficiently large matrices.

A standard approach for computing eigenvectors of a non-symmetric matrix reduced to real Schurform relies on a variant of backward substitution. Backward substitution is prone to overflow. To avoid overflow, the LAPACK eigenvector routine DTREVC3 associates every eigenvector with a scaling factor and dynamically rescales an entire eigenvector during the backward substitution such that overflow cannot occur. When many eigenvectors are computed, DTREVC3 applies backward substitution successively for every eigenvector. This corresponds to level-2 BLAS operations and constitutes a bottleneck. This paper redesigns the backward substitution such that the entire computation is cast as tile operations (level-3 BLAS). By replacing LAPACK’s scaling factor with tile-local scaling factors, our solver decouples the tiles and sustains parallel scalability even when a lot of numerical scaling is necessary.

A standard approach for computing eigenvectors of a non-symmetric matrix reduced to real Schur form relies on a variant of backward substitution. Backward substitution is prone to overflow. To avoid overflow, the LAPACK eigenvector routine DTREVC3 associates every eigenvector with a scaling factor and dynamically rescales an entire eigenvector during the backward substitution such that overflow cannot occur. When many eigenvectors are computed, DTREVC3 applies backward substitution successively for every eigenvector. This corresponds to level-2 BLAS operations and constitutes a bottleneck. This paper redesigns the backward substitution such that the entire computation is cast as tile operations (level-3 BLAS). By replacing LAPACK’s scaling factor with tile-local scaling factors, our solver decouples the tiles and sustains parallel scalability even when a lot of numerical scaling is necessary.

The Bartels-Stewart algorithm is a standard approach to solving the dense Sylvester equation. It reduces the problem to the solution of the triangular Sylvester equation. The triangular Sylvester equation is solved with a variant of backward substitution. Backward substitution is prone to overflow. Overflow can be avoided by dynamic scaling of the solution matrix. An algorithm which prevents overflow is said to be robust. The standard library LAPACK contains the robust scalar sequential solver dtrsyl. This paper derives a robust, level-3 BLAS-based task-parallel solver. By adding overflow protection, our robust solver closes the gap between problems solvable by LAPACK and problems solvable by existing non-robust task-parallel solvers. We demonstrate that our robust solver achieves a performance similar to non-robust solvers.