Schur Complement Computations in Intel® Math Kernel Library PARDISO

Abstract

This paper describes a method of calculating the Schur complement of a sparse positive definite matrix A. The main idea of this approach is to represent matrix A in the form of an elimination tree using a reordering algorithm like METIS and putting columns/rows for which the Schur complement is needed into the top node of the elimination tree. Any problem with a degenerate part of the initial matrix can be resolved with the help of iterative refinement. The proposed approach is close to the "multifrontal" one which was implemented by Ian Duff and others in 1980s. Schur complement computations described in this paper are available in Intel ® Math Kernel Library (Intel ® MKL). In this paper we present the algorithm for Schur complement computations, experiments that demonstrate a negligible increase in the number of elements in the factored matrix, and comparison with existing alternatives.

Accelerated Cyclic Reduction: A distributed-memory fast solver for structured linear systems

“…The source code is written in C and compiled with the Intel C compiler v15. External libraries utilized in the reference implementation include HLIBpro v2.2 with Intel TBB [31,37], and the sequential version of the Intel Math Kernel Library [38]. Experiments are conducted on the Cray XC40 Shaheen supercomputer at the King Abdullah University of Science & Technology.…”

Intel<sup>®</sup> Math Kernel Library PARDISO<sup>*</sup> for Intel<sup>®</sup> Xeon Phi<sup>TM</sup> Manycore Coprocessor

“…The Intel Math Kernel Library now includes several new features such as the Schur complement of a sparse matrix [13] and a cluster version of the direct sparse solver [11]. At the same time, Intel MKL continues to provide high performance functionality on modern Intel processors.…”

Implicit Schur complement iterative modal solvers for multiphysics model order reduction

“…We implement the explicit and implicit Schur approaches in Julia 1.8.5 [13] on a HPC server with 503 GB of RAM and two 20 cores/40 threads CPUs (Intel Xeon Gold 6138). All the BLAS/LAPACK linear algebra operations are performed using the widespread and optimized Intel Math Kernel Library To simplify the implementation of the explicit Schur approach, we compute the Schur complement matrix S uu 𝑐𝑐 using the simple algorithm described in [14]. Because this is inefficient compared to optimized algorithms implemented in PARDISO [14] or MUMPS [15], the Schur complement cost is ignored to not skew the measured computational costs.…”

“…All the BLAS/LAPACK linear algebra operations are performed using the widespread and optimized Intel Math Kernel Library To simplify the implementation of the explicit Schur approach, we compute the Schur complement matrix S uu 𝑐𝑐 using the simple algorithm described in [14]. Because this is inefficient compared to optimized algorithms implemented in PARDISO [14] or MUMPS [15], the Schur complement cost is ignored to not skew the measured computational costs. However, this cost is significant for large problems even for optimized implementations.…”