## Embree/ifiss_mat

Fluid stability problem, illustrates need for iter. refinement
Name ifiss_mat Embree 2830 96,307 96,307 3,599,932 3,599,932 Computational Fluid Dynamics No 2015 M. Embree T. Davis
Structural Rank 96,307 true 1 1 0 100% 28.5% no no real
Download Fluid stability problem from IFISS package, Mark Embree, Virginia Tech The matrix comes from a shift-invert eigenvalue computation that arises during linear stability analysis of a 2D backward facing step. The discretizations were created with IFISS, which we also use to compute the steady-state flow whose stability we are assessing. Created by the IFISS package, by David Silvester (School of Mathematics, Univ. of Manchester), Howard Elman (Computer Science, Univ. of Maryland), and Alison Ramage (Dept. of Mathematics and Statistics, Univ. of Strathclyde). http://www.maths.manchester.ac.uk/~djs/ifiss/ This matrix requires one step of iterative refinement after LU factorization. x=A\b in MATLAB (using UMFPACK) does iterative refinement by default but with just lu, no iterative refinement is done: In MATLAB R2014a: x = randn(96307,1); b = A*x; x1 = A\b; norm(x-x1) is 1.1045984e-12 norm(b-A*x1) is 9.3538018e-15 [L,U,P,Q,R] = lu(A); x2 = Q*(U\(L\(P*(R\b)))); norm(x-x2) is 4.9150874e-05 norm(b-A*x2) is 4.0055911e-07 The matrix is well conditioned, with singular values ranging from 1.003 to 2e-5. The singluar values themselves are in Problem.aux.singular_values. The need for iterative refinement comes from the threshold partial pivoting in UMFPACK, which tries to balance reduction in fill with finding good numerical pivots. Thus UMFPACK uses iterative refinement with sparse backward error using the method described in Arioli, Demmel, and Duff, "Sovling sparse linear systems with sparse backward error", SIAM J. Matrix Analysis & Applic, vol 10, no 2, pp 165-190, Apr 1989). The matrix itself is described in this paper: https://arxiv.org/abs/1601.00044 Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations Mark Embree, Blake Keeler (Submitted on 1 Jan 2016 (v1), last revised 27 Jun 2017 (this version, v3)) To understand the solution of a linear, time-invariant differential-algebraic equation, one must analyze a matrix pencil (A,E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left-half plane), the solution can exhibit transient growth before its inevitable decay. When the equation results from the linearization of a nonlinear system, this transient growth gives a mechanism that can promote nonlinear instability. One might hope to enrich the conventional large-scale eigenvalue calculation used for linear stability analysis to signal the potential for such transient growth. Toward this end, we introduce a new definition of the pseudospectrum of a matrix pencil, use it to bound transient growth, explain how to incorporate a physically-relevant norm, and derive approximate pseudospectra using the invariant subspace computed in conventional linear stability analysis. We apply these tools to several canonical test problems in fluid mechanics, an important source of differential-algebraic equations.