RKPACK A numerical package for solving large eigenproblems Che-Rung Lee

University of Maryland, College Park Outline Introduction RKPACK Experiments Conclusion 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Introduction The residual Krylov method Shift-invert enhancement Properties and examples 2019/7/28 University of Maryland, College Park

The residual Krylov method Basic algorithm Let be a selected eigenpair approximation of A. Compute the residual . Use r in subspace expansion. 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Properties The selected approximation (candidate) can converge even with errors. The allowed error || f || must be less than  ||r||, for a constant  <1. The residual Krylov method can work with an initial subspace that contains good Ritz approximations. 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Example A 100x100 matrix with eigenvalues 1, 0.95, …,0.9599. 2019/7/28 University of Maryland, College Park

Shift-invert enhancement Algorithm: (shift value =  ) Let be a selected eigenpair approximation of A. Compute the residual . Solve the equation . Use v in subspace expansion. Equation in step 3 can be solved in low accuracy, such as 103. 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Example 0 5 10 15 20 25 30 35 40 100 10-5 10-10 10-15 The same matrix Shift value is 1.3 Linear systems are solved to 10 3. 2019/7/28 University of Maryland, College Park

University of Maryland, College Park RKPACK Features Computation modes Memory requirement Time complexity 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Features Can compute several selected eigenpairs Allow imprecise computational results with shift-invert enhancement Can start with an appropriate initial subspace Use the Krylov-Schur restarting algorithm Use reverse communication 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Computation modes Two computation modes The normal mode: needs matrix vector multiplication only The imprecise shift-invert mode: needs matrix vector multiplication and linear system solving (with low accuracy requirement) can change the shift value Both can be initialized with a subspace. 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Memory requirement Use the Krylov-Schur restarting algorithm to control the maximum dimension of subspace Required memory: O(nm)+O(m2) n: the order of matrix A m: the maximum dimension of subspace 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Time complexity The normal mode: kf (n)+kO(nm)+kO(m3) f (n): the time for matrix vector multiplication. k: the number of iterations The imprecise shift-invert mode kf (n) + kO(nm) + kO(m3) + kg(n, ) g(n, ) : the time for solving linear system to the precision . 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Experiments Test problem Performance of RKPACK The inexact residual Krylov method The successive inner-outer process 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Test problem Let A be a 1000010000 matrix with first 100 eigenvalues 1, 0.95, …, 0.9599, and the rest randomly distributed in (0.25, 0.75). Eigenvectors are randomly generated. Maximum dimension of subspace is 20. Stopping criterion: when the norm of residual is smaller than 1013. 2019/7/28 University of Maryland, College Park

Performance of the normal mode Compute six dominant eigenpairs. Compare to the mode 1 of ARPACK Etime: elapse time (second) MVM: number of matrix vector multiplications Iteration: number of subspace expansions ARPACK RKPACK Etime 25.93 24.41 MVM (Iteration) 117 142 2019/7/28 University of Maryland, College Park

The imprecise shift-invert mode Compute six smallest eigenvalues. Use GMRES to solve linear system. (shift = 0) Compare to the mode 3 of ARPACK Prec: precision requirement of solution ARPACK RKPACK Iteration 68 153 ETime 4246.77 623.46 MVM 14552 4932 Prec 1013 103 2019/7/28 University of Maryland, College Park

Inexact residual Krylov method Allow increasing errors in the computation Use the normal mode with matrix A1. The required precision of solving A1.  is the desired precision of computed eigenpairs m is the maximum dimension of subspace 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Experiment and result Compute six smallest eigenpairs. The required precision (using GMRES) Etime: 910.11 second MVM: 6282 Iteration: 67 20 30 40 50 60 70 80 10-2 10-4 10-6 10-8 10-10 2019/7/28 University of Maryland, College Park

Successive inner-outer process Use the convergence properties of Krylov subspace (superlinear) to minimize total number of MVM. (Golub, Zhang and Zha, 2000) Divide the process into stages, with increasing precision requirement. The original algorithm can only compute a single eigenpair 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Experiment and result Compute six smallest eigenpairs. Four stages with required precision (GMRES) 103,106,109,1012. Etime : 1188.12 MVM : 13307 Iteration : 163 20 40 60 80 100 120 140 160 100 10-2 10-4 10-6 10-8 10-10 10-12 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Conclusion Summary Future work 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Summary The residual Krylov method for eigenproblems allows errors in the computation, and can work on an appropriate initial subspace. RKPACK can solve eigenproblems rapidly when uses the imprecise shift-invert enhancement, and is able to integrate other algorithms easily. 2019/7/28 University of Maryland, College Park

University of Maryland, College Park Future work … Parallelization Data parallelism Block version of the residual Krylov method Other eigenvector approximations Refine Ritz vector or Harmonic Ritz vector New algorithms Inexact methods, residual power method … 2019/7/28 University of Maryland, College Park