Award Abstract # 1819097
New Preconditioned Solvers for Large and Complex Eigenvalue Problems

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: CLEMSON UNIVERSITY
Initial Amendment Date: July 12, 2018
Latest Amendment Date: July 12, 2018
Award Number: 1819097
Award Instrument: Standard Grant
Program Manager: Yuliya Gorb
ygorb@nsf.gov
 (703)292-2113
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: July 15, 2018
End Date: June 30, 2022 (Estimated)
Total Intended Award Amount: $200,000.00
Total Awarded Amount to Date: $200,000.00
Funds Obligated to Date: FY 2018 = $200,000.00
History of Investigator:
  • Fei Xue (Principal Investigator)
    fxue@clemson.edu
Recipient Sponsored Research Office: Clemson University
201 SIKES HALL
CLEMSON
SC  US  29634-0001
(864)656-2424
Sponsor Congressional District: 03
Primary Place of Performance: Clemson University
220 Parkway Dr.
Clemson
SC  US  29634-0001
Primary Place of Performance
Congressional District:
03
Unique Entity Identifier (UEI): H2BMNX7DSKU8
Parent UEI:
NSF Program(s): COMPUTATIONAL MATHEMATICS
Primary Program Source: 01001819DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 9150, 9263
Program Element Code(s): 127100
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

This project aims to develop new algorithms that will help enable large-scale eigenvalue-related modeling and simulations in many scientific and engineering areas, including linear stability analysis of dynamical systems, mechanics of materials, optimization of acoustic emissions, study of superconductivity, vibrations under conditions of uncertainty, and many more. Certain methods are also of significance to other areas; for example, a fast exponential matrix-vector product is essential to the exponential integrator for solving stiff time-dependent differential equations that are difficult to tackle by traditional methods. New textbook writing and graduate student mentoring will help cultivate qualified researchers and industrial professionals to generate further impact in future.

Eigenvalues and closely related mathematical tools (e.g., pseudospectra) are fundamentally descriptive in many areas of applied mathematics and scientific computing. This project concerns systematic development and analysis of innovative preconditioned solvers for several important classes of large-scale and complex eigenvalue-related problems. For large linear symmetric eigenproblems, variants of preconditioned eigensolvers have been thoroughly investigated and widely used with great success in many applications. The plan is to show that the preconditioning and the solver framework can both be generalized significantly and integrated with great flexibility to solve a much broader class of challenging eigenvalue-related problems. The methods to be developed will be reliable, efficient and flexible. The specific research topics include (i) preconditioning (spectral filtering) with matrix functions, (ii) solving nonlinear eigenproblems, and (iii) computing spectra and pseudospectra of large structured matrices.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Dominique Forbes, Leo Rebholz "Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation" Advances in Applied Mathematics and Mechanics , v.13 , 2021 https://doi.org/10.4208/aamm.OA-2020-0270 Citation Details
Rostami, Minghao W. and Xue, Fei "Robust Linear Stability Analysis and a New Method for Computing the Action of the Matrix Exponential" SIAM Journal on Scientific Computing , v.40 , 2018 10.1137/17M1132537 Citation Details
Wang, Guanjie and Xue, Fei and Liao, Qifeng "LOCALIZED STOCHASTIC GALERKIN METHODS FOR HELMHOLTZ PROBLEMS CLOSE TO RESONANCE" International Journal for Uncertainty Quantification , v.11 , 2021 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2021034247 Citation Details
Wu, Lingfei and Xue, Fei and Stathopoulos, Andreas "TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems" SIAM Journal on Scientific Computing , v.41 , 2019 10.1137/17M1157568 Citation Details
Xue, Fei "A Block Preconditioned Harmonic Projection Method for Large-Scale Nonlinear Eigenvalue Problems" SIAM Journal on Scientific Computing , v.40 , 2018 10.1137/17M112141X Citation Details
Xue, Fei "One-step convergence of inexact Anderson acceleration for contractive and non-contractive mappings" ETNA - Electronic Transactions on Numerical Analysis , v.55 , 2021 https://doi.org/10.1553/etna_vol55s285 Citation Details
Xu, Shengjie and Xue, Fei "Inexact rational Krylov subspace method for eigenvalue problems" Numerical Linear Algebra with Applications , 2022 https://doi.org/10.1002/nla.2437 Citation Details

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

This project concerns development and analysis of new numerical methods for solving large-scale algebraic eigenvalue problems. Specifically, the proposal aims to address three topics: preconditioning eigenvalue problems with matrix functions, solving nonlinear eigenvalue problems of the form T(λ)v = 0 and T(v)v = λv, and computing the spectra and pseudospectra of large structured matrices.

The PI has completed several studies of numerical methods for solving a wide range of algebraic eigenproblems, including computing a few lowest eigenvalues of the symmetric positive definite matrix pair (A,B) by thick-restarted preconditioned Lanczos+k method, the rightmost eigenvalues of a nonsymmetric matrix pair (A,B) by matrix exponential transformation, a few eigenvalues near a specified value σ of a nonsymmetric matrix pair (A,B) by a preconditioned harmonic projection method and by inexact rational Krylov subspace methods (RKSM), the ground state solution of the stationary Gross-Pitaevskii equation by Anderson acceleration, and solving the system of linear equations arising from modeling parameter-dependent Helmholtz equations close to resonance by multi-element generalized polynomial chaos (ME-gPC). Other issues not generically related to eigenvalue problems but can be used for tackling them, such as convergence of inexact Anderson acceleration for contractive or non-contractive mapping x = g(x), have also been studied.

This project also funded the PI's coauthoship on an undergraduate textbook entitled "Numerical Analysis: An Introduction", published by De Gruyter in 2019. It also funded a Ph.D student in the School of Mathematical and Statistical Sciences at Clemson University as a research assistant conducting research described in the project proposal.

The funded research has made an important step forward to the development of a new set of preconditioned solvers for large-scale complex eigenvalue-related problems and insight into characteristics of these problems. Upon completion of this project, we have substantially enhanced our understanding of these methods and expand our capabilities to reliably tackle a much broader class of computationally intensive eigenvalue problems. The new algorithms are expected to help enable large-scale eigenvalue-related modeling and simulations in many areas, including linear stability analysis, mechanics of materials, optimization of acoustic emissions, study of superconductivity, vibrations under conditions of uncertainty, and many more.


Last Modified: 10/29/2022
Modified by: Fei Xue

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