
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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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: |
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History of Investigator: |
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Recipient Sponsored Research Office: |
201 SIKES HALL CLEMSON SC US 29634-0001 (864)656-2424 |
Sponsor Congressional District: |
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Primary Place of Performance: |
220 Parkway Dr. Clemson SC US 29634-0001 |
Primary Place of
Performance Congressional District: |
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Unique Entity Identifier (UEI): |
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Parent UEI: |
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NSF Program(s): | COMPUTATIONAL MATHEMATICS |
Primary Program Source: |
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Program Reference Code(s): |
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Program Element Code(s): |
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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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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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