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A fast algorithm for parabolic PDE-based inverse problems based on Laplace transforms and flexible Krylov solvers

Tania Bakhos, Arvind K. Saibaba, Peter K. Kitanidis
2015 Journal of Computational Physics  
We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently  ...  We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series  ...  We would also like to thank James Lambers and Anil Damle for their careful reading of the manuscript and useful suggestions.  ... 
doi:10.1016/ fatcat:supyau6px5b4zlshklyqqko7hi

An improved Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves [article]

Brendan Gavin, Eric Polizzi
2017 arXiv   pre-print
The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration in the complex plane to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user-defined  ...  We show that this iterative FEAST algorithm (which we call IFEAST) is mathematically equivalent to a block Krylov subspace method for solving eigenvalue problems.  ...  This material is supported by Intel and NSF under Grant #CCF-1510010.  ... 
arXiv:1706.00692v1 fatcat:whvqv36zprfijjsmfbif6qsopq

Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems

Georgios Tzounas, Ioannis Dassios, Muyang Liu, Federico Milano
2020 Applied Sciences  
These kinds of eigenvalue problems are particularly difficult for most numerical methods to handle.  ...  The paper focuses, in particular, on methods and software libraries that are able to handle the large-scale, non-symmetric matrices that arise in power system eigenvalue problems.  ...  the following classes of eigenvalue numerical methods: vector iteration methods, Schur decomposition methods, Krylov subspace methods, and contour integration methods.  ... 
doi:10.3390/app10217592 fatcat:twyqzug2j5c3jhroisuq4uidae

Computing resonant modes of accelerator cavities by solving nonlinear eigenvalue problems via rational approximation

Roel Van Beeumen, Osni Marques, Esmond G. Ng, Chao Yang, Zhaojun Bai, Lixin Ge, Oleksiy Kononenko, Zenghai Li, Cho-Kuen Ng, Liling Xiao
2018 Journal of Computational Physics  
Existing methods The existing approaches to solve a general nonlinear eigenvalue problem can, roughly speaking, be grouped into three main classes [38] : Newton-based techniques, contour integration methods  ...  , the rational linearization method [34] , the infinite Arnoldi method [15] , the Newton rational Krylov method [38] , the fully rational Krylov method [14] , and the generic class of CORK methods  ... 
doi:10.1016/ fatcat:sartf4mhkjhancou7i67zbextm

Computing the Weighted Geometric Mean of Two Large-Scale Matrices and Its Inverse Times a Vector

Massimiliano Fasi, Bruno Iannazzo
2018 SIAM Journal on Matrix Analysis and Applications  
On the other hand, we adapt several existing Krylov subspace techniques to the computation of the weighted geometric mean.  ...  We derive two novel algorithms, based on Gauss-Jacobi quadrature, and tailor an existing technique based on contour integration.  ...  , and Mario Berljafa, for fruitful discussions about the RKFIT algorithm and the use of the Rational Krylov Toolbox.  ... 
doi:10.1137/16m1073315 fatcat:qa66qkm3h5gafc5t4emkjkzyki

A rational QZ method [article]

Daan Camps, Karl Meerbergen, Raf Vandebril
2018 arXiv   pre-print
Whereas the QZ method performs nested subspace iteration driven by a polynomial, the rational QZ method allows for nested subspace iteration driven by a rational function, this creates the additional freedom  ...  The link with rational Krylov subspaces allows us to prove essential uniqueness (implicit Q theorem) of the rational QZ iterates as well as convergence of the proposed method.  ...  This approach is inspired by the link between contour integration methods [17, 22] and rational filtering techniques [24, 25] .  ... 
arXiv:1802.04094v2 fatcat:4xk5f4ntjfhyhpfakvs7hodbx4

Solving stochastic chemical kinetics by Metropolis Hastings sampling [article]

Azam S. Zavar Moosavi, Paul Tranquilli, Adrian Sandu
2014 arXiv   pre-print
algorithm to accelerate the accuracy of the tau-leap method.  ...  Samples generated by this technique have the same distribution as SSA and the histogram of samples show it's convergence to SSA.  ...  Acknowledgements This work was partially supported by awards NSF DMS-1419003, NSF CCF-1218454, AFOSR FA9550-12-1-0293-DEF, AFOSR 12-2640-06, and by the Computational Science Laboratory at Virginia Tech  ... 
arXiv:1410.8155v1 fatcat:bx5unzt2orempgyey5pfrx4oju

Deflated Restarting for Matrix Functions

M. Eiermann, O. G. Ernst, S. Güttel
2011 SIAM Journal on Matrix Analysis and Applications  
We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector.  ...  An approximation to the subspace to be deflated is successively refined in the course of the underlying restarted Arnoldi process by extracting Ritz vectors and using those closest to the spectral region  ...  Methods based on rational Krylov spaces (cf.  ... 
doi:10.1137/090774665 fatcat:thqryhld3vawla62zwsrepzzvm

Limited-memory polynomial methods for large-scale matrix functions [article]

Stefan Güttel, Daniel Kressner, Kathryn Lund
2020 arXiv   pre-print
We review various limited-memory methods for the approximation of the action of a large-scale matrix function on a vector.  ...  Methods based on explicit polynomial approximation or interpolation, as well as restarted Arnoldi methods, are treated in detail.  ...  Parts of this survey were prepared during visits of the last two authors to the Department of Mathematics at The University of Manchester, whose hospitality is gratefully acknowledged.  ... 
arXiv:2002.01682v3 fatcat:4m7lsekh4vhyxodkrxcapxmui4

Time Integration and Steady-State Continuation for 2d Lubrication Equations

Philippe Beltrame, Uwe Thiele
2010 SIAM Journal on Applied Dynamical Systems  
On the other hand, we consider the depinning of drops pinned by a wettability defect.  ...  In particular, we present a time integration algorithm based on exponential propagation and an algorithm for steady-state continuation.  ...  We thank the Max-Planck-Institut für Physik komplexer Systeme in Dresden (Germany) that hosted us during the early stage of the project. P. B. is also grateful to L. S.  ... 
doi:10.1137/080718619 fatcat:qgxfyveodrhgjnlk4ey7bhlz34

A map of contour integral-based eigensolvers for solving generalized eigenvalue problems [article]

Akira Imakura, Lei Du, Tetsuya Sakurai
2016 arXiv   pre-print
From the analysis, we conclude that all contour integral-based eigensolvers can be regarded as projection methods and can be categorized based on their subspace used, the type of projection and the problem  ...  In this paper, we reconsider the algorithms of the five typical contour integral-based eigensolvers from the viewpoint of projection methods, and then map the relationships among these methods.  ...  Kensuke Aishima, The University of Tokyo for his valuable comments. The authors are also grateful to an anonymous referee for useful comments.  ... 
arXiv:1510.02572v2 fatcat:fcqzcfzylfhuvhzxnsv6vv7s4e

Beyond AMLS: Domain decomposition with rational filtering [article]

Vassilis Kalantzis, Yuanzhe Xi, Yousef Saad
2017 arXiv   pre-print
Compared to rational filtering projection methods applied to the original matrix pencil, the proposed technique integrates only a part of the matrix resolvent while it applies any orthogonalization necessary  ...  Numerical experiments performed in distributed memory architectures illustrate the competitiveness of the proposed technique against rational filtering Krylov approaches.  ...  To reduce the orthonormalization and memory costs, it is typical to enhance the convergence rate of the Krylov projection method of choice by a filtering acceleration technique so that eigenvalues located  ... 
arXiv:1711.09487v1 fatcat:pkp2pkms3bd3xptpxkuavpn4oq

Fast Matrix Square Roots with Applications to Gaussian Processes and Bayesian Optimization [article]

Geoff Pleiss, Martin Jankowiak, David Eriksson, Anil Damle, Jacob R. Gardner
2020 arXiv   pre-print
Our method combines Krylov subspace methods with a rational approximation and typically achieves 4 decimal places of accuracy with fewer than 100 MVMs.  ...  We demonstrate our method's applicability on matrices as large as 50,000 × 50,000 - well beyond traditional methods - with little approximation error.  ...  Acknowledgments and Disclosure of Funding  ... 
arXiv:2006.11267v2 fatcat:twbpjfca2nc5npww4q442itdoq

Approximation of functions of large matrices with Kronecker structure

Michele Benzi, Valeria Simoncini
2016 Numerische Mathematik  
Our findings are illustrated by numerical experiments with typical functions used in applications.  ...  of the problem and for many ways of numerically approximating its solution.  ...  Rational Krylov subspace approximation.  ... 
doi:10.1007/s00211-016-0799-9 fatcat:rnidatkf3bcopmdny6ev66kgme

Approximation of functions of large matrices with Kronecker structure [article]

Michele Benzi, Valeria Simoncini
2015 arXiv   pre-print
Our findings are illustrated by numerical experiments with typical functions used in applications.  ...  We consider the numerical approximation of f( A)b where b∈ R^N and A is the sum of Kronecker products, that is A=M_2 ⊗ I + I ⊗ M_1∈ R^N× N.  ...  Rational Krylov subspace approximation.  ... 
arXiv:1503.02615v1 fatcat:agko7bdxgnhbrfnvquc2qkqlsq
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