@online{CorrentyFreitagSoodhalter:2023:XXA,
title = {Sparse grid based Chebyshev HOPGD for parameterized linear systems},
author = {Siobh\'{a}n Correnty and Melina A. Freitag and Kirk M. Soodhalter},
url = {https://arxiv.org/abs/2309.14178},
year  = {2023},
date = {2023-09-29},
urldate = {2023-09-29},
note = {Submitted to Journal},
keywords = {tech},
pubstate = {published},
tppubtype = {online}
}
@online{MG-stopping-crit.2022,
title = {The effect of approximate coarsest-level solves on the convergence of multigrid V-cycle methods},
author = {Petr Vacek and Erin Carson and Kirk M. Soodhalter},
url = {https://doi.org/10.48550/arXiv.2306.06182},
doi = {10.48550/arXiv.2306.06182},
year  = {2023},
date = {2023-01-01},
note = {In revision with journal},
keywords = {tech},
pubstate = {published},
tppubtype = {online}
}
@unpublished{actuator-disc-basu.2023,
title = {An iterative method for solving sparse linear algebraic systems with continuum solution dependent right-hand side},
author = {Sudipta Basu and Kirk M. Soodhalter and Breiffni Fitzgerald and Biswajit Basu},
year  = {2023},
date = {2023-01-01},
keywords = {tech},
pubstate = {published},
tppubtype = {unpublished}
}
@article{CorrentyJarlebringSoodhalter:2022:1,
title = {Preconditioned Infinite GMRES for Parameterized Linear Systems},
author = {Siobh\'{a}n Correnty and Elias Jarlebring and Kirk M. Soodhalter},
url = {https://doi.org/10.1137%2F22m1502380},
doi = {10.1137/22m1502380},
year  = {2023},
date = {2023-01-01},
pages = {S120\textendashS141},
publisher = {Society for Industrial \& Applied Mathematics (SIAM)},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@article{HintererHubmerJethwaSoodhalterVanDeVenRamlau:2022:1,
title = {A projected Nesterov-Kaczmarz approach to stellar population-kinematic distribution reconstruction in extragalactic archaeology},
author = {Fabian Hinterer and Simon Hubmer and Prashin Jethwa and Kirk M. Soodhalter and Glenn Ven and Ronny Ramlau},
url = {https://doi-org.elib.tcd.ie/10.1137/22M1503002},
doi = {10.1137/22M1503002},
year  = {2023},
date = {2023-01-01},
urldate = {2023-01-01},
journal = {SIAM Journal on Imaging Sciences},
volume = {16},
number = {1},
pages = {192\textendash222},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@online{SoodhalterWilsonPham:2022:1,
title = {Fast solution of Sylvester-structured systems for spatial source separation of the Cosmic Microwave Background},
author = {Kirk M. Soodhalter and Simon Wilson and Dung Pham},
url = {https://arxiv.org/abs/2204.08057},
year  = {2023},
date = {2023-01-01},
note = {Revisions with Journal},
keywords = {tech},
pubstate = {published},
tppubtype = {online}
}
@inproceedings{electro-mag-scattering-deflation-ssor.2021,
title = {Computation of scattering from rough surfaces using successive symmetric over relaxation and eigenvalue deflation},
author = {Conor Brennan and Imtiaz Islam and Jason Basquill and Kirk M Soodhalter},
year  = {2022},
date = {2022-01-01},
booktitle = {2022 16th European Conference on Antennas and Propagation (EuCAP)},
pages = {1\textendash5},
organization = {IEEE},
abstract = {The problem of computing 2D EM wave scattering from rough surfaces is addressed using an integral equation formulation discretised using the method of moments. Successive Symmetric Over-Relaxation (SSOR) is applied to the governing matrix equations along with eigenvalue deflation which is de- signed to separately account for the effect of those eigenvectors of the iteration matrix that have eigenvalues greater than 1. Numerical results are presented applying the method to a variety of scattering profiles and examining the resultant convergence performance.},
note = {Submitted to Conference},
keywords = {published},
pubstate = {published},
tppubtype = {inproceedings}
}
@unpublished{recycilng-review.2020,
title = {A unifying framework for recycling-based iterative methods (working title)},
author = {Eric Sturler and Kirk M. Soodhalter},
year  = {2022},
date = {2022-01-01},
keywords = {prep},
pubstate = {published},
tppubtype = {unpublished}
}
@article{BasuSoodhalterFitzgeraldBasu:2022:1,
title = {Flow in a large wind field with multiple actuators in the presence of constant vorticity},
author = {Sudipta Lal Basu and Kirk M. Soodhalter and Breiffni Fitzgerald and Biswajit Basu},
url = {https://doi.org/10.1063%2F5.0104902},
doi = {10.1063/5.0104902},
year  = {2022},
date = {2022-01-01},
journal = {Physics of Fluids},
volume = {34},
number = {10},
pages = {103603},
publisher = {AIP Publishing},
abstract = {Study of wind farms is an area of active research. Over the years researchers have proposed simplified wind farm models which define the wake structure in a wind farm and how they affect the performance of the wind turbines. Interestingly these models do not take into account an important aspect of fluid flow i.e. the fluid-structure interaction (FSI) between the wind turbines and the wind, which has an important role to play as well. This motivated researchers to implement numerical analysis tools to model the geometry of the wind turbines in computational fluid dynamics (CFD) based models of a wind farm in order to better understand the wake structure and study the performance of the wind farms. But, modelling the complex geometry of the blades and the turbines makes these models computationally expensive. In this paper, we propose an FSI methdology which can simplify the blade resolving CFD models and eliminate the requirement for modelling these complex geometries during preliminary engineering phase. As an example, we present simulations of up to three back-to-back wind turbines and compare the results with those obtained from wind engineering software, FLORIS. The proposed methodology demonstrates how the approach can be used to develop a relatively less computationally expensive wind farm model. The approach formulated in this paper follows an intermediate way between the analytical wind farm models and CFD models by introducing modifications to one of the most basic wind farm models (Jensen’s model) and using it to develop a simplified CFD model using the Decomposed Immersed Interface Method strategy.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@online{BurkeSoodhalter:2022:1,
title = {Augmented unprojected Krylov subspace methods from an alternative view of an existing framework},
author = {Liam Burke and Kirk M. Soodhalter},
url = {https://arxiv.org/abs/2206.12315},
year  = {2022},
date = {2022-01-01},
keywords = {tech},
pubstate = {published},
tppubtype = {online}
}
@online{BurkeFrommerRamirezSoodhalter:2022:1,
title = {Krylov subspace recycling for matrix functions},
author = {Liam Burke and Andreas Frommer and Gustavo Ramirez Hidalgo and Kirk M. Soodhalter},
url = {https://arxiv.org/abs/2209.14163},
year  = {2022},
date = {2022-01-01},
urldate = {2022-01-01},
keywords = {tech},
pubstate = {published},
tppubtype = {online}
}
@article{Soodhalter:2022:1,
title = {A note on augmented unprojected Krylov subspace methods},
author = {Kirk M. Soodhalter},
url = {https://doi.org/10.1553%2Fetna_vol55s532},
doi = {10.1553/etna_vol55s532},
year  = {2022},
date = {2022-01-01},
volume = {55},
pages = {532\textendash546},
publisher = {Osterreichische Akademie der Wissenschaften, Verlag},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@unpublished{CMB-kron.2021,
title = {A fast linear system solution with application to spatial source separation for the Cosmic Microwave Background},
author = {Kirk M. Soodhalter and Simon Wilson and Jiajin Zhao},
year  = {2021},
date = {2021-01-01},
keywords = {prep},
pubstate = {published},
tppubtype = {unpublished}
}
@unpublished{actuator-disc-basu.2021,
title = {A Decomposed Immersed Interface Method for Simulating a Large Wind Field with An Actuator Disc},
author = {Sudipta Basu and Kirk M. Soodhalter and Breiffni Fitzgerald and Biswajit Basu},
year  = {2021},
date = {2021-01-01},
urldate = {2021-01-01},
abstract = {Fluid-structure interaction (FSI) in a large wind field is still an active area of research with several computational challenges related to the physics of fluid flow yet to be resolved.  In the past, numerical methods involving FSI have been applied to problems having physical domains which are small in size. 
 In this paper, we formulate an approach to simulate two dimensional (2D) wind fields over a domain which is large in size, including the interaction due to the presence of a wind turbine modelled as an actuator disc. We have considered a fluid flow model with constant vorticity and in this connection a reformulation of the potential flow equation has been proposed, with the flow velocity as a primary variable instead of potential function.  In the development of the approach, we use an FSI scheme, Decomposed Immersed Interface Method (DIIM), coupled with an Message Passing Interface-based (MPI) Parallel Numerical Scheme. A modified pre-conditioned bi-conjugate gradient stable (PBiCGStab) algorithm and a red-black coloring scheme have been implemented to adapt the DIIM to the parallel setting, with an aim to address FSI problems in a time and memory efficient way, for such large domains. The proposed approach is suitable for application in studying flows in the wake of wind turbines.},
note = {Submitted to Journal},
keywords = {tech},
pubstate = {published},
tppubtype = {unpublished}
}
@article{regularized-recyc.2020,
title = {Subspace Recycling\textendashBased Regularization Methods},
author = {Ronny Ramlau and Kirk M. Soodhalter and Victoria Hutterer},
url = {https://doi.org/10.1137%2F20m1379617},
doi = {10.1137/20m1379617},
year  = {2021},
date = {2021-01-01},
urldate = {2021-01-01},
journal = {SIAM Journal on Matrix Analysis and Applications},
volume = {42},
number = {4},
pages = {1480\textendash1505},
publisher = {Society for Industrial \& Applied Mathematics (SIAM)},
abstract = {Subspace recycling techniques have been used quite successfully for the acceleration of iterative methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace generated iteratively by a known algorithm with a fixed subspace of vectors which are ``useful'' for solving the problem. Often, this has the effect of inducing a projected version of the original linear system to which the known iterative method is then applied, and this projection can act as a deflation preconditioner, accelerating convergence. Most often, these methods have been applied for the solution of well-posed problems. However, they have also begun to be considered for the solution of ill-posed problems. 
In this paper, we consider subspace augmentation-type iterative schemes applied to linear ill-posed problems in a continuous Hilbert space setting, based on a recently developed framework describing these methods. We show that under suitable assumptions, a recycling method satisfies the formal definition of a regularization, as long as the underlying scheme is itself a regularization. We then develop an augmented subspace version of the gradient descent method and demonstrate its effectiveness, both on an academic Gaussian blur model and on problems arising from the adaptive optics community for the resolution of large sky images by ground-based extremely large telescopes.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@article{RRSW.2017,
title = {Lanczos-based fast blind deconvolution methods},
author = {Laura Dykes and Ronny Ramlau and Lothar Reichel and Kirk M. Soodhalter and Roland Wagner},
url = {http://www.sciencedirect.com/science/article/pii/S0377042720303587},
doi = {https://doi.org/10.1016/j.cam.2020.113067},
issn = {0377-0427},
year  = {2021},
date = {2021-01-01},
journal = {Journal of Computational and Applied Mathematics},
volume = {382},
pages = {113067},
abstract = {The task of restoring an image that has been contaminated by blur and noise arises in many applications. When the blurring matrix (or equivalently, the point-spread function) is explicitly known, this task commonly is referred to as deconvolution. In many applications only an approximation of the blurring matrix is available. The restoration task then is referred to as blind deconvolution. This paper describes a family of blind deconvolution methods that allow a user to adjust the blurring matrix used in the computation to achieve an improved restoration. The methods are inexpensive to use; the major computational effort required for large-scale problems is the partial reduction of an available large symmetric approximate blurring matrix by a few steps of the symmetric Lanczos process. A real-time application to adaptive optics that requires fast blind deconvolution is described.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@online{Soodhalter:2021:1,
title = {A note on augmented unprojected Krylov subspace methods},
author = {Kirk M. Soodhalter},
url = {https://arxiv.org/abs/2106.10050},
year  = {2021},
date = {2021-01-01},
urldate = {2021-01-01},
abstract = {Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in accelerating convergence or conveying knowledge of the solution. Recently, a survey was published, in which a framework describing the vast majority of such methods was proposed [Soodhalter et al, GAMM-Mitt. 2020]. In many of these methods, the Krylov subspace is one generated by the system matrix composed with a projector that depends on the augmentation space. However, it is not a requirement that a projected Krylov subspace be used. There are augmentation methods built on using Krylov subspaces generated by the original system matrix, and these methods also fit into the general framework. 
In this note, we observe that one gains implementation benefits by considering such augmentation methods with unprojected Krylov subspaces in the general framework. We demonstrate this by applying the idea to the R3GMRES method proposed in [Dong et al. ETNA 2014] to obtain a simplified implementation and to connect that algorithm to early augmentation schemes based on flexible preconditioning [Saad. SIMAX 1997].},
note = {Revision with Journal; arXiv preprint available},
keywords = {tech},
pubstate = {published},
tppubtype = {online}
}
@misc{Soodhalter:2021:2,
title = {kirkmsoodhalter/r3gmres-simplified: R3GMRES Simplified Implementation},
author = {Kirk M. Soodhalter},
url = {https://zenodo.org/record/4975990},
doi = {10.5281/ZENODO.4975990},
year  = {2021},
date = {2021-01-01},
publisher = {Zenodo},
note = {urlhttps://zenodo.org/record/4975990},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@article{BAGS.2019h,
title = {Error Analysis and Improving the Accuracy of Winograd Convolution for Deep Neural Networks},
author = {Barbara Barabasz and Andrew Anderson and Kirk M. Soodhalter and David Gregg},
url = {https://doi.org/10.1145/3412380},
doi = {10.1145/3412380},
issn = {0098-3500},
year  = {2020},
date = {2020-11-01},
journal = {ACM Trans. Math. Softw.},
volume = {46},
number = {4},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
abstract = {Popular deep neural networks (DNNs) spend the majority of their execution time computing convolutions. The Winograd family of algorithms can greatly reduce the number of arithmetic operations required and is present in many DNN software frameworks. However, the performance gain is at the expense of a reduction in floating point (FP) numerical accuracy. In this paper, we analyse the worst case FP error and prove the estimation of norm and conditioning of the algorithm. We show that the bound grows exponentially with the size of the convolution, but the error bound of the textitmodified algorithm is smaller than the original one. We propose several methods for reducing FP error. We propose a canonical evaluation ordering based on Huffman coding that reduces summation error. We study the selection of sampling "points" experimentally and find empirically good points for the most important sizes. We identify the main factors associated with good points. In addition, we explore other methods to reduce FP error, including mixed-precision convolution, and pairwise summation across DNN channels. Using our methods we can significantly reduce FP error for a given block size, which allows larger block sizes and reduced computation.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@article{SdSK.2020-GAMM,
title = {A survey of subspace recycling iterative methods},
author = {Kirk M. Soodhalter and Eric Sturler and Misha E. Kilmer},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/gamm.202000016},
doi = {https://doi.org/10.1002/gamm.202000016},
year  = {2020},
date = {2020-01-01},
journal = {GAMM-Mitteilungen},
volume = {43},
number = {4},
pages = {e202000016},
abstract = {This survey concerns subspace recycling methods, a popular class of iterative methods that enable effective reuse of subspace information in order to speed up convergence and find good initial guesses over a sequence of linear systems with slowly changing coefficient matrices, multiple right-hand sides, or both. The subspace information that is recycled is usually generated during the run of an iterative method (usually a Krylov subspace method) on one or more of the systems. Following introduction of definitions and notation, we examine the history of early augmentation schemes along with deflation preconditioning schemes and their influence on the development of recycling methods. We then discuss a general residual constraint framework through which many augmented Krylov and recycling methods can both be viewed. We review several augmented and recycling methods within this framework. We then discuss some known effective strategies for choosing subspaces to recycle before taking the reader through more recent developments that have generalized recycling for (sequences of) shifted linear systems, some of them with multiple right-hand sides in mind. We round out our survey with a brief review of application areas that have seen benefit from subspace recycling methods.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@article{2019arXiv190703677K,
title = {Admissible and Attainable Convergence Behavior of Block Arnoldi and GMRES},
author = {Marie Kub\'{i}nov\'{a} and Kirk M. Soodhalter},
url = {https://arxiv.org/abs/1907.03677},
doi = {10.1137/19M1272469},
year  = {2020},
date = {2020-01-01},
journal = {SIAM Journal on Matrix Analysis and Applications},
volume = {41},
number = {2},
pages = {464-486},
abstract = {It is well-established that any non-increasing convergence curve is possible for GMRES and a family of pairs $(A,b)$ can be constructed for which GMRES exhibits a given convergence curve with A having arbitrary spectrum. No analog of this result has been established for block GMRES, wherein multiple right-hand sides are considered. By reframing the problem as a single linear system over a ring of square matrices, we develop convergence results for block Arnoldi and block GMRES. In particular, we show what convergence behavior is admissible for block GMRES and how the matrices and right-hand sides producing any admissible behavior can be constructed. Moreover, we show that the convergence of the block Arnoldi method for eigenvalue approximation can be almost fully independent of the convergence of block GMRES for the same coefficient matrix and the same starting vectors.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@techreport{HJS.2018,
title = {Fast computation of the magnetization of an air-gapped transformer using a boundary element method.},
author = {Peter Hamberger and Stefan Janecek and Kirk M. Soodhalter},
year  = {2018},
date = {2018-01-01},
urldate = {2018-01-01},
institution = {Private/Industrial},
abstract = {The computation of the magnetic field generated by a transformer with an air-gapped core is considered. We describe how 
this can be done using a boundary element method to obtain an approximate solution rapidly enough to be used in 
the fabrication setting.},
keywords = {tech},
pubstate = {published},
tppubtype = {techreport}
}
@article{HS.2016,
title = {A Modified Implementation of MINRES to Monitor Residual Subvector Norms for Block Systems},
author = {Roland Herzog and Kirk M. Soodhalter},
url = {https://arxiv.org/abs/1603.04475},
doi = {10.1137/16M1093021},
year  = {2017},
date = {2017-01-01},
journal = {SIAM Journal on Scientific Computing},
volume = {39},
number = {6},
pages = {A2645-A2663},
abstract = {Saddle-point systems, i.e., structured linear systems with symmetric matrices are considered. A modified implementation of (preconditioned) MINRES is derived which allows to monitor the norms of the subvectors individually. Compared to the implementation from the textbook of [Elman, Sylvester and Wathen, Oxford University Press, 2014], our method requires one extra vector of storage and no additional applications of the preconditioner. Numerical experiments are included.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@article{NRS.2015,
title = {Solution of coupled differential equations arising from imbalance problems},
author = {Jenny Niebsch and Ronny Ramlau and Kirk M. Soodhalter},
url = {http://etna.mcs.kent.edu/vol.46.2017/pp89-106.dir/pp89-106.pdf},
year  = {2017},
date = {2017-01-01},
journal = {Electronic Transactions on Numerical Analysis},
volume = {46},
pages = {89\textendash106},
abstract = {We investigate the efficient solution of a set of coupled ordinary differential equations arising from a model describing vibrations of a wind turbine induced by imbalances of its spinning blades. The Forward Problem (computing vibrations from imbalances) admits a coupled integral equation formulation. Each integral equation is solved over the same underlying Hilbert space $$mathbbH$$. We observe that these coupled integral equations can be represented as one compact operator acting on the tensor product space $$mathbbR^N\otimes mathbbH$$, where N is the number of coupled equations. A Galerkin discretization leads to a linear system of dimension nN with corresponding Kronecker product structure, where n is the number of basis elements used to discretize $$mathbbH$$. Such systems can be solved using a variety of techniques which exploit the Kronecker structure. We demonstrate the effectiveness of exploiting the tensor structure with numerical experiments and show that our results agree with data recorded from actual wind turbines.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@article{S.2014-3,
title = {Stagnation of block GMRES and its relationship to block FOM},
author = {Kirk M. Soodhalter},
url = {http://arxiv.org/abs/1411.7801},
year  = {2017},
date = {2017-01-01},
urldate = {2017-01-01},
journal = {Electronic Transactions on Numerical Analysis},
volume = {46},
pages = {162\textendash189},
abstract = {We examine the the convergence behavior of block GMRES and block FOM 
	and characterize the phenomenon of stagnation in block GMRES. Stagnation 
	is then related to the behavior of the block FOM method. As in previous 
	work, we generalize the block FOM method to generate well-defined 
	approximations in the case that block FOM would normally break down, 
	and these generalized solutions are used in our analysis. Two toy 
	numerical examples are given to illustrate what we have proven and 
	we also apply both block methods to a small application problem to 
	demonstrate the validity of the analysis in non-pathological cases.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@misc{soodhalter_2016_56157,
title = {Shift Block Recycled GMRES: A Sylvester equation 
 based minimum residual Krylov subspace method for 
 solving multiple shifted systems},
author = {Kirk M. Soodhalter},
url = {https://doi.org/10.5281/zenodo.56157},
doi = {10.5281/zenodo.56157},
year  = {2016},
date = {2016-05-01},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@misc{michael_l_parks_2016_48836,
title = {Block GCRO-DR: A version of the recycled GMRES 
 method using block Krylov subspaces and harmonic 
 Ritz vectors},
author = {Michael L. Parks and Kirk M. Soodhalter and Daniel B. Szyld},
url = {https://doi.org/10.5281/zenodo.48836},
doi = {10.5281/zenodo.48836},
year  = {2016},
date = {2016-04-01},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@article{S.2014,
title = {Two recursive GMRES-type methods for shifted linear systems with general preconditioning},
author = {Kirk M. Soodhalter},
url = {http://etna.mcs.kent.edu/vol.45.2016/pp499-523.dir/pp499-523.pdf},
year  = {2016},
date = {2016-03-01},
journal = {Electronic Transactions on Numerical Analysis)},
volume = {45},
pages = {499\textendash523},
abstract = {We present two minimum residual methods for solving sequences of shifted linear systems, the right-preconditioned shifted GMRES and shifted Recycled GMRES algorithms which use a seed projection strategy often employed to solve multiple related problems. These methods are compatible with a general preconditioning of all systems, and, when restricted to right preconditioning, require no extra applications of the operator or preconditioner. These seed projection methods perform a minimum residual iteration for the base system while improving the approximations for the shifted systems at little additional cost. The iteration continues until the base system approximation is of satisfactory quality. The method is then recursively called for the remaining unconverged systems. We present both methods inside of a general framework which allows these techniques to be extended to the setting of flexible preconditioning and inexact Krylov methods. We present some analysis of such methods and numerical experiments demonstrating the effectiveness of the proposed algorithms.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@misc{herzog_2016_47393,
title = {SUBMINRES. A modified implementation of MINRES to 
 monitor residual subvector norms for block systems},
author = {Roland Herzog and Kirk M. Soodhalter},
url = {https://doi.org/10.5281/zenodo.47393},
doi = {10.5281/zenodo.47393},
year  = {2016},
date = {2016-03-01},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@article{S2.2014,
title = {Block Krylov Subspace Recycling for Shifted Systems with Unrelated Right-Hand Sides},
author = {Kirk M. Soodhalter},
url = {https://arxiv.org/abs/1412.0393},
doi = {10.1137/140998214},
year  = {2016},
date = {2016-01-01},
journal = {SIAM Journal on Scientific Computing},
volume = {38},
number = {1},
pages = {A302-A324},
abstract = {Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial residuals must be collinear and this collinearity must be maintained at restart. Thus we cannot simultaneously solve shifted systems with unrelated right-hand sides using this strategy, and all shifted residuals cannot be simultaneously minimized over a Krylov subspace such that collinearity is maintained. It has been shown that this renders them generally incompatible with techniques of subspace recycling [Soodhalter et al. APNUM '14]. 
 This problem, however, can be overcome. By interpreting a family of shifted systems as one Sylvester equation, we can take advantage of the known "shift invariance" of the Krylov subspace generated by the Sylvester operator. Thus we can simultaneously solve all systems over one block Krylov subspace using FOM or GMRES type methods, even when they have unrelated right-hand sides. Because residual collinearity is no longer a requirement at restart, these methods are fully compatible with subspace recycling techniques. Furthermore, we realize the benefits of block sparse matrix operations which arise in the context of high-performance computing applications. 
 In this paper, we discuss exploiting this Sylvester equation point of view which has yielded methods for shifted systems which are compatible with unrelated right-hand sides. From this, we propose a recycled GMRES method for simultaneous solution of shifted systems.Numerical experiments demonstrate the effectiveness of the methods.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@techreport{PSS.2016,
title = {A block Recycled GMRES method with investigations into aspects of solver performance},
author = {Michael L. Parks and Kirk M. Soodhalter and Daniel B. Szyld},
year  = {2016},
date = {2016-01-01},
abstract = {We propose a block Krylov subspace version of the GCRO-DR method proposed in [Parks et al. SISC 2005], which is an iterative method allowing for the efficient minimization of the the residual over an augmented block Krylov subspace. We offer a clean derivation of the method and discuss methods of selecting recycling subspaces at restart as well as implementation decisions in the context of high-performance computing. Numerical experiments are split into those demonstrating convergence properties and those demonstrating the data movement and cache efficiencies of the dominant operations of the method, measured using processor monitoring code from Intel.},
note = {arXiv preprint: urlhttps://arxiv.org/abs/1604.01713},
keywords = {tech},
pubstate = {published},
tppubtype = {techreport}
}
@article{S.2013,
title = {A block MINRES algorithm based on the banded Lanczos method},
author = {Kirk M. Soodhalter},
url = {http://arxiv.org/abs/1301.2102},
doi = {10.1007/s11075-014-9907-z},
issn = {1017-1398},
year  = {2015},
date = {2015-01-01},
urldate = {2015-01-01},
journal = {Numerical Algorithms},
volume = {69},
issue = {3},
pages = {473-494},
abstract = {We present a block MINRES algorithm for symmetric indefinite matrices 
	based on an alternate method for 
	constructing the block Lanczos vectors. This method allows us to compare 
	performance with the 
	standard MINRES algorithm iteration for iteration, and it handles 
	removal of dependent Lanczos vectors 
	more gracefully. We describe both a theoretical derivation of the 
	algorithm as well as practical implementation 
	details. Some numerical results are shown to illustrate performance 
	on some sample problems. We also 
	present some experiments to show how the relationship between right-hand 
	sides affects the performance 
	of this method.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@misc{recyc-gmres-shift-precond,
title = {Recursive recycled GMRES methods for shifted systems with preconditioning},
author = {Kirk M. Soodhalter},
year  = {2015},
date = {2015-01-01},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@article{SSX.2014,
title = {Krylov subspace recycling for sequences of shifted linear systems},
author = {Kirk M. Soodhalter and Daniel B. Szyld and Fei Xue},
url = {http://arxiv.org/abs/1301.2650},
doi = {10.1016/j.apnum.2014.02.006},
year  = {2014},
date = {2014-01-01},
urldate = {2014-01-01},
journal = {Applied Numerical Mathematics},
volume = {81C},
pages = {105\textendash118},
abstract = {We study the use of Krylov subspace recycling for the solution of 
	a sequence of slowly-changing families of linear systems, where each 
	family consists of shifted linear systems that differ in the coefficient 
	matrix only by multiples of the identity. Our aim is to explore the 
	simultaneous solution of each family of shifted systems within the 
	framework of subspace recycling, using one augmented subspace to 
	extract candidate solutions for all the shifted systems. The ideal 
	method would use the same augmented subspace for all systems and 
	have fixed storage requirements, independent of the number of shifted 
	systems per family. We show that a method satisfying both requirements 
	cannot exist in this framework. 
	As an alternative, we introduce two schemes. One constructs a separate 
	deflation space for each shifted system but solves each family of 
	shifted systems simultaneously. The other builds only one recycled 
	subspace and constructs approximate corrections to the solutions 
	of the shifted systems at each cycle of the iterative linear solver 
	while only minimizing the base system residual. At convergence of 
	the base system solution, we apply the method recursively to the 
	remaining unconverged systems. We present numerical examples involving 
	systems arising in lattice quantum chromodynamics.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@misc{shift-recyc-gmres,
title = {Recycled GMRES for Shifted Linear Systems},
author = {Kirk M. Soodhalter and Daniel B. Szyld and Fei Xue},
year  = {2014},
date = {2014-01-01},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@misc{bminres,
title = {Block MINRES based on Banded Lanczos},
author = {Kirk M. Soodhalter},
year  = {2014},
date = {2014-01-01},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@techreport{HSS.2013,
title = {Automatic Detection of Weather Fronts},
author = {Meredith Hegg and Benjamin Seibold and Kirk M. Soodhalter},
year  = {2013},
date = {2013-01-01},
urldate = {2013-01-01},
abstract = {We propose a new algorithm for the detection of weather fronts from 
	discrete weather data generated by weather forecasting software. 
	Given data for a snapshop in time, the algorithm detects cold and 
	warm fronts and draws them on a weather map. Given multiple time 
	frames, the algorithm creates a smooth animation of the weather fronts 
	as they travel. Detection of occluded fronts (formed when a fast-moving 
	cold front overtakes a warm front) is also treated.},
keywords = {tech},
pubstate = {published},
tppubtype = {techreport}
}
@article{ESSSX.2012,
title = {Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian 
	Matrices},
author = {Mark Embree and Josef A. Sifuentes and Kirk M. Soodhalter and Daniel B. Szyld and Fei Xue},
url = {files/pdfs/Nearly-Herm.pdf},
doi = {10.1137/110825327},
year  = {2012},
date = {2012-01-01},
urldate = {2012-01-01},
journal = {SIAM. J. Matrix Anal. and Appl.},
volume = {33-2},
pages = {480-500},
abstract = {The progressive GMRES algorithm, introduced by Beckermann and Reichel 
	in 2008, is a residual-minimizing short-recurrence Krylov subspace 
	method for solving a linear system in which the coefficient matrix 
	has a low-rank skew-Hermitian part. We analyze this algorithm, observing 
	a critical instability that makes the method unsuitable for some 
	problems. To work around this issue we introduce a different short-term 
	recurrence method based on Krylov subspaces for such matrices, which 
	can be used as either a solver or a preconditioner. Numerical experiments 
	compare this method to alternative algorithms.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@phdthesis{S.2012,
title = {Krylov subspace methods with fixed memory requirements: Nearly Hermitian 
	linear systems and subspace recycling},
author = {Kirk M. Soodhalter},
year  = {2012},
date = {2012-01-01},
urldate = {2012-01-01},
school = {Temple University},
abstract = {Krylov subspace iterative methods provide an effective tool for reducing 
	the solution of large linear systems to a size for which a direct 
	solver may be applied. However, the problems of limited storage and 
	speed are still a concern. Therefore, in this dissertation work, 
	we present iterative Krylov subspace algorithms for non-Hermitian 
	systems which do have fixed memory requirements and have favorable 
	convergence characteristics.},
note = {PhD Thesis, Supervisor: Daniel B. Szyld},
keywords = {thesis},
pubstate = {published},
tppubtype = {phdthesis}
}
@article{MRS.2010,
title = {The action of Hecke operators on hypergeometric functions},
author = {Victor H. Moll and Sinai Robins and Kirk Soodhalter},
url = {http://arxiv.org/abs/1005.2946},
doi = {10.1017/S1446788710001461},
issn = {1446-7887},
year  = {2010},
date = {2010-01-01},
urldate = {2010-01-01},
journal = {J. Aust. Math. Soc.},
volume = {89},
number = {1},
pages = {51\textendash74},
abstract = {We study the action of the Hecke operators Un on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set n^a with a an integer and n a positive integer, and that the polylogarithms play a dominant role in the study of the eigenfunctions of the Hecke operators Un on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apriori unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}
@misc{schur-compl-meth,
title = {Schur Complement Method (for nearly-Hermitian linear systems)},
author = {Josef Sifuentes and Kirk M. Soodhalter},
year  = {2010},
date = {2010-01-01},
keywords = {software},
pubstate = {published},
tppubtype = {misc}
}
@other{S.2004,
title = {An analysis of the Landen transformation},
author = {Kirk M. Soodhalter},
year  = {2004},
date = {2004-01-01},
urldate = {2004-01-01},
school = {Tulane University},
note = {Tulane University, Supervisor: Victor H. Moll},
keywords = {thesis},
pubstate = {published},
tppubtype = {other}
}
@unpublished{HS.2020,
title = {Revisiting GMRES from the Hilbert space perspective (working title)},
author = {Roland Herzog and Kirk M. Soodhalter},
keywords = {prep},
pubstate = {published},
tppubtype = {unpublished}
}
@unpublished{KMS.2020,
title = {Augmented Arnoldi-Tikhonov methods (working title)},
author = {Kirk M. Soodhalter},
keywords = {prep},
pubstate = {published},
tppubtype = {unpublished}
}
@unpublished{enriched-galerkin.2023,
title = {A general approach for enriching Galerkin finite element methods},
author = {Niall Madden and Kirk M. Soodhalter},
keywords = {prep},
pubstate = {published},
tppubtype = {unpublished}
}
@unpublished{BurkeGuettelSoodhalter:2023:1,
title = {GMRES-SDR: GMRES with randomized sketching and deflated restarting},
author = {Liam Burke and Stefan G\"{u}ttel and Kirk M. Soodhalter},
keywords = {prep},
pubstate = {published},
tppubtype = {unpublished}
}