I work primarily in the field of numerical linear algebra and Krylov subspace iterative methods. I enjoy work in the entire spectrum, from mathematical theory and algorithm design to the more practical implementation of an algorithm for high performance computing applications. My current focuses are:
Augmented/deflated/recycled Krylov subspace methods
- development of methods for specific applications
- analysis of augmentation subspace choices
Block Krylov subspace methods: analysis and high-performance implementations
- analysis and characterization of the behavior of these methods
- development of high-performance implementations
Parameterized linear systems (focusing primarily on shifted systems) and matrix equations
- development of techniques compatible with preconditioning
- iterative solvers for Sylvester and Lyapunov equations
- high-performance implementations
- using augmented Krylov subspace techniques
- development of methods compatible with a wider range of parameterized problems
- analysis of newer Krylov subspace technologies in the ill-posed problem setting
- analysis of other types of iterative schemes
- development of algorithms which leverage more recently proposed iterative techniques
- matrix equation formulations of large-scale ill-posed problems
My research papers are
||| A modified implementation of MINRES to monitor residual subvector norms for block systems , In ArXiv e-prints (To appear in SIAM Journal on Scientific Computing)), 2017. (Matlab code)
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.
||| Stagnation of block GMRES and its relationship to block FOM , In Electronic Transactions on Numerical Analysis, volume 46, 2017.
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.
||| Lanczos-based fast blind deconvolution methods , Technical report, Johann Radon Institute for Computation and Applied Mathematics, 2017.
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.
||| Solution of coupled differential equations arising from imbalance problems , In Electronic Transactions on Numerical Analysis, volume 46, 2017.
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 $$\mathbbR^Nøtimes \mathbbH$$, 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.
||| Two recursive GMRES-type methods for shifted linear systems with general preconditioning , In Electronic Transactions on Numerical Analysis), volume 45, 2016.
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.
|||Block Krylov Subspace Recycling for Shifted Systems with Unrelated Right-Hand Sides , In SIAM Journal on Scientific Computing, volume 38, 2016. (Matlab code) [bib] [pdf] [doi]|
||| A block Recycled GMRES method with investigations into aspects of solver performance , In ArXiv e-prints (and submitted for publication), 2016. (Supplementary Materials)(Matlab code)(Trilinos code)
We extend the GCRODR algorithm (GMRES with recycling) of Parks et al [SIAM J. Sci. Comput. 2006] to the block Krylov subspace setting for solving systems with multiple physical right-hand sides and for acceleration through the introduction of fictitious right-hand sides. We demonstrate the method's effectiveness in reducing the cost of a Newton iteration arising from fluid Density Functional Theory.
||| A block MINRES algorithm based on the banded Lanczos method , In Numerical Algorithms, volume 69, 2015.
[bib] [pdf][abstract] [doi]|
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.
||| Krylov subspace recycling for sequences of shifted linear systems , In Applied Numerical Mathematics, volume 81C, 2014.
[bib] [pdf][abstract] [doi]|
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.
||| Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian Matrices , In SIAM. J. Matrix Anal. and Appl., volume 33-2, 2012.
[bib] [pdf][abstract] [doi]|
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.
||| The action of Hecke operators on hypergeometric functions , In J. Aust. Math. Soc., volume 89, 2010.
[bib] [pdf][abstract] [doi]|
We study the action of the Hecke operators \(\lbrace U_n \rbrace_1^\infty\) 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 \(\lbrace na:n\in \mathbb N , a\in \mathbb Z \rbrace\), and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators \(\lbrace U_n \rbrace_1^\infty\) on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apparently unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.
|||Automatic Detection of Weather Fronts , In Preparation, 20xx.
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.
My theses (bachelors and doctoral)
|||Krylov subspace methods with fixed memory requirements: Nearly Hermitian linear systems and subspace recycling , PhD thesis, Temple University, 2012.
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.
|||An analysis of the Landen transformation , 2004. (Bachelor's Thesis, Tulane University) [bib] [pdf]|