# Eugene Vecharynski

**Position**

Project Scientist, May 2016-present

Postdoctoral Research Fellow, Computational Research Division, January 2013 - May 2016.

**Education**

- Ph.D. Applied Mathematics, University of Colorado Denver, 2011.
- M.S. Applied Mathematics, Belarusian State University, 2006.
- B.S. Applied Mathematics and Computer Science, Belarusian State University, 2005.

**Short Bio**

Eugene Vecharynski is a project scientist with the Scalable Solvers Group in the Computational Research Division at the Lawrence Berkeley National Laboratory. He is currently working on numerical solution of extreme-scale eigenvalue problems arising in electronic structure calculations.

Vecharynski's research is primarily in the field of preconditioned iterative solvers for large algebraic systems, including linear and nonlinear systems and eigenvalue problems. He has contributed to various aspects of matrix computations, such as development of new solvers, theoretical analysis of Krylov subspace methods, and novel preconditioning techniques. Results of his work have been used in a number of applications in physical sciences and data analytics.

Before joining LBNL, Eugene Vecharynski was a postdoc with Yousef Saad and Masha Sosonkina at the Department of Computer Science and Engineering, University of Minnesota, and spent a semester as a Visiting Assistant Professor at the Department of Mathematics and Statistics, Georgia State University. He has a Ph.D. from the Department of Mathematical and Statistical Sciences, University of Colorado Denver, advised by Andrew Knyazev.

## Journal Articles

### E. Vecharynski and C. Yang, "Preconditioned iterative methods for eigenvalue counts", Lecture Notes in Computational Science, January 1, 2017,

### E. Vecharynski, A. Knyazev, "Preconditioned steepest descent-like methods for symmetric indefinite systems", Linear Algebra and its Applications, Vol. 511, pp. 274–295, 2016,

We construct preconditioned steepest descent (PSD)-like methods for iterative solution of symmetric indefinite linear systems using symmetric and positive definite (SPD) preconditioners. Our construction is based on a locally optimal residual minimization over two-dimensional subspaces, mathematically equivalent in exact arithmetic to preconditioned MINRES (PMINRES) restarted after every two steps. A convergence bound is derived. If certain information on the spectrum of the preconditioned system is available, we present a simpler PSD-like algorithm that performs only one-dimensional residual minimization. Search direction randomization for accelerating this algorithm is discussed. Our primary goal is to bridge the theoretical gap between the optimal (PMINRES) and PSD-like methods for solving symmetric indefinite systems. We also demonstrate situations where the suggested PSD-like schemes can be preferable to the optimal PMINRES iteration.

### R. Li, Y. Xi, E. Vecharynski, C. Yang, and Y. Saad, "A Thick-Restart Lanczos algorithm with polynomial filtering for Hermitian eigenvalue problems", SIAM Journal on Scientific Computing, Vol. 38, Issue 4, pp. A2512–A2534, 2016, doi: 10.1137/15M1054493

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a Thick-Restart version of the Lanczos algorithm with deflation ('locking') and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a 'spectrum-slicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.

### J. R. Jones, F.-H. Rouet, K. V. Lawler, E. Vecharynski, K. Z. Ibrahim, S. Williams, B. Abeln, C. Yang, C. W. McCurdy, D. J. Haxton, X. S. Li, T. N. Rescigno, "An efficient basis set representation for calculating electrons in molecules", Journal of Molecular Physics, 2016, doi: 10.1080/00268976.2016.1176262

The method of McCurdy, Baertschy, and Rescigno, J. Phys. B, 37, R137 (2004) is generalized to obtain a straightforward, surprisingly accurate, and scalable numerical representation for calculating the electronic wave functions of molecules. It uses a basis set of product sinc functions arrayed on a Cartesian grid, and yields 1 kcal/mol precision for valence transition energies with a grid resolution of approximately 0.1 bohr. The Coulomb matrix elements are replaced with matrix elements obtained from the kinetic energy operator. A resolution-of-the-identity approximation renders the primitive one- and two-electron matrix elements diagonal; in other words, the Coulomb operator is local with respect to the grid indices. The calculation of contracted two-electron matrix elements among orbitals requires only O(N log(N)) multiplication operations, not O(N^4), where N is the number of basis functions; N = n^3 on cubic grids. The representation not only is numerically expedient, but also produces energies and properties superior to those calculated variationally. Absolute energies, absorption cross sections, transition energies, and ionization potentials are reported for one- (He^+, H_2^+ ), two- (H_2, He), ten- (CH_4) and 56-electron (C_8H_8) systems.

### E. Vecharynski, C. Yang, and F. Xue, "Generalized preconditioned locally harmonic residual method for non-Hermitian eigenproblems", SIAM Journal on Scientific Computing, Vol. 38, No. 1, pp. A500–A527, 2016, doi: 10.1137/15M1027413

We introduce the Generalized Preconditioned Locally Harmonic Residual (GPLHR) method for solving standard and generalized non-Hermitian eigenproblems. The method is particularly useful for computing a subset of eigenvalues, and their eigen- or Schur vectors, closest to a given shift. The proposed method is based on block iterations and can take advantage of a preconditioner if it is available. It does not need to perform exact shift-and-invert transformation. Standard and generalized eigenproblems are handled in a unified framework. Our numerical experiments demonstrate that GPLHR is generally more robust and efficient than existing methods, especially if the available memory is limited.

### E. Vecharynski, "A generalization of Saad's bound on harmonic Ritz vectors of Hermitian matrices", Linear Algebra and its Applications, Vol. 494, pp. 219-235, 2016, doi: 10.1016/j.laa.2016.01.013

We prove a Saad's type bound for harmonic Ritz vectors of a Hermitian matrix. The new bound reveals a dependence of the harmonic Rayleigh-Ritz procedure on the condition number of a shifted problem operator. Several practical implications are discussed. In particular, the bound motivates incorporation of preconditioning into the harmonic Rayleigh-Ritz scheme.

### D. B. Szyld, E. Vecharynski, and F. Xue, "Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. II. Interior eigenvalues.", SIAM Journal on Scientific Computing, Vol. 37, Issue 6, pp. A2969-A2997, 2015,

We consider the solution of large-scale nonlinear algebraic Hermitian eigenproblems of the form $T(\lambda)v=0$ that admit a variational characterization of eigenvalues. These problems arise in a variety of applications and are generalizations of linear Hermitian eigenproblems $Av\!=\!\lambda Bv$. In this paper, we propose a Preconditioned Locally Minimal Residual (PLMR) method for efficiently computing interior eigenvalues of problems of this type. We discuss the development of search subspaces, preconditioning, and eigenpair extraction procedure based on the refined Rayleigh-Ritz projection. Extension to the block methods is presented, and a moving-window style soft deflation is described. Numerical experiments demonstrate that PLMR methods provide a rapid and robust convergence towards interior eigenvalues. The approach is also shown to be efficient and reliable for computing a large number of extreme eigenvalues, dramatically outperforming standard preconditioned conjugate gradient methods.

### E. Vecharynski, A. Knyazev, "Preconditioned Locally Harmonic Residual Method for computing interior eigenpairs of certain classes of Hermitian matrices", SIAM Journal on Scientific Computing, Vol. 37, Issue 5, pp. S3–S29, 2015,

We propose a Preconditioned Locally Harmonic Residual (PLHR) method for computing several interior eigenpairs of a generalized Hermitian eigenvalue problem, without traditional spectral transformations, matrix factorizations, or inversions. PLHR is based on a short-term recurrence, easily extended to a block form, computing eigenpairs simultaneously. PLHR can take advantage of Hermitian positive definite preconditioning, e.g., based on an approximate inverse of an absolute value of a shifted matrix, introduced in [SISC, 35 (2013), pp. A696–A718]. Our numerical experiments demonstrate that PLHR is efficient and robust for certain classes of large-scale interior eigenvalue problems, involving Laplacian and Hamiltonian operators, especially if memory requirements are tight.

### E. Vecharynski, C. Yang, J. E. Pask, "A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix", Journal of Computational Physics, Vol. 290, pp. 73–89, 2015,

We present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix *A*. We are interested in the case in which the dimension of the invariant subspace is large (e.g., over several hundreds or thousands) even though it may still be small relative to the dimension of *A*. These problems arise from, for example, density functional theory (DFT) based electronic structure calculations for complex materials. The key feature of our algorithm is that it performs fewer Rayleigh–Ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the Davidson algorithm. It is a block algorithm, and hence can take advantage of efficient BLAS3 operations and be implemented with multiple levels of concurrency. We discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer.

### D. Zuev, E. Vecharynski, C. Yang, N. Orms, and A.I. Krylov, "New algorithms for iterative matrix-free eigensolvers in quantum chemistry", Journal of Computational Chemistry, Vol. 36, Issue 5, pp. 273–284, 2015,

New algorithms for iterative diagonalization procedures that solve for a small set of eigen-states of a large matrix are described. The performance of the algorithms is illustrated by calculations of low and high-lying ionized and electronically excited states using equation-of-motion coupled-cluster methods with single and double substitutions (EOM-IP-CCSD and EOM-EE-CCSD). We present two algorithms suitable for calculating excited states that are close to a specified energy shift (interior eigenvalues). One solver is based on the Davidson algorithm, a diagonalization procedure commonly used in quantum-chemical calculations. The second is a recently developed solver, called the “Generalized Preconditioned Locally Harmonic Residual (GPLHR) method.” We also present a modification of the Davidson procedure that allows one to solve for a specific transition. The details of the algorithms, their computational scaling, and memory requirements are described. The new algorithms are implemented within the EOM-CC suite of methods in the Q-Chem electronic structure program.

### E. Vecharynski and Y. Saad, "Fast updating algorithms for latent semantic indexing", SIAM Journal on Matrix Analysis and Applications, Vol. 35, Issue 3, pp. 1105–1131, 2014,

This paper discusses a few algorithms for updating the approximate singular value decomposition (SVD) in the context of information retrieval by latent semantic indexing (LSI) methods. A unifying framework is considered which is based on Rayleigh–Ritz projection methods. First, a Rayleigh–Ritz approach for the SVD is discussed and it is then used to interpret the Zha and Simon algorithms [SIAM J. Sci. Comput., 21 (1999), pp. 782–791]. This viewpoint leads to a few alternatives whose goal is to reduce computational cost and storage requirement by projection techniques that utilize subspaces of much smaller dimension. Numerical experiments show that the proposed algorithms yield accuracies comparable to those obtained from standard ones at a much lower computational cost.

### E. Vecharynski, Y. Saad, and M. Sosonkina, "Graph partitioning using matrix values for preconditioning symmetric positive definite systems", SIAM Journal on Scientific Computing Vol. 36, Issue 1, pp. A63-A87, 2014,

Prior to the parallel solution of a large linear system, it is required to perform a partitioning of its equations/unknowns. Standard partitioning algorithms are designed using the considerations of the efficiency of the parallel matrix-vector multiplication, and typically disregard the information on the coefficients of the matrix. This information, however, may have a significant impact on the quality of the preconditioning procedure used within the chosen iterative scheme. In the present paper, we suggest a spectral partitioning algorithm, which takes into account the information on the matrix coefficients and constructs partitions with respect to the objective of enhancing the quality of the nonoverlapping additive Schwarz (block Jacobi) preconditioning for symmetric positive definite linear systems. For a set of test problems with large variations in magnitudes of matrix coefficients, our numerical experiments demonstrate a noticeable improvement in the convergence of the resulting solution scheme when using the new partitioning approach.

### E. Vecharynski and A. Knyazev, "Absolute value preconditioning for symmetric indefinite linear systems", SIAM Journal on Scientific Computing Vol. 35, Issue 2, pp. A696-A718, 2013,

We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite matrices. The strategy, called absolute value preconditioning, is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the matrix nor its exact inverse are computationally feasible to construct in general. However, we provide a practical example of an SPD preconditioner that is based on the suggested approach. In this example we consider a model problem with a shifted discrete negative Laplacian and suggest a geometric multigrid (MG) preconditioner, where the inverse of the matrix absolute value appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new MG preconditioner, which leads to a robust iterative scheme with minimalist memory requirements.

### E. Vecharynski and J. Langou, "Any admissible cycle-convergence behavior is possible for restarted GMRES at its initial cycles", Numerical Linear Algebra with Applications Vol. 18, Issue 3, pp. 499-511, 2011,

We show that any admissible cycle-convergence behavior is possible for restarted GMRES at a number of initial cycles, moreover the spectrum of the coefficient matrix alone does not determine this cycle-convergence. The latter can be viewed as an extension of the result of Greenbaum, Pták and Strakosˇ (SIAM Journal on Matrix Analysis and Applications 1996; 17(3):465–469) to the case of restarted GMRES.

### E. Vecharynski, J. Langou, "The cycle-convergence of restarted GMRES for normal matrices is sublinear", SIAM Journal on Scientific Computing Vol. 32, Issue 1, pp. 186-196, 2010,

We prove that the cycle-convergence of the restarted GMRES applied to a system of linear equations with a normal coefficient matrix is sublinear.

### L. Pilipchuk, E. Vecharynski, Yu. H. Pesheva, "Solution of Large Linear Systems with Embedded Network Structure for a Non-Homogeneous Network Flow Programming Problem", Mathematica Balkanica Vol. 22, Fasc. 3-4, p. 235-254, 2008,

In the paper the linear underdetermined system of a special type is considered. Systems of this type appear in non-homogeneous network flow programming problems in the form of systems of constraints and can be characterized as systems with a large sparse submatrix representing the embedded network structure. A direct method for finding solutions of the system is developed. The algorithm is based on the theoretic-graph specificities for the structure of the support and properties of the basis of a solution space of a homogeneous system. One of the key steps is decomposition of the system. A simple example is regarded at the end of the paper.

## Conference Papers

### E. Vecharynski and C. Yang, "Preconditioned iterative methods for eigenvalue counts", to appear in Proceedings of International Workshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, in Lecture Notes in Computational Science and Engineering, Springer, 2016,

We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications.In particular, it can be used to develop an efficient spectrum-slicing strategy to compute many eigenpairs of a Hermitian matrix. Our method is based on the Lanczos- and Arnoldi-type of iterations. We show that with a properly defined preconditioner, only a few iterations may be needed to obtain a good estimate of the number of eigenvalues within a prescribed interval. We also demonstrate that the number of iterations required by the proposed preconditioned schemes is independent of the size and condition number of the matrix. The efficiency of the methods is illustrated on several problems arising from density functional theory based electronic structure calculations.

### L. Pilipchuk and E. Vecharynski, "Solution of Large Underdetermined Linear Systems for a Generalized Non-Homogeneous Network Flow Programming Problem", Proceedings of I International Conference "Mathematical Modeling and Differential Equations", Minsk, 2007,

## Reports

### E. Vecharynski, J. Brabec, M. Shao, N. Govind, C. Yang, "Efficient Block Preconditioned Eigensolvers for Linear Response Time-dependent Density Functional Theory", submitted to JCC, 2015,

We present two efficient iterative algorithms for solving the linear response eigenvalue problem arising fromthe time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously.The efficiency of the new algorithms is demonstrated by numerical examples.