arXiv subjects

Zydrunas Gimbutas

Publications and source records attributed to Zydrunas Gimbutas.

12 recordsLinked to original sources

arXiv2026

Algorithm librla: A library of randomized linear algebra routines

The library \texttt{librla} is a randomized linear algebra library that is specifically designed for the intermediate-sized matrices (of dimension up to roughly 10,000) that arise in applications such as reduced order modeling, fast direct solvers, least squares solves and, in some settings, data compression. \texttt{librla} is the first software package that is both stable and efficient in several high-level languages: MATLAB, Python and Julia. It also provides increased functionality over existing software. Specifically, it allows the user to choose to create a factorization based on a fixed rank or a desired tolerance. The factorization options include QR, SVD and the interpolative decomposition. Additionally, the factorization can be generated either with access to the matrix or access to a matrix-vector multiplication routine. Numerical results compare the Python implementation with the available PyTorch and SciPy randomized factorizations. Performance of \texttt{librla} in the three languages is comparable.

cs.MS↗

arXiv2026

Precise 2D electric field density simulations for superconducting quantum devices

Dielectric loss due to two-level systems is a limiting factor for superconducting qubit relaxation times. These losses arise mostly from nanometer-scale interfacial defect regions in superconducting devices with planar dimensions of microns to millimeters, thus making it resource intensive to accurately simulate the electric field density in these regions with traditional electromagnetic solvers. In this work, we demonstrate a fast boundary integral equation solver that allows precise simulation of electric field density in these thin regions, showing a speedup of around two orders of magnitude over traditional solvers, with relative errors around $10^{-7}$ for a ten-minute solution runtime. By computing participation ratios through Green's first identity without squaring the electric field, our approach is less susceptible to the field singularities near conductor corners. We apply this solver to a basic untrenched coplanar waveguide cross-section, showing that the common assumption of participation ratio linearity with dielectric constant holds well for some interfaces and not others; in particular, while the metal-air (MA) top and corner follow this linear relationship strongly, the MA sidewall does not. We then compare isotropic and anisotropic etching, showing that the MA sidewall and the metal-air-substrate triple junction are the most strongly affected. We are currently leveraging this solver to explore geometries that will uniquely isolate the participation ratios of the different dielectrics. Finally, we are working to combine this solver framework with a full 3D microwave solver to accurately calculate participation ratios for the thin dielectrics that are known sources of loss in superconducting qubits.

quant-ph↗

arXiv2026

A parametrix for the surface Stokes equation

We introduce an integral equation formulation of the surface Stokes equations, constructed using two-dimensional Stokeslets. The resulting integral equations are Fredholm integral equations of the second kind and can be discretized to high order using standard tools. Since the resulting discrete linear systems are dense, we describe and analyze a proxy shell method to construct fast direct solvers for these systems. The properties of our integral equation, and the performance of the resulting numerical scheme, are illustrated with several representative numerical examples.

math.NA↗

arXiv2019

A fast simple algorithm for computing the potential of charges on a line

We present a fast method for evaluating expressions of the form $$ u_j = \sum_{i = 1,i \not = j}^n \frac{α_i}{x_i - x_j}, \quad \text{for} \quad j = 1,\ldots,n, $$ where $α_i$ are real numbers, and $x_i$ are points in a compact interval of $\mathbb{R}$. This expression can be viewed as representing the electrostatic potential generated by charges on a line in $\mathbb{R}^3$. While fast algorithms for computing the electrostatic potential of general distributions of charges in $\mathbb{R}^3$ exist, in a number of situations in computational physics it is useful to have a simple and extremely fast method for evaluating the potential of charges on a line; we present such a method in this paper, and report numerical results for several examples.

math.NA↗

arXiv2019

Evaluation of Abramowitz functions in the right half of the complex plane

A numerical scheme is developed for the evaluation of Abramowitz functions $J_n$ in the right half of the complex plane. For $n=-1,\, \ldots,\, 2$, the scheme utilizes series expansions for $|z|<1$ and asymptotic expansions for $|z|>R$ with $R$ determined by the required precision, and modified Laurent series expansions which are precomputed via a least squares procedure to approximate $J_n$ accurately and efficiently on each sub-region in the intermediate region $1\le |z| \le R$. For $n>2$, $J_n$ is evaluated via a recurrence relation. The scheme achieves nearly machine precision for $n=-1, \ldots, 2$, with the cost about four times of evaluating a complex exponential per function evaluation.

math.NA↗

arXiv2016

A Fast Summation Method for Oscillatory Lattice Sums

We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge. They were discovered by Wood in 1902 as singularities in the spectral response. The primary tools in our approach are the Euler-Maclaurin formula and a steepest descent argument. The resulting algorithm has super-algebraic convergence and requires only milliseconds of CPU time.

math.NA↗

arXiv2015

Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space

We derive new formulas for the fundamental solutions of slow, viscous flow, governed by the Stokes equations, in a half-space. They are simpler than the classical representations obtained by Blake and collaborators, and can be efficiently implemented using existing fast solvers libraries. We show, for example, that the velocity field induced by a Stokeslet can be annihilated on the boundary (to establish a zero slip condition) using a single reflected Stokeslet combined with a single Papkovich-Neuber potential that involves only a scalar harmonic function. The new representation has a physically intuitive interpretation.

physics.flu-dyn↗

arXiv2014

The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering

We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials ${\bf A},ϕ$ in the Lorenz gauge, we establish boundary conditions on the potentials themselves, rather than on the field quantities. This permits the development of a well-conditioned second kind Fredholm integral equation which has no spurious resonances, avoids low frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, $ϕ^{Sc}$, is determined entirely by the incident scalar potential $ϕ^{In}$. Likewise, the unknown vector potential defining the scattered field, ${\bf A}^{Sc}$, is determined entirely by the incident vector potential ${\bf A}^{In}$. This decoupled formulation is valid not only in the static limit but for arbitrary $ω\ge 0$.

math-ph↗

arXiv2014

A fast multipole method for the evaluation of elastostatic fields in a half-space with zero normal stress

In this paper, we present a fast multipole method (FMM) for the half-space Green's function in a homogeneous elastic half-space subject to zero normal stress, for which an explicit solution was given by Mindlin (1936). The image structure of this Green's function is unbounded, so that standard outgoing representations are not easily available. We introduce two such representations here, one involving an expansion in plane waves and one involving a modified multipole expansion. Both play a role in the FMM implementation.

math-ph↗

arXiv2014

Randomized methods for rank-deficient linear systems

We present a simple, accurate method for solving consistent, rank-deficient linear systems, with or without addi- tional rank-completing constraints. Such problems arise in a variety of applications, such as the computation of the eigenvectors of a matrix corresponding to a known eigenvalue. The method is based on elementary linear algebra combined with the ob- servation that if the matrix is rank-k deficient, then a random rank-k perturbation yields a nonsingular matrix with probability 1.

math.NA↗

arXiv2012

A consistency condition for the vector potential in multiply-connected domains

A classical problem in electromagnetics concerns the representation of the electric and magnetic fields in the low-frequency or static regime, where topology plays a fundamental role. For multiply connected conductors, at zero frequency the standard boundary conditions on the tangential components of the magnetic field do not uniquely determine the vector potential. We describe a (gauge-invariant) consistency condition that overcomes this non-uniqueness and resolves a longstanding difficulty in inverting the magnetic field integral equation.

physics.class-ph↗

arXiv2011

Fast multi-particle scattering: a hybrid solver for the Maxwell equations in microstructured materials

A variety of problems in device and materials design require the rapid forward modeling of Maxwell's equations in complex micro-structured materials. By combining high-order accurate integral equation methods with classical multiple scattering theory, we have created an effective simulation tool for materials consisting of an isotropic background in which are dispersed a large number of micro- or nano-scale metallic or dielectric inclusions.

math.NA↗