arXiv subjects

Jay Gopalakrishnan

Publications and source records attributed to Jay Gopalakrishnan.

At least 19 recordsLinked to original sources

arXiv2026

Fortin operators for DPG advection discretizations

We construct Fortin operators for discontinuous Petrov-Galerkin (DPG) discretizations of the advection equation with a piecewise constant divergence-free advection vector $\beta$, on simplicial meshes of any spatial dimension $N$ and for any polynomial degree $k \ge 1$ of the trial space. A minimal test space is built on each element from facet and interior bubbles and later augmented. The Fortin operator is shown to be bounded, uniformly over shape-regular mesh families, in the natural $\beta$-weighted broken test graph norm built on $L_q$ for every $1 < q < \infty$, where $q$ is the exponent conjugate to the trial exponent $p$. A non-characteristic facet condition is assumed when $k \ge 2$, while the lowest-order case requires no such condition and admits characteristic facets. As applications we prove that the practical fully discrete residual minimization method is quasioptimal in the DPG energy norm for every $1 < p < \infty$, with a quasioptimality constant governed solely by the Fortin operator, that its computable residual is a globally reliable and efficient a posteriori error estimator, and that augmenting the test space and changing the test norm improves the results in the lowest-order case.

math.NA↗

arXiv2026

Reliable eigenspace error estimation using source error estimators

We introduce a framework for repurposing error estimators for source problems to compute an estimator for the gap between eigenspaces and their discretizations. Of interest are eigenspaces of finite clusters of eigenvalues of unbounded nonselfadjoint linear operators with compact resolvent. Eigenspaces and eigenvalues of rational functions of such operators are studied as a first step. Under an assumption of convergence of resolvent approximations in the operator norm and an assumption on global reliability of source problem error estimators, we show that the gap in eigenspace approximations can be bounded by a globally reliable and computable error estimator. Also included are applications of the theoretical framework to first-order system least squares (FOSLS) discretizations and discontinuous Petrov-Galerkin (DPG) discretizations, both yielding new estimators for the error gap. Numerical experiments with a selfadjoint model problem and with a leaky nonselfadjoint waveguide eigenproblem show that adaptive algorithms using the new estimators give refinement patterns that target the cluster as a whole instead of individual eigenfunctions.

math.NA↗

arXiv2025

Accelerated Coupled Mode Model for Fiber Laser Amplifiers as an Averaged Dynamical System

We apply a known theorem for simplifying dynamical systems with bounded error to a specific optical fiber waveguide problem, supplementing the physical intuition and heuristics used in the optics community with proper mathematical justification. Using techniques from averaging theory of dynamical systems, a reliable accelerated model based on the coupled mode theory (CMT) approach for a common fiber laser amplifier application is derived. Computational testing reveals that this accelerated model achieves an ${\sim}4000$x increase in computational speed compared to the CMT model while preserving a high accuracy in key figures-of-merit such as output power and amplification efficiency. Further, we argue that by adopting our recommended approximations within the reduced model framework enables the model to be applied a wider set of amplifier types and configurations than can the current (comparable) reduced models found in the literature.

physics.optics↗

arXiv2025

Sufficient conditions for QMC analysis of finite elements for parametric differential equations

Parametric regularity of discretizations of flux vector fields satisfying a balance law is studied under some assumptions on a random parameter that links the flux with an unknown primal variable (often through a constitutive law). In the primary example of the stationary diffusion equation, the parameter corresponds to the inverse of the diffusivity. The random parameter is modeled here as a Gevrey-regular random field. Specific focus is on random fields expressible as functions of countably infinite sequences of independent random variables, which may be uniformly or normally distributed. Quasi-Monte Carlo (QMC) error bounds for some quantity of interest that depends on the flux are then derived using the parametric regularity. It is shown that the QMC method achieves a dimension-independent, faster-than-Monte Carlo convergence rate if the quantity of interest depends continuously on the primal variable, its flux, or its gradient. A series of assumptions are introduced with the goal of encompassing a broad class of discretizations by various finite element methods. The assumptions are verified for the diffusion equation discretized using conforming finite elements, mixed methods, and hybridizable discontinuous Galerkin schemes. Numerical experiments confirm the analytical findings, highlighting the role of accurate flux approximation in QMC methods.

math.NA↗

arXiv2025

A 2-complex containing Sobolev spaces of matrix fields

Using a generalization of complexes, called 2-complexes, this paper defines and analyzes new Sobolev spaces of matrix fields and their interrelationships within a commuting diagram. These spaces have very weak second-order derivatives. An example is the space of matrix fields of square-integrable components whose row-wise divergence followed by yet another divergence operation yield a function in a standard negative-order Sobolev space. Similar spaces where the double divergence is replaced by a curl composed with divergence, or a double curl operator (the incompatibility operator), are also studied. Stable decompositions of such spaces in terms of more regular component functions (which are continuous in natural norms) are established. Appropriately ordering such Sobolev spaces with and without boundary conditions (in a weak sense), we discover duality relationships between them. Motivation to study such Sobolev spaces, from a finite element perspective and implications for weak well-posed variational formulations are pointed out.

math.AP↗

arXiv2025

DPG loss functions for learning parameter-to-solution maps by neural networks

We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance prediction capability of resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishing the variational correctness of a loss function from an ultraweak Discontinuous Petrov Galerkin (DPG) discretization are worked out. Despite the focus on the example, the proposed concepts apply to a much wider scope of problems, namely problems for which stable DPG formulations are available. The issue of {high-contrast} diffusion fields and ensuing difficulties with degrading ellipticity are discussed. Both numerical results and theoretical arguments illustrate that for high-contrast diffusion parameters the proposed DPG loss functions deliver much more robust performance than simpler least-squares losses.

math.NA↗

arXiv2025

Guided modes of helical waveguides

This paper studies guided transverse scalar modes propagating through helically coiled waveguides. Modeling the modes as solutions of the Helmholtz equation within the three-dimensional (3D) waveguide geometry, a propagation ansatz transforms the mode-finding problem into a 3D quadratic eigenproblem. Through an untwisting map, the problem is shown to be equivalent to a 3D quadratic eigenproblem on a straightened configuration. Next, exploiting the constant torsion and curvature of the Frenet frame of a circular helix, the 3D eigenproblem is further reduced to a two-dimensional (2D) eigenproblem on the waveguide cross section. All three eigenproblems are numerically treated. As expected, significant computational savings are realized in the 2D model. A few nontrivial numerical techniques are needed to make the computation of modes within the 3D geometry feasible. They are presented along with a procedure to effectively filter out unwanted non-propagating eigenfunctions. Computational results show that the geometric effect of coiling is to shift the localization of guided modes away from the coiling center. The variations in modes as coiling pitch is changed are reported considering the example of a coiled optical fiber.

physics.optics↗

arXiv2024

PT-symmetry enabled stable modes in multi-core fiber

Open systems with balanced gain and loss, described by parity-time PT-symmetric Hamiltonians have been deeply explored over the past decade. Most explorations are limited to finite discrete models (in real or reciprocal spaces) or continuum problems in one dimension. As a result, these models do not leverage the complexity and variability of two-dimensional continuum problems on a compact support. Here, we investigate eigenvalues of the Schrodinger equation on a disk with zero boundary condition, in the presence of constant, PT-symmetric, gain-loss potential that is confined to two mirror-symmetric disks. We find a rich variety of exceptional points, re-entrant PT-symmetric phases, and a non-monotonic dependence of the PT-symmetry breaking threshold on the system parameters. By comparing results of two model variations, we show that this simple model of a multi-core fiber supports propagating modes in the presence of gain and loss.

physics.optics↗

arXiv2024

Adaptive resolution of fine scales in modes of microstructured optical fibers

An adaptive algorithm for computing eigenmodes and propagation constants of optical fibers is proposed. The algorithm is built using a dual-weighted residual error estimator. The residuals are based on the eigensystem for leaky hybrid modes obtained from Maxwell equations truncated to a finite domain after a transformation by a perfectly matched layer. The adaptive algorithm is then applied to compute practically interesting modes for multiple fiber microstructures. Emerging microstructured optical fibers are characterized by complex geometrical features in their transverse cross-section. Their leaky modes, useful for confining and propagating light in their cores, often exhibit fine scale features. The adaptive algorithm automatically captures these features without any expert input. The results also show that confinement losses of these modes are captured accurately on the adaptively found meshes.

math.NA↗

arXiv2024

The Johnson-Krizek-Mercier elasticity element in any dimensions

Mixed methods for linear elasticity with strongly symmetric stresses of lowest order are studied in this paper. On each simplex, the stress space has piecewise linear components with respect to its Alfeld split (which connects the vertices to barycenter), generalizing the Johnson--Mercier two-dimensional element to higher dimensions. Further reductions in the stress space in the three-dimensional case (to 24 degrees of freedom per tetrahedron) are possible when the displacement space is reduced to local rigid displacements. Proofs of optimal error estimates of numerical solutions and improved error estimates via postprocessing and the duality argument are presented.

math.NA↗

arXiv2024

On the improved convergence of lifted distributional Gauss curvature from Regge elements

Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.

math.NA↗

arXiv2023

Generalizing Riemann curvature to Regge metrics

In this paper, we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components is assumed. While linear derivatives of the metric can be generalized to Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization that combines the classical angle defect and jumps in the second fundamental form across element interfaces, and argue its correctness. Specifically, if a piecewise smooth metric approximates a globally smooth metric, then our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor associated with the latter. Moreover, we show that if the metric approximation converges at some rate in a mesh-dependent norm equivalent to the $L^2$ norm, then the curvature approximation converges in the negative Sobolev space $H^{-2}$, the dual space of $H^2_0$, at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any dimension.

math.NA↗

arXiv2023

Discrete Elasticity Exact Sequences on Worsey-Farin Splits

We construct conforming finite element elasticity complexes on Worsey-Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey-Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space.

math.NA↗

arXiv2023

Divergence-conforming velocity and vorticity approximations for incompressible fluids obtained with minimal facet coupling

We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

math.NA↗

arXiv2023

Sensitivity of Confinement Losses in Optical Fibers to Modeling Approach

A prime objective of modeling optical fibers is capturing mode confinement losses correctly. This paper demonstrates that specific modeling choices, especially regarding the outer fiber cladding regions and the placement of the computational boundary, have significant impacts on the calculated mode losses. Our results illustrate that one can obtain disparate mode confinement loss profiles for the same optical fiber design simply by moving the boundary to a new material region. We conclude with new recommendations for how to better model these losses.

physics.optics↗

arXiv2022

Stability of structure-aware Taylor methods for tents

Structure-aware Taylor (SAT) methods are a class of timestepping schemes designed for propagating linear hyperbolic solutions within a tent-shaped spacetime region. Tents are useful to design explicit time marching schemes on unstructured advancing fronts with built-in locally variable timestepping for arbitrary spatial and temporal discretization orders. The main result of this paper is that an $s$-stage SAT timestepping within a tent is weakly stable under the time step constraint $Δt \leq Ch^{1+1/s}$, where $Δt$ is the time step size and $h$ is the spatial mesh size. Improved stability properties are also presented for high order SAT time discretizations coupled with low order spatial polynomials. A numerical verification of the sharpness of proven estimates is also included.

math.NA↗

arXiv2022

Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

math.NA↗

arXiv2021

Computing leaky modes of optical fibers using a FEAST algorithm for polynomial eigenproblems

An efficient contour integral technique to approximate a cluster of nonlinear eigenvalues of a polynomial eigenproblem, circumventing certain large inversions from a linearization, is presented. It is applied to the nonlinear eigenproblem that arises from a frequency-dependent perfectly matched layer. This approach is shown to result in an accurate method for computing leaky modes of optical fibers. Extensive computations on an antiresonant fiber with a complex transverse microstructure are reported. This structure is found to present substantial computational difficulties: Even when employing over one million degrees of freedom, the fiber model appears to remain in a preasymptotic regime where computed confinement loss values are likely to be off by orders of magnitude. Other difficulties in computing mode losses, together with practical techniques to overcome them, are detailed.

physics.optics↗