arXiv subjects

Nilima Nigam

Publications and source records attributed to Nilima Nigam.

At least 19 recordsLinked to original sources

arXiv2025

At the intersection of Numerical Analysis and Spectral Geometry

How do the geometric properties of a domain impact the spectrum of an operator defined on it? How do we compute accurate and reliable approximations of these spectra? The former question is studied in spectral geometry, and the latter is a central concern in numerical analysis. In this short expository survey we revisit the process of eigenvalue approximation, from the perspective of computational spectral geometry. Over the years a multitude of methods -- for discretizing the operator and for the resultant discrete system -- have been developed and analyzed in the field of numerical analysis. High-accuracy and provably convergent discretization approaches can be used to examine the interplay between the spectrum of an operator and the geometric properties of the spatial domain or manifold it is defined on. While computations have been used to guide conjectures in spectral geometry, in recent years approximation-theoretic tools and validated computations are also being used as part of proof strategies in spectral geometry. Given a particular spectral feature of interest, should we discretize the original problem, or seek a reformulation? Of the many possible approximation strategies, which should we choose? These choices are inextricably linked to the objective: on the one hand, rapid, specialized methods are often ideal for conjecture formulation (prioritizing efficiency and accuracy), whereas schemes with guaranteed, computable error bounds are needed when computation is incorporated into a proof strategy. We also review instances where the demanding requirements of spectral geometry -- the need for rigorous error control or the robust calculation of higher eigenvalues -- motivate new developments in numerical analysis.

math.NA↗

arXiv2025

On the dynamical stability of skeletal muscle

There has been debate for over 70-years about whether active skeletal muscle is dynamically stable at lengths greater than its optimal length. The stability of computational muscle models is a critical issue, as it directly affects our ability to simulate muscle deformation across different operating lengths, especially at lengths where muscles are known to remain functional despite model-predicted instabilities. In this study, we revisit the question of dynamical stability of ODE-based models of skeletal muscle. In particular, we investigate whether activation-independent tissue properties can provide stability to contractions along the dip region of the total force-length curve. First, using a combination of analytical tools (eigenvalue analysis and non-dimensionalization) and numerical simulations, we confirm that traditional Hill-type muscle models can display divergent dynamics in this region. Then, we propose a stabilized version of a 1D Hill-type muscle model that incorporates the 3D nature of skeletal muscle deformation. This results in a completely convex force-length relationship that can bring robustness to numerical simulations, while preserving the computational efficiency of 1D models. Our findings suggest that activation-independent intrinsic mechanical properties of muscle are sufficient to stabilize contractions even in the dip region, offering new insight into how muscles maintain functional integrity during active stretch.

physics.med-ph↗

arXiv2025

The whys and hows of conditioning of DG plane wave Trefftz methods: a single element

Plane-wave Trefftz methods (PWB) for the Helmholtz equation offer significant advantages over standard discretization approaches whose implementation employs more general polynomial basis functions. A disadvantage of these methods is the poor conditioning of the system matrices. In the present paper, we carefully examine the conditioning of the plane-wave discontinuous Galerkin method with reference to a single element. The properties of the mass and stiffness matrices depend on the size and geometry of the element. We study the mass and system matrices arising from a PWB on a single disk-shaped element. We then examine some preconditioning strategies, and present results showing their behaviour with three different criteria: conditioning, the behaviour of GMRES residuals, and impact on the $L^2$-error.

math.NA↗

arXiv2025

The spectrum of the Steklov-Helmholtz operator

We present a wavenumber-robust strategy for computing Steklov eigenpairs of the Helmholtz operator $-Δ-μ^2$. As the wavenumber $μ\rightarrow μ_D$ from below (where $μ_D^2 $ is a Dirichlet- Laplace eigenvalue of multiplicity $\ell$), the lowest $\ell$ Steklov-Helmholtz eigenvalues diverge to $-\infty$. Computationally, the Steklov-Helmholtz eigenvalue problem becomes severely ill-conditioned when $μ\approx μ_D$. We first reformulate the problem in terms of a suitably-defined Dirichlet-to-Neumann map. We then use an indirect approach based on a single layer ansatz. The discrete single layer matrix is nearly singular close to exceptional wavenumbers, and we use a reduced singular value decomposition to avoid the consequent ill-conditioning. For smooth domains, convergence of our eigenvalue solver is spectral. We use this method (called the BIO-MOD approach) for shape optimization of scale-invariant Steklov-Helmholtz problems and prove that the disk maximizes the second eigenvalue under appropriate scaling. For curvilinear polygons, we use polynomially-graded meshes rather than uniform meshes. As a proof of concept, we also implemented BIO-MOD using RCIP quadratures (using the ChunkIE implementation). The BIO-MOD approach successfully removes ill-conditioning near exceptional wavenumbers, and very high eigenvalue accuracy (up to 10 digits for polygons, arbitrary precision accuracy for smooth domains) is observed. We deploy our approach to computationally study the spectral geometry of the Steklov-Helmholtz operator, including some questions about spectral asymptotics and spectral optimization.

math.NA↗

arXiv2023

Conforming Finite Element Function Spaces in Four Dimensions, Part 1: Foundational Principles and the Tesseract

The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four-dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed `Finite Element Exterior Calculus'. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom.

math.NA↗

arXiv2023

Conforming Finite Element Function Spaces in Four Dimensions, Part II: The Pentatope and Tetrahedral Prism

In this paper, we present explicit expressions for conforming finite element function spaces, basis functions, and degrees of freedom on the pentatope and tetrahedral prism elements. More generally, our objective is to construct finite element function spaces that maintain conformity with infinite-dimensional spaces of a carefully chosen de Rham complex. This paper is a natural extension of the companion paper entitled "Conforming Finite Element Function Spaces in Four Dimensions, Part I: Foundational Principles and the Tesseract" by Nigam and Williams, (2023). In contrast to Part I, in this paper we focus on two of the most popular elements which do not possess a full tensor-product structure in all four coordinate directions. We note that these elements appear frequently in existing space-time finite element methods. In order to build our finite element spaces, we utilize powerful techniques from the recently developed 'Finite Element Exterior Calculus'. Subsequently, we translate our results into the well-known language of linear algebra (vectors and matrices) in order to facilitate implementation by scientists and engineers.

math.NA↗

arXiv2023

Characterization of singular flows of zeroth-order pseudo-differential operators via elliptic eigenfunctions: a numerical study

The propagation of internal gravity waves in stratified media, such as those found in ocean basins and lakes, leads to the development of geometrical patterns called "attractors". These structures accumulate much of the wave energy and make the fluid flow highly singular. In more analytical terms, the cause of this phenomenon has been attributed to the presence of a continuous spectrum in some nonlocal zeroth-order pseudo-differential operators. In this work, we analyze the generation of these attractors from a numerical analysis perspective. First, we propose a high-order pseudo-spectral method to solve the evolution problem (whose long-term behaviour is known to be not square-integrable). Then, we use similar tools to discretize the corresponding eigenvalue problem. Since the eigenvalues are embedded in a continuous spectrum, we compute them using viscous approximations. Finally, we explore the effect that the embedded eigenmodes have on the long-term evolution of the system.

math.SP↗

arXiv2022

A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems

In this paper we revisit an approach pioneered by Auchmuty to approximate solutions of the Laplace- Robin boundary value problem. We demonstrate the efficacy of this approach on a large class of non-tensorial domains, in contrast with other spectral approaches for such problems. We establish a spectral approximation theorem showing an exponential fast numerical evaluation with regards to the number of Steklov eigenfunctions used, for smooth domains and smooth boundary data. A polynomial fast numerical evaluation is observed for either non-smooth domains or non-smooth boundary data. We additionally prove a new result on the regularity of the Steklov eigenfunctions, depending on the regularity of the domain boundary. We describe three numerical methods to compute Steklov eigenfunctions.

math.NA↗

arXiv2022

A three-dimensional model of skeletal muscle tissues

Skeletal muscles are living tissues that can undergo large deformations in short periods of time and that can be activated to produce force. In this paper we use the principles of continuum mechanics to propose a dynamic, fully non-linear, and three-dimensional model to describe the deformation of these tissues. We model muscles as a fibre-reinforced composite and transversely isotropic material. We introduce a flexible computational framework to approximate the deformations of skeletal muscle to provide new insights into the underlying mechanics of these tissues. The model parameters and mechanical properties are obtained through experimental data and can be specified locally. A semi-implicit in time, conforming finite element in space scheme is used to approximate the solutions to the governing nonlinear dynamic model. We provide a series of numerical experiments demonstrating the application of this framework to relevant problems in biomechanics, and also discuss questions around model validation.

math.NA↗

arXiv2020

A simple extrapolation method for clustered eigenvalues

This paper introduces a simple variant of the power method. It is shown analytically and numerically to accelerate convergence to the dominant eigenvalue/eigenvector pair; and, it is particularly effective for problems featuring a small spectral gap. The introduced method is a one-step extrapolation technique that uses a linear combination of current and previous update steps to form a better approximation of the dominant eigenvector. The provided analysis shows the method converges exponentially with respect to the ratio between the two largest eigenvalues, which is also approximated during the process. An augmented technique is also introduced, and is shown to stabilize the early stages of the iteration. Numerical examples are provided to illustrate the theory and demonstrate the methods.

math.NA↗

arXiv2019

Korn's inequality and Jones eigenpairs

In this paper we show that Korn's inequality \cite{ref:korn1906} holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains. We further show that the validity of this inequality depends on the geometry of this subset of the boundary. We then consider the {\it Jones eigenvalue problem} which consists of the usual traction eigenvalue problem for the Lamé operator for linear elasticity coupled with a zero normal trace of the displacement on a non-empty part of the boundary. Here we extend previous works in the literature to show the Jones eigenpairs exist on a broad variety of domains even when the normal trace of the displacement is constrained only on a subset of the boundary. We further show that one can have eigenpairs of a modified eigenproblem in which the constraint on the normal trace on a subset of the boundary is replaced by one on the tangential trace.

math.AP↗

arXiv2019

Revisiting the Jones eigenproblem in fluid-structure interaction

The Jones eigenvalue problem first described by D.S. Jones in 1983 concerns unusual modes in bounded elastic bodies: time-harmonic displacements whose tractions and normal components are both identically zero on the boundary. This problem is usually associated with a lack of unique solvability for certain models of fluid-structure interaction. The boundary conditions in this problem appear, at first glance, to rule out {\it any} non-trivial modes unless the domain possesses significant geometric symmetries. Indeed, Jones modes were shown to not be possible in most $C^\infty$ domains (see article by T. Hargé 1990). However, we should in this paper that while the existence of Jones modes sensitively depends on the domain geometry, such modes {\it do} exist in a broad class of domains. This paper presents the first detailed theoretical and computational investigation of this eigenvalue problem in Lipschitz domains. We also analytically demonstrate Jones modes on some simple geometries.

math.NA↗

arXiv2019

Wave Enhancement through Optimization of Boundary Conditions

It is well known that changing boundary conditions for the Laplacian from Dirichlet to Neumann can result in significant changes to the associated eigenmodes, while keeping the eigenvalues close. We present a new and efficient approach for optimizing the transmission signal between two points in a cavity at a given frequency, by changing boundary conditions. The proposed approach makes use of recent results on the monotonicity of the eigenvalues of the mixed boundary value problem and on the sensitivity of the Green s function to small changes in the boundary conditions. The switching of the boundary condition from Dirichlet to Neumann can be performed through the use of the recently modeled concept of metasurfaces which are comprised of coupled pairs of Helmholtz resonators. A variety of numerical experiments are presented to show the applicability and the accuracy of the proposed new methodology.

math-ph↗

arXiv2019

Optimization of Steklov-Neumann eigenvalues

This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem.

math-ph↗

arXiv2016

Well-posed Bayesian Inverse Problems: Priors with Exponential Tails

We consider the well-posedness of Bayesian inverse problems when the prior measure has exponential tails. In particular, we consider the class of convex (log-concave) probability measures which include the Gaussian and Besov measures as well as certain classes of hierarchical priors. We identify appropriate conditions on the likelihood distribution and the prior measure which guarantee existence, uniqueness and stability of the posterior measure with respect to perturbations of the data. We also consider consistent approximations of the posterior such as discretization by projection. Finally, we present a general recipe for construction of convex priors on Banach spaces which will be of interest in practical applications where one often works with spaces such as $L^2$ or the continuous functions.

math.PR↗

arXiv2015

Nearly radial Neumann eigenfunctions on symmetric domains

We study the existence of Neumann eigenfunctions which do not change sign on the boundary of some special domains. We show that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 5 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. We use analytic methods combined with symmetry arguments to prove the result for polygons with six or more sides. The case for the regular pentagon is harder. We develop a validated numerical method to prove this case, which involves iteratively bounding eigenvalues for a sequence of subdomains of the triangle. We use a learning algorithm to find and optimize this sequence of subdomains, making it straightforward to check our computations with standard software.

math.SP↗

arXiv2015

Structural Changes of Active Skeletal Muscles: Modelling, Validation and Numerical Experiments

The purpose of this study was to report numerical validation of a 3D finite element model of contracting muscle. The model was based on continuum theory for fibre-reinforced composite materials. Here we simulated contractions for an idealized medial gastrocnemius muscle in man, using the model. Simulations were performed to test the force-length relation of the whole muscle, to evaluate the changes in internal fascicle geometry during contractions, and to assess the importance of material formulations for the aponeurosis and tendon. The simulation results were compared to previously published experimental values. The force-length profile for the whole muscle showed a realistic profile. As the muscle contracted the fascicles curved into S-shaped trajectories and curled around 3D paths, both of which matched previous experimental findings. As the fascicles shortened they increased in their cross-sectional area, but this increase was asymmetric with the smaller increase occurring within the fascicle-plane: the Poisson's ratio in this plane matched that previously shown from ultrasound imaging. The distribution of strains in the aponeurosis and tendon was shown to be a function of their material properties. This study demonstrated that the model could replicate realistic patterns of whole muscle-force, and changes to the internal muscle geometry, and so will be useful for testing mechanisms that affect the structural changes within contracting muscle.

q-bio.QM↗

arXiv2014

On regularizations of the delta distribution

In this article we consider regularizations of the Dirac delta distribution with applications to prototypical elliptic and hyperbolic partial differential equations (PDEs). We study the convergence of a sequence of distributions $\mathcal{S}_H$ to a singular term $\mathcal{S}$ as a parameter $H$ (associated with the {support size} of $\mathcal{S}_H$) shrinks to zero. We characterize this convergence in both the weak-$\ast$ topology of distributions, as well as in a weighted Sobolev norm. These notions motivate a framework for constructing regularizations of the delta distribution that includes a large class of existing methods in the literature. This framework allows different regularizations to be compared. The convergence of solutions of PDEs with these regularized source terms is then studied in various topologies such as pointwise convergence on a deleted neighborhood and weighted Sobolev norms. We also examine the lack of symmetry in tensor product regularizations and effects of dissipative error in hyperbolic problems.

math.NA↗