arXiv subjects

Harbir Antil

Publications and source records attributed to Harbir Antil.

At least 37 records · Page 2Linked to original sources

arXiv2025

Well-Posedness of Pseudo-Parabolic Gradient Systems with State-Dependent Dynamics

This paper develops a general mathematical framework for pseudo-parabolic gradient systems with state-dependent dynamics. The state dependence is induced by variable coefficient fields in the governing energy functional. Such coefficients arise naturally in scientific and technological models, including state-dependent mobilities in KWC-type grain boundary motion and variable orientation-adaptation operators in anisotropic image denoising. We establish two main results: the existence of energy-dissipating solutions, and the uniqueness and continuous dependence on initial data. The proposed framework yields a general well-posedness theory for a broad class of nonlinear evolutionary systems driven by state-dependent operators. As illustrative applications, we present an anisotropic image-denoising model and a new pseudo-parabolic KWC-type model for anisotropic grain boundary motion, and prove that both fit naturally within the abstract structure of $(\mathrm{S})_ν$.

math.AP↗

arXiv2025

Convection Effects and Optimal Insulation: Modelling and Analysis

In this paper, we study an insulation problem that seeks to determine the optimal distribution of a given amount $m>0$ of insulating material coating an insulated boundary part $Γ_I\subseteq \partialΩ$ of a thermally conducting body $Ω\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$, subject to convective heat transfer. The `$\textit{thickness}$' of the insulating layer $Σ_{I}^{\varepsilon}\subseteq \mathbb{R}^d$ is given locally via $\varepsilon \mathtt{d}$, where $\varepsilon>0$ denotes the (arbitrarily small) conductivity and $\mathtt{d}\colon Γ_{I}\to [0,+\infty)$ the (to be determined) distribution of the insulating material. Then, the physical process is modelled by the stationary heat equation in the insulated thermally conducting body $Ω_{I}^{\varepsilon}:= Ω\cupΣ_{I}^{\varepsilon}$ with Robin-type boundary conditions on the interacting insulation boundary $Γ_I^{\varepsilon}\subseteq \partialΩ_{I}^{\varepsilon}$ (reflecting convective heat transfer between the thermally conducting body $Ω$ and its surrounding medium) as well as Dirichlet and Neumann boundary conditions at the remaining boundary parts, $\textit{i.e.}$, $\partialΩ_{I}^{\varepsilon}\setminus Γ_I^{\varepsilon}$. More precisely, we establish $Γ(L^2(\mathbb{R}^d))$-convergence of the heat loss formulation (as ${\varepsilon \to 0^+}$), in the case that the thermally conducting body $Ω$ is a bounded Lipschitz domain having a $C^{1,1}$-regular or piece-wise flat insulated boundary $Γ_I$.

math.AP↗

arXiv2025

Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty

Stochastic optimization problems are generally known to be ill-conditioned to the form of the underlying uncertainty. A framework is introduced for optimal control problems with partial differential equations as constraints that is robust to inaccuracies in the precise form of the problem uncertainty. The framework is based on problem relaxation and involves optimizing a bivariate, "Rockafellian" objective functional that features both a standard control variable and an additional perturbation variable that handles the distributional ambiguity. In the presence of distributional corruption, the Rockafellian objective functionals are shown in the appropriate settings to $Γ$-converge to uncorrupted objective functionals in the limit of vanishing corruption. Numerical examples illustrate the framework's utility for outlier detection and removal and for variance reduction.

math.OC↗

arXiv2025

Well-Posedness and Approximation of Weak Solutions to Time Dependent Maxwell's Equations with $L^2$-Data

We study Maxwell's equations in conducting media with perfectly conducting boundary conditions on Lipschitz domains, allowing rough material coefficients and $L^2$-data. Our first contribution is a direct proof of well-posedness of the first-order weak formulation, including solution existence and uniqueness, an energy identity, and continuous dependence on the data. The argument uses interior-in-time mollification to show uniqueness while avoiding reflection techniques. Existence is via the well-known Galerkin method (cf.~Duvaut and Lions \cite[Eqns.~(4.31)--(4.32), p.~346; Thm.~4.1]{GDuvaut_JLLions_1976a}). For completeness, and to make the paper self-contained, a complete proof has been provided. Our second contribution is a structure-preserving semi-discrete finite element method based on the Nédélec/Raviart--Thomas de Rham complex. The scheme preserves a discrete Gauss law for all times and satisfies a continuous-in-time energy identity with stability for nonnegative conductivity. With a divergence-free initialization of the magnetic field (via potential reconstruction or constrained $L^2$ projection), we prove convergence of the semi-discrete solutions to the unique weak solution as the mesh is refined. The analysis mostly relies on projector consistency, weak-* compactness in time-bounded $L^2$ spaces, and identification of time derivatives in dual spaces.

math.NA↗

arXiv2025

Optimal Boundary Control of Diffusion on Graphs via Linear Programming

We propose a linear programming (LP) framework for steady-state diffusion and flux optimization on geometric networks. The state variable satisfies a discrete diffusion law on a weighted, oriented graph, where conductances are scaled by edge lengths to preserve geometric fidelity. Boundary potentials act as controls that drive interior fluxes according to a linear network Laplacian. The optimization problem enforces physically meaningful sign and flux-cap constraints at all boundary edges, derived directly from a gradient bound. This yields a finite-dimensional LP whose feasible set is polyhedral, and whose boundedness and solvability follow from simple geometric or algebraic conditions on the network data. We prove that under the absence of negative recession directions--automatically satisfied in the presence of finite box bounds, flux caps, or sign restrictions--the LP admits a global minimizer. Several sufficient conditions guaranteeing boundedness of the feasible region are identified, covering both full-rank and rank-deficient flux maps. The analysis connects classical results such as the Minkowski--Weyl decomposition, Hoffman's bound, and the fundamental theorem of linear programming with modern network-based diffusion modeling. Two large-scale examples illustrate the framework: (i) A typical large stadium in a major modern city, which forms a single connected component with relatively uniform corridor widths, and a (ii) A complex street network emanating from a large, historical city center, which forms a multi-component system.

math.OC↗

arXiv2025

Randomized Matrix Sketching for Neural Network Training and Gradient Monitoring

Neural network training relies on gradient computation through backpropagation, yet memory requirements for storing layer activations present significant scalability challenges. We present the first adaptation of control-theoretic matrix sketching to neural network layer activations, enabling memory-efficient gradient reconstruction in backpropagation. This work builds on recent matrix sketching frameworks for dynamic optimization problems, where similar state trajectory storage challenges motivate sketching techniques. Our approach sketches layer activations using three complementary sketch matrices maintained through exponential moving averages (EMA) with adaptive rank adjustment, automatically balancing memory efficiency against approximation quality. Empirical evaluation on MNIST, CIFAR-10, and physics-informed neural networks demonstrates a controllable accuracy-memory tradeoff. We demonstrate a gradient monitoring application on MNIST showing how sketched activations enable real-time gradient norm tracking with minimal memory overhead. These results establish that sketched activation storage provides a viable path toward memory-efficient neural network training and analysis.

cs.LG↗

arXiv2025

Tensor train solution to uncertain optimization problems with shared sparsity penalty

We develop both first and second order numerical optimization methods to solve non-smooth optimization problems featuring a shared sparsity penalty, constrained by differential equations with uncertainty. To alleviate the curse of dimensionality we use tensor product approximations. To handle the non-smoothness of the objective function we employ a smoothed version of the shared sparsity objective. We consider both a benchmark elliptic PDE constraint, and a more realistic topology optimization problem in engineering. We demonstrate that the error converges linearly in iterations and the smoothing parameter, and faster than algebraically in the number of degrees of freedom, consisting of the number of quadrature points in one variable and tensor ranks. Moreover, in the topology optimization problem, the smoothed shared sparsity penalty actually reduces the tensor ranks compared to the unpenalised solution. This enables us to find a sparse high-resolution design under a high-dimensional uncertainty.

math.OC↗

arXiv2025

OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case

This paper presents a novel, mathematically rigorous framework for autoencoder-type deep neural networks that combines optimal control theory and low-rank tensor methods to yield memory-efficient training and automated architecture discovery. The learning task is formulated as an optimization problem constrained by differential equations representing the encoder and decoder components of the network and the corresponding optimality conditions are derived via a Lagrangian approach. Efficient memory compression is enabled by approximating differential equation solutions on low-rank tensor manifolds using an adaptive explicit integration scheme. These concepts are combined to form OCTANE (Optimal Control for Tensor-based Autoencoder Network Emergence) -- a unified training framework that yields compact autoencoder architectures, reduces memory usage, and enables effective learning, even with limited training data. The framework's utility is illustrated with application to image denoising and deblurring tasks and recommendations regarding governing hyperparameters are provided.

math.OC↗

arXiv2025

A smoothed proximal trust-region algorithm for nonconvex optimization problems with $L^p$-regularization, $p\in (0,1)$

We investigate a trust-region algorithm to solve a nonconvex optimization problem with $L^p$-regularization for $p\in(0,1)$. The algorithm relies on descent properties of a so-called generalized Cauchy point that can be obtained efficiently by a line search along a suitable proximal path. To handle the nonconvexity and nonsmoothness of the $L^p$-pseudonorm, we replace it by a smooth approximation and construct a convex upper bound of that approximation. This enables us to use results of a trust-region method for composite problems with a convex nonsmooth term. We prove convergence properties of the resulting smoothed proximal trust-region algorithm and investigate its performance in some numerical examples. Furthermore, approximate subproblem solvers for the arising trust-region subproblems are considered.

math.OC↗

arXiv2025

Modeling and Analysis of an Optimal Insulation Problem on Non-Smooth Domains

In this paper, we study an insulation problem that seeks the optimal distribution of a fixed amount $m>0$ of insulating material coating an insulated boundary $Γ_I\subseteq \partialΩ$ of a thermally conducting body $Ω\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$. The thickness of the thin insulating layer $Σ_{I}^{\varepsilon}$ is given locally via $\varepsilon \mathtt{d}$, where $\mathtt{d}\colon Γ_{I}\to [0,+\infty)$ specifies the (to be determined) distribution of the insulating material. We establish $Γ(L^2(\mathbb{R}^d))$-convergence of the problem (as $\varepsilon\to 0^+$). Different from the existing literature, which predominantly assumes that the thermally conducting body $Ω$ has a $C^{1,1}$-boundary, we merely assume that $Γ_I$ is piece-wise flat. To overcome this lack of boundary regularity, we define the thin insulating boundary layer $Σ_{I}^{\varepsilon}$ using a Lipschitz continuous transversal vector field rather than the outward unit normal vector field. The piece-wise flatness condition on $Γ_I$ is only needed to prove the $\liminf$-estimate. In fact, for the $\limsup$-estimate is enough that the thermally conducting body $Ω$ has a $C^{0,1}$-boundary.

math.AP↗

arXiv2025

A posteriori and a priori error estimates for linearized thin sheet folding

We describe a posteriori error analysis for a discontinuous Galerkin method for a fourth order elliptic interface problem that arises from a linearized model of thin sheet folding. The primary contribution is a local efficiency bound for an estimator that measures the extent to which the interface conditions along the fold are satisfied, which is accomplished by constructing a novel edge bubble function. We subsequently conduct a medius analysis to obtain improved a priori error estimates under the minimal regularity assumption on the exact solution. The performance of the method is illustrated by numerical experiments.

math.NA↗

arXiv2025

Optimal Control of Pseudo-Parabolic KWC Systems for Grain Boundary Motion

The KWC system is a well-known generic framework for phase-field models of grain boundary motion, whose original formulation is given as a parabolic gradient flow of a free energy. In the original KWC system, the results of uniqueness have been relatively scarce compared to other issues, such as existence, qualitative behavior, and numerics. This lack of progress has posed a significant challenge for more advanced topics, including optimal control. To overcome this, the authors have recently introduced the pseudo-parabolic structure to simultaneously preserve the gradient flow nature of free-energy, and to ensure the well-posedness including the uniqueness. The goal of this paper is to study an optimization problem constrained by pseudo-parabolic KWC system. The theory will be developed through a series of Main Theorems concerning the existence and semi-continuous dependence of optimal controls, and first-order necessary conditions for optimality.

math.OC↗

arXiv2025

Duality-Based Algorithm and Numerical Analysis for Optimal Insulation Problems on Non-Smooth Domains

This article develops a numerical approximation of a convex non-local and non-smooth minimization problem. The physical problem involves determining the optimal distribution, given by $h\colon Γ_I\to [0,+\infty)$, of a given amount $m\in \mathbb{N}$ of insulating material attached to a boundary part $Γ_I\subseteq \partialΩ$ of a thermally conducting body $Ω\subseteq \mathbb{R}^d$, $d \in \mathbb{N}$, subject to conductive heat transfer. To tackle the non-local and non-smooth character of the problem, the article introduces a (Fenchel) duality framework: (a) At the continuous level, using (Fenchel) duality relations, we derive an a posteriori error identity that can handle arbitrary admissible approximations of the primal and dual formulations of the convex non-local and non-smooth minimization problem; (b) At the discrete level, using discrete (Fenchel) duality relations, we derive an a priori error identity that applies to a Crouzeix--Raviart discretization of the primal formulation and a Raviart--Thomas discretization of the dual formulation. The proposed framework leads to error decay rates that are optimal with respect to the specific regularity of a minimizer. In addition, we prove convergence of the numerical approximation under minimal regularity assumptions. Since the discrete dual formulation can be written as a quadratic program, it is solved using a primal-dual active set strategy interpreted as semismooth Newton method. A solution of the discrete primal formulation is reconstructed from the solution of the discrete dual formulation by means of an inverse generalized Marini formula. This is the first such formula for this class of convex non-local and non-smooth minimization problems.

math.NA↗

arXiv2025

On Techniques for Barely Coupled Multiphysics

A technique to combine codes to solve barely coupled multiphysics problems has been developed. Each field is advanced separately until a stop is triggered. This could be due to a preset time increment, a preset number of timesteps, a preset decrease of residuals, a preset change in unknowns, a preset change in geometry, or any other physically meaningful quantity. The technique allows for a simple implementation in coupled codes using the loose coupling approach. Examples from evaporative cooling of electric motors, a problem that has come to the forefront with the rise of electric propulsion in the aerospace sector (drones and air taxis in particular) shows the viability and accuracy of the proposed procedure.

cs.CE↗

arXiv2024

Adjoint-based Recovery of Thermal Fields from Displacement or Strain Measurements

A finite-element method dependant adjoint-based procedure to determine the temperature field of structures based on measured displacements or strains and a set of standard loads is developed and tested. Given a series of force and deformation measurements, the temperature field is obtained by minimizing the adequately weighted differences between the measured and computed values. Three numerical examples - a Plate With a Hole, a Bridge, and a Hoover Dam example - each with multiple sensors distributed in different configurations, demonstrate the procedure's capabilities. A target temperature distribution is prescribed in all cases, and the displacement sensor data is recorded. The optimization algorithm (here, steepest descent with Barzilai-Borwein step) uses this data to optimize the temperatures such that the same deformation is obtained at the sensor locations. Vertex Morphing is used as a filter to mitigate the ill-conditioning. Results show that the proposed approach can accurately reconstruct the target thermal distribution, especially when more sensors are used. Additionally, it is observed that the sensors do not need to be positioned in the region of interest; the method remains effective as long as the sensors can detect changes related to that area. A comparison with standard spatial interpolation techniques, namely, k-nearest neighbors and ordinary and universal kriging, is performed using temperature sensors in the same configurations. The proposed approach performs remarkably better than the interpolation techniques with a reduction in the root-mean-squared error of up to 38.4%, 94%, and 40%, for the Plate With a Hole, the Bridge, and the Dam examples, respectively.

math.OC↗

arXiv2024

A Generic MATLAB Toolbox to Approximate PDEs Using Computational Geometry

This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from computational geometry. They may enable experimentation with novel mesh generation techniques and could simplify the implementation of methods such as multigrid. We also implement quadrature methods following the work of Grundmann and Moller. These methods have been overlooked in the past but are more efficient than traditional tensor product methods. The proposed framework is applied to several standard examples.

math.NA↗

arXiv2024

A gradient system based on anisotropic monochrome image processing with orientation auto-adjustment

This paper is devoted to the mathematical analysis of a system of pseudo-parabolic partial differential equations governed by an energy functional, associated with anisotropic monochrome image processing. The energy functional is based on the one proposed by [Berkels et al. Cartoon extraction based on anisotropic image classification, SFB 611, 2006], which incorporates an orientation auto-adjustment mechanism, and our energy is a simplified version which reduces the order of derivatives to improve computational efficiency. The aim of this paper is to establish a stable minimization process that addresses some instability of the algorithm caused by the reduction of derivative order. As a part of the study, we here prove Main Theorems concerned with the well-posedness of the pseudo-parabolic system, and provide a mathematically rigorous guarantee for the stability of the image denoising process derived from our system.

math.AP↗

arXiv2024

Dynamic Reconstruction from Neuromorphic Data

Unlike traditional cameras which synchronously register pixel intensity, neuromorphic sensors only register `changes' at pixels where a change is occurring asynchronously. This enables neuromorphic sensors to sample at a micro-second level and efficiently capture the dynamics. Since, only sequences of asynchronous event changes are recorded rather than brightness intensities over time, many traditional image processing techniques cannot be directly applied. Furthermore, existing approaches, including the ones recently introduced by the authors, use traditional images combined with neuromorphic event data to carry out reconstructions. The aim of this work is introduce an optimization based approach to reconstruct images and dynamics only from the neuromoprhic event data without any additional knowledge of the events. Each pixel is modeled temporally. The experimental results on real data highlight the efficacy of the presented approach, paving the way for efficient and accurate processing of neuromorphic sensor data in real-world applications.

cs.CV↗