arXiv subjects

Koondanibha Mitra

Publications and source records attributed to Koondanibha Mitra.

14 recordsLinked to original sources

arXiv2026

An adaptive, space-time discretized linear iterative scheme for doubly-degenerate parabolic problems

Degenerate diffusion problems, where the governing parabolic equation can change type to either an ordinary differential equation or an elliptic equation, model many real life applications. Due to the presence of free-boundaries, accurate numerical simulation of such problems require extremely small mesh and time step sizes locally. To remediate this issue, in this work, we consider a space-time formulation of the problem based on an efficient splitting of the nonlinearities. First, an iterative linearization scheme is proposed to resolve the nonlinearities that effectively reduces to solving a sequence of heat equations. Unconditional convergence of the scheme is proven even for double degenerate cases with linear convergence achieved if the problem is non-degenerate. Next, the dual norm of the nonlinear residual is decomposed into a linearization error component and a discretization error component corresponding to the heat equation. This leads to reliable and fully computable a posteriori estimates for the problem that are robust with respect to the nonlinearities/degeneracies. These estimates are used then in a fully adaptive (discretization + linearization) space-time solver. Numerical experiments for multiple test cases (one and two dimensions in space) demonstrate that this solver efficiently allocates the computational resources in the space-time domain, resulting in a rapid decay of error in terms of total degrees of freedom spent.

math.NA↗

arXiv2026

Neural-network methods for two-dimensional finite-source reflector design

We address the inverse problem of designing two-dimensional reflectors that transform light from a finite, extended source into a prescribed far-field distribution. The reflector height is represented by a neural network and optimized with two objective functions: a direct change-of-variables loss based on the closed-form inverse ray map, and a mesh-based loss that maps target cells back to the source and remains usable for discontinuous sources. Gradients are computed by automatic differentiation and minimized with a robust quasi-Newton method. As a baseline, we adapt a deconvolution pipeline built on a simplified finite-source approximation: a one-dimensional monotone map is recovered from flux balance, converted to a reflector by an integrating-factor ODE solve, and embedded in a modified Van Cittert iteration with nonnegativity clipping and ray-traced feedback. Across four benchmarks, covering continuous and discontinuous sources and minimum-height constraints, accuracy is measured by ray-traced normalized mean absolute error. On the two main benchmarks, the neural method reaches errors of about 2e-5 and 5e-5 within a few seconds on one NVIDIA RTX 4090 GPU, compared with 4e-3 and 5e-2 for the deconvolution baseline after several hundred seconds. The results show that the neural formulation is both more accurate and substantially faster, while still supporting practical height constraints. We also discuss extensions to rotationally symmetric and full three-dimensional reflector design through iterative correction schemes.

cs.LG↗

arXiv2026

Recent Analytical and Computational Developments on the Advection-Diffusion-Reaction Wildfire Model

Wildfires represent a problem for ecosystems, human activities, and economies, driven by the climate crisis and land-use changes. Predicting wildfire propagation through mathematical modelling is essential for damage mitigation and risk assessment. This paper provides a comprehensive review of a physics-based Advection-Diffusion-Reaction (ADR) model, focusing on the balance between physical accuracy and computational efficiency. We analyze the ability of the ADR model to estimate fire front speed and behaviour and discuss its preliminary mathematical properties. Additionally, we discuss some modelling improvements which enhance the physical realism of the model. Furthermore, we address the challenge of reducing computational costs, emphasizing the need for inexpensive but precise numerical schemes. We report recent findings outlining open challenges in model discretization and technological solutions. All these developments highlight the potential of ADR models as powerful tools for efficient wildfire simulation and risk assessment.

math.AP↗

arXiv2025

An inverse design method for generalized zero-étendue sources and two targets

We present an inverse method to compute freeform optical surfaces that transform a light distribution, parameterized by two source planes, into two separate target distributions. The surfaces can be reflectors or lenses, and control both the spatial and directional source and target coordinates of light rays. From energy conservation we derive Jacobian equations for optical mappings, and the optical path length provides generating functions for the optical surfaces. A three-stage least-squares algorithm numerically solves the resulting equations. We present examples with complex source and target distributions.

physics.optics↗

arXiv2025

Robust, fast, and adaptive splitting schemes for nonlinear doubly-degenerate diffusion equations

We consider linear iterative schemes for the time-discrete equations stemming from a class of nonlinear, doubly-degenerate parabolic equations. More precisely, the diffusion is nonlinear and may vanish or become multivalued for certain values of the unknown, so the parabolic equation becomes hyperbolic or elliptic, respectively. After performing an Euler implicit time-stepping, a splitting strategy is applied to the time-discrete equations. This leads to a formulation that is more suitable for dealing with the degeneracies. Based on this splitting, different iterative linearization strategies are considered, namely the Newton scheme, the L-scheme, and the modified L-scheme. We prove the convergence of the latter two schemes even for the double-degenerate case. In the non-degenerate case, we prove that the scheme is contractive, and the contraction rate is proportional to a non-negative exponent of the time-step size. Moreover, an a posteriori estimator-based adaptive algorithm is developed to select the optimal parameters for the M-scheme, which accelerates its convergence. Numerical results are presented, showing that the M- and the M-adaptive schemes are more stable than the Newton scheme, as they converge irrespective of the mesh. Moreover, the adaptive M-scheme consistently out-competes not only the M/L-schemes, but also the Newton scheme showing quadratic convergence behavior.

math.NA↗

arXiv2025

Design of a three-dimensional parallel-to-point imaging system based on inverse methods

We present an inverse method for designing a three-dimensional imaging system comprising of freeform optical surfaces. We impose an imaging condition on the optical map and combine it with the law of conservation of energy to conclude that the ratio of energy distributions at the source and target of an imaging system must be constant. A mathematical model for the design of a parallel-to-point system consisting of two freeform reflectors is presented. A Schwarzschild telescope, a classical design known for maximum correction of third-order aberrations, is utilized to specify the design parameters in the mathematical model, enabling us to compute an inverse freeform imaging system. The performance of both designs is compared by ray tracing various parallel beams of light and determining the corresponding spot sizes of the image. We demonstrate that our inverse freeform design is superior to the classical design.

physics.optics↗

arXiv2025

Inverse freeform design of a parallel-to-two-target reflector system

We present an inverse method for transforming a given parallel light emittance to two light distributions at different parallel target planes using two freeform reflectors. The reflectors control both the spatial and directional target coordinates of light rays. To determine the shape and position of the reflectors, we derive generating functions and use Jacobian equations to find the optical mappings to the two targets. The model is solved numerically by a three-stage least-squares algorithm. A feasibility condition is derived, which ensures that the reflectors are not self-intersecting. Several examples validate this condition and demonstrate the algorithm's capability of realizing complex distributions.

physics.optics↗

arXiv2024

A neural network approach for solving the Monge-Ampère equation with transport boundary condition

This paper introduces a novel neural network-based approach to solving the Monge-Ampère equation with the transport boundary condition, specifically targeted towards optical design applications. We leverage multilayer perceptron networks to learn approximate solutions by minimizing a loss function that encompasses the equation's residual, boundary conditions, and convexity constraints. Our main results demonstrate the efficacy of this method, optimized using L-BFGS, through a series of test cases encompassing symmetric and asymmetric circle-to-circle, square-to-circle, and circle-to-flower reflector mapping problems. Comparative analysis with a conventional least-squares finite-difference solver reveals the competitive, and often superior, performance of our neural network approach on the test cases examined here. A comprehensive hyperparameter study further illuminates the impact of factors such as sampling density, network architecture, and optimization algorithm. While promising, further investigation is needed to verify the method's robustness for more complicated problems and to ensure consistent convergence. Nonetheless, the simplicity and adaptability of this neural network-based approach position it as a compelling alternative to specialized partial differential equation solvers.

cs.LG↗

arXiv2024

Studying wildfire fronts using advection-diffusion-reaction models

In this work, we study the propagation of wildfires using an advection--diffusion--reaction model which also includes convective and radiative heat loss. An existing model is discussed \cite{asensio_2002} and a physically consistent modification of the model is proposed. Using this, the existence of travelling waves (TWs) in the one-dimensional case is investigated. Prior numerical studies reveal the existence of TWs \cite{reisch_2023}. Under the travelling wave ansatz and certain approximation, the model is reduced to a semi-autonomous dynamical system with three unknowns which can be analyzed by a shooting algorithm. It is hypothesized that under mild wind speeds, TWs in both directions exist, and under strong tailwinds only TWs in the direction of wind are possible. The theoretical implications are investigated using both solvers for the PDE models and the shooting algorithm. The results match, and unveil the dependence of the fronts on the parameters consistent with the predictions.

math.AP↗

arXiv2023

Well-posedness and qualitative properties of quasilinear degenerate evolution systems

We analyze nonlinear degenerate coupled PDE-PDE and PDE-ODE systems that arise, for example, in the modelling of biofilm growth. One of the equations, describing the evolution of a biomass density, exhibits degenerate and singular diffusion. The other equations are either of advection-reaction-diffusion type or ordinary differential equations. Under very general assumptions, the existence of weak solutions is proven by considering regularized systems, deriving uniform bounds, and using fixed point arguments. Assuming additional structural assumptions we also prove the uniqueness of solutions. Global-in-time well-posedness is established for Dirichlet and mixed boundary conditions, whereas, only local well-posedness can be shown for homogeneous Neumann boundary conditions. Using a suitable barrier function and comparison theorems we formulate sufficient conditions for finite-time blow-up or uniform boundedness of solutions. Finally, we show that solutions of the degenerate parabolic equation inherit additional global spatial regularity if the diffusion coefficient has a power-law growth.

math.AP↗

arXiv2023

An adaptive solution strategy for Richards' equation

Flow in variably saturated porous media is typically modelled by the Richards equation, a nonlinear elliptic-parabolic equation which is notoriously challenging to solve numerically. In this paper, we propose a robust and fast iterative solver for Richards' equation. The solver relies on an adaptive switching algorithm, based on rigorously derived a posteriori indicators, between two linearization methods: L-scheme and Newton. Although a combined L-scheme/Newton strategy was introduced previously in [List & Radu (2016)], here, for the first time we propose a reliable and robust criteria for switching between these schemes. The performance of the solver, which can be in principle applied to any spatial discretization and linearization methods, is illustrated through several numerical examples.

math.NA↗

arXiv2022

Travelling waves in a PDE-ODE coupled system with nonlinear diffusion

We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. Finally, numerical simulations are presented that affirm the theoretical predictions on the existence, stability, and parametric dependence of the TWs.

math.AP↗

arXiv2019

Fronts in two-phase porous flow problems: effects of hysteresis and dynamic capillarity

In this work, we study the behaviour of saturation fronts for two phase flow through a long homogeneous porous column. In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses travelling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong to a much broader class compared to the standard Oleinik solutions, where hysteresis and dynamic effects are neglected. The relevant cases are examined and the corresponding solutions are categorized. They include non-monotone profiles, multiple shocks and self-developing stable saturation plateaus. Numerical results are presented that illustrate the mathematical analysis. Finally, we compare experimental results with our theoretical findings.

math.DS↗

arXiv2017

A linear domain decomposition method for partially saturated flow in porous media

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $Γ$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $Γ$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme.

math.NA↗