arXiv subjects

B. V. Rathish Kumar

Publications and source records attributed to B. V. Rathish Kumar.

At least 19 recordsLinked to original sources

arXiv2025

Finite Element Analysis of Nash Equilibrium of Bi-objective Optimal Control Problem Governed by Stokes Equation with $L^2$-norm State-Constraints

This paper investigates the Nash equilibrium of a bi-objective optimal control problem governed by the Stokes equations. A multi-objective Nash strategy is formulated, and fundamental theoretical results are established, including the existence, uniqueness, and analytical characterization of the equilibrium. A finite element framework is developed to approximate the coupled optimal control system, and the corresponding optimality conditions for both the continuous and discrete formulations are rigorously derived and analyzed. Furthermore, \textit{a priori} finite element error estimates are obtained for the discrete problem, ensuring convergence and stability of the proposed method. The theoretical results are corroborated by numerical experiments, which demonstrate the accuracy and computational efficiency of the finite element approach.

math.OC↗

arXiv2025

Nash Equilibrium of Bi-objective Optimal Control of Fractional Space-Time Parabolic PDE

This work investigates the existence and uniqueness of the Nash equilibrium (solutions to competitive problems in which individual controls aim at separate desired states) for a bi-objective optimal control problem governed by a fractional space-time parabolic partial differential equation. The governing equation involves a Caputo fractional derivative with respect to time of order $γ$ in (0,1) and a fractional Laplacian in the spatial variables of order $s$ in (0,1). The system is associated with two independent controls, each aiming at different targets. The problem is formulated as a distributed optimal control system with quadratic cost functionals. Existence and uniqueness of the Nash equilibrium are established under convexity and coercivity assumptions. The solution is computed using conjugate gradient algorithms applied iteratively to the discretized optimal control problems. The numerical experiments agree with the theoretical estimates and demonstrate the efficiency of the proposed scheme.

math.OC↗

arXiv2025

CBINNS: Cancer Biology-Informed Neural Network for Unknown Parameter Estimation and Missing Physics Identification

The dynamics of tumor-immune interactions within a complex tumor microenvironment are typically modeled using a system of ordinary differential equations or partial differential equations. These models introduce some unknown parameters that need to be estimated accurately and efficiently from the limited and noisy experimental data. Moreover, due to the intricate biological complexity and limitations in experimental measurements, tumor-immune dynamics are not fully understood, and therefore, only partial knowledge of the underlying physics may be available, resulting in unknown or missing terms within the system of equations. In this study, we develop a cancer biology-informed neural network model(CBINN) to infer the unknown parameters in the system of equations as well as to discover the missing physics from sparse and noisy measurements. We test the performance of the CBINN model on three distinct nonlinear compartmental tumor-immune models and evaluate its robustness across multiple synthetic noise levels. By harnessing these highly nonlinear dynamics, our CBINN framework effectively estimates the unknown model parameters and uncovers the underlying physical laws or mathematical structures that govern these biological systems, even from scattered and noisy measurements. The models chosen here represent the dynamic patterns commonly observed in compartmental models of tumor-immune interactions, thereby validating the generalizability and efficacy of our methodology.

q-bio.QM↗

arXiv2025

Bridging Accuracy and Interpretability: Deep Learning with XAI for Breast Cancer Detection

In this study, we present an interpretable deep learning framework for the early detection of breast cancer using quantitative features extracted from digitized fine needle aspirate (FNA) images of breast masses. Our deep neural network, using ReLU activations, the Adam optimizer, and a binary cross-entropy loss, delivers state-of-the-art classification performance, achieving an accuracy of 0.992, precision of 1.000, recall of 0.977, and an F1 score of 0.988. These results substantially exceed the benchmarks reported in the literature. We evaluated the model under identical protocols against a suite of well-established algorithms (logistic regression, decision trees, random forests, stochastic gradient descent, K-nearest neighbors, and XGBoost) and found the deep model consistently superior on the same metrics. Recognizing that high predictive accuracy alone is insufficient for clinical adoption due to the black-box nature of deep learning models, we incorporated model-agnostic Explainable AI techniques such as SHAP and LIME to produce feature-level attributions and human-readable visualizations. These explanations quantify the contribution of each feature to individual predictions, support error analysis, and increase clinician trust, thus bridging the gap between performance and interpretability for real-world clinical use. The concave points feature of the cell nuclei is found to be the most influential feature positively impacting the classification task. This insight can be very helpful in improving the diagnosis and treatment of breast cancer by highlighting the key characteristics of breast tumor.

cs.CV↗

arXiv2025

Adaptive SIPG method for approximations of boundary control problems governed by parabolic PDEs

This study presents an aposteriori error analysis of adaptive finite element approximations of parabolic boundary control problems with bilateral box constraints that act on a Neumann boundary. The control problem is discretized using the symmetric interior penalty Galerkin (SIPG) technique. We derive both reliable and efficient type residual-based error estimators coupling with the data oscillations. The implementation of these error estimators serves as a guide for the adaptive mesh refinement process, indicating whether or not more refinement is required. Although the control error estimator effectively captured control approximation errors, it had limitations in guiding refinement localization in critical cases. To overcome this, an alternative control indicator was used in numerical tests. The results demonstrated the clear superiority of adaptive refinements over uniform refinements, confirming the proposed approach's effectiveness in achieving accurate solutions while optimizing computational efficiency. numerical experiment showcases the effectiveness of the derived error estimators.

math.NA↗

arXiv2025

Optimal control of fractional Poisson equation from non-local to local

In this article, the limiting behavior of the solution $\bar u_s$ of the optimal control problem subjected to the fractional Poisson equation $$(-Δ)^s u_s(x)=f_s(x), \quad x\in Ω$$ defined on domain $Ω$ bounded by smooth boundary with zero exterior boundary conditions $u_s(x)\equiv 0, \quad x \in Ω^c $ is established. We will prove that $\lim_{s\to 1^-} \bar u_s= \bar u$, where $\bar u$ is a solution of the optimal control problem subjected to classical Poisson equation $-Δu(x)=f(x), \quad x \in Ω$ and $u(x)=0, \quad x\in \partial Ω.$

math.NA↗

arXiv2023

Finite element analysis of modified N-S equations coupled with energy transfer for Hybrid Nanofluid flow in complex domains

A theoretical and computational finite element study of modified Navier-Stokes Equations coupled with energy conservation governing the flow and heat transfer in complex domains with hybrid nanofluid$(HNF)$ is carried out. The apriori error estimates providing the convergence analysis for the finite element scheme is derived in the $H^1$ norm. A detailed parametric analysis on the physics related to flow and heat tranfer for different governing parameters such as: hybrid nano-particle volume fraction $(ϕ),$ Rayleigh Number $(Ra)$, Prandtl Number$(Pr)$, and the length of heat source has been reported. The use of hybrid nano-particles leads to the enhancement of heat conduction both in L and H shaped domains. While a butterfly flow pattern with a multi-cellular structure is noticed in H-shaped domain, the flow in L-Shaped domain depicts a boundary layer feature with primary, secondary and tertiary co-rotating and counter rotating cells.

math.NA↗

arXiv2023

An efficient Haar wavelet method for the coupled non-linear transient PDE-ODEs system with discontinuous coefficients

In this work, the Haar wavelet method for the coupled non-linear transient PDE-ODEs system with the Neumann boundary condition has been proposed. The capability of the method in handling multiple jump discontinuities in the coefficients and parameters of the transient and coupled PDE-ODEs system is brought out through a series of 1D/2D/3D test problems. The method is easy to implement, computationally efficient, and compares well with conventional methods like the finite element method. Convergence analysis of the method has been carried out and apriori error is found to be exponential order i.e. ||u-u_H||_X<= o(2^j ). ILU-GMRES is found to better accelerate the numerical solution convergence than other Krylov solvers for the class of transient non-linear PDE-ODE system.

math.NA↗

arXiv2022

A Fractional Image Inpainting Model Using a Variant of Mumford-Shah Model

In this paper, we propose a fourth order PDE model for image inpainting based on a variant of the famous Mumford-Shah (MS) image segmentation model. Convexity splitting is used to discrtised the time and we replace the Laplacian by its fractional counterpart in the time discretised scheme. Fourier spectral method is used for space discretization. Consistency, stability and convergence of the time discretised model has been carried out. The model is tested on some standard test images and compared them with the result of some models existed in the literature.

math.NA↗

arXiv2021

Myocardial ischemic effects on cardiac electro-mechanical activity

In this work, we investigated the effect of varying strength of Hyperkalemia and Hypoxia, in human cardiac tissue with a local ischemic subregion, on the electrical and mechanical activity of healthy and ischemic zones of the cardiac muscle. The Monodomain model in a deforming domain is taken with the addition of mechanical feedback and stretch-activated channel current coupled with the ten Tusscher human ventricular membrane model. The equations of finite elasticity are used to describe the deformation of the cardiac tissue. The resulting coupled electro-mechanical PDEs-ODEs non-linear system is solved numerically using finite elements in space and finite difference method in time. We examined the effect of local ischemia on cardiac electrical and mechanical activity in different cases. We concluded that the spread of Hyperkalemic or Hypoxic region alters the electro-mechanical coupling in terms of the action potential ($v$), intracellular calcium ion concentration $[Ca^{+2}]_i$, active tension, ($T_A$), stretch ($λ$), stretch rate ($ \frac{d λ}{dt}$). With the increase in the size of the ischemic region by a factor of five, approximately $45\%$ variation in the stretch rate $\frac{d λ}{dt}$ is noticed. It is also shown that ischemia affects the deformation (expansion and contraction) of the heart.

math.NA↗

arXiv2021

Subgrid multiscale stabilized finite element analysis of non-Newtonian Casson model fully coupled with Advection-Diffusion-Reaction equations

In this paper we have studied subgrid multiscale stabilized formulation with dynamic subscales for non-Newtonian Casson fluid flow model tightly coupled with variable coefficients ADR ($VADR$) equation. The Casson viscosity coefficient is taken to be dependent upon solute mass concentration. This paper presents the stability and convergence analyses of the stabilized finite element solution. The proposed expressions of the stabilization parameters helps in obtaining optimal order of convergences. Appropriate numerical experiments have been provided.

math.NA↗

arXiv2021

Subgrid multiscale stabilized finite element analysis of non-Newtonian Power-law model fully coupled with Advection-Diffusion-Reaction equations

This article presents stability and convergence analyses of subgrid multiscale stabilized finite element formulation of non-Newtonian power-law fluid flow model strongly coupled with variable coefficients Advection-Diffusion-Reaction ($VADR$) equation. Considering the highly non-linear viscosity coefficient as solute concentration dependent makes the coupling two way. The stabilized formulation of the transient coupled system is developed based upon time dependent subscales, which ensures inherent consistency of the method. The proposed algebraic expressions of the stabilization parameters appropriately shape up the apriori and aposteriori error estimates. Both the shear thinning and shear thickening properties, indicated by different power-law indices are properly highlighted in theoretical derivations as well as in numerical validations. The numerical experiments carried out for different combinations of small and large Reynolds numbers and power-law indices establish far better performance of time dependent $ASGS$ method in approximating the solution of this coupled system for all the cases over the other well known stabilized finite element methods.

math.NA↗

arXiv2020

On the Existence Uniqueness and Numerical Computation of Non-linear Coupled Elliptic PDE System with its application

In this study we prove the existence-uniqueness of a coupled non-linear elliptic PDE system using Lax-Milgram theorem, Galerkin Method, Brouwer's fixed point theorem. Later we derive the finite element scheme for the numerical solution of the PDE system and also carry out the convergence analysis for the derived scheme. Further successfully apply the scheme to an application related to free convection phenomena.

math.AP↗

arXiv2020

Apriori and aposteriori error estimation of Subgrid multiscale stabilized finite element method for fully coupled Navier-Stokes Transport model

In this paper a fully coupled system of transient $Navier$-$Stokes$ ($NS$) fluid flow model and variable coefficient unsteady Advection-Diffusion-Reaction ($VADR$) transport model has been studied through subgrid multiscale stabilized finite element method. In particular algebraic approach of approximating the subscales has been considered to arrive at stabilized variational formulation of the coupled system. This system is strongly coupled since viscosity of the fluid depends upon the concentration, whose transportation is modelled by $VADR$ equation. Fully implicit schemes have been considered for time discretisation. Further more elaborated derivations of both $apriori$ and $aposteriori$ error estimates for stabilized finite element scheme have been carried out. Credibility of the stabilized method is also established well through various numerical experiments, presented before concluding.

math.NA↗

arXiv2020

Apriori and aposteriori error estimation of Subgrid multiscale stabilized finite element method for coupled unified Stokes-Brinkman/Transport model

In this study, we present a stabilized finite element analysis for completely unified Stokes-Brinkman problems fully coupled with variable coefficient transient Advection-Diffusion-Reaction equation(VADR). As well we have carried out the stabilized finite element analysis for Stokes-Brinkman model with interface conditions fully coupled with VADR. The viscosity of the fluid, involved in flow problem, depends on the concentration of the solute, whose transport is described by VADR equation. The algebraic subgrid multiscale approach has been employed to arrive at the stabilized coupled variational formulation. For the time discretization the fully implicit Euler scheme has been used. A detailed derivation of both the apriori and aposteriori estimates for the stabilized subgrid multiscale finite element scheme have been presented. Few numerical experiments have been carried out to verify the credibility of the method.

math.NA↗

arXiv2019

Subgrid multiscale stabilized finite element analysis of fully-coupled unified Stokes-Darcy-Brinkman/Transport model

In this study, a stabilized finite element analysis of unified Stokes-Darcy-Brinkman system fully coupled with variable coefficient Advection-Diffusion-Reaction equation(VADR) has been carried out. The viscosity of the fluid, involved in Stokes-Darcy flow, depends on the concentration of the solute, whose transport is described by VADR equation. The algebraic subgrid multiscale approach has been employed to arrive at the stabilized coupled variational formulation. For the time discretization the fully implicit Euler scheme has been used. A detailed derivation of both the apriori and aposteriori estimates for the stabilized subgrid multiscale finite element scheme have been presented. Few numerical experiments have been carried out to verify the credibility of the method.

math.AP↗

arXiv2019

Stabilized subgrid multiscale finite element formulation for advection-diffusion-reaction equation with variable coefficients coupled with Stokes-Darcy equation

In this paper subgrid multiscale stabilized finite element method for Advection-Diffusion-Reaction (ADR) equation coupled with Stokes-Darcy flow problem has been studied. Here the advection velocity involved in ADR equation obeys Stokes-Darcy flow equation. In this study the approach of algebraic approximation of stabilization parameter has been considered. Further apriori error estimation has been elaborately carried out.

math.AP↗

arXiv2019

Haar wavelet method for the coupled degenerate reaction-diffusion PDEs and the ODEs having a non-linear source

In this work, we propose the Haar wavelet method for the coupled degenerate reaction-diffusion PDEs and the ODEs having non-linear a source with Neumann boundary, applicable in various fields of the natural sciences, engineering, and economics, for example in gas dynamics, certain biological models, assets pricing in economics, composite media etc. Convergence analysis of the proposed numerical scheme has been carried out. We use the GMRES solver to solve the linear system of equations. Numerical solutions for the model problems of medical significance have been successfully solved.

math.NA↗