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Abstract This paper presents a novel methodological framework to obtain superior reconstructions in limited data photoacoustic tomography. The proposed framework exploits the presence of Cauchy data on an accessible part of the observation domain and uses a Nash game-theoretic framework to complete the missing data on the inaccessible region. To solve the game-theoretic problem, a gradient-free sequential quadratic Hamiltonian scheme, which is based on Pontryagin’s maximum principle characterization, is combined with physics-informed neural networks to obtain the initial guess, leading to a robust and accurate reconstruction scheme. Numerical simulations with various phantoms, choice of accessible observation domains, and noise, demonstrate the effectiveness of our proposed framework to obtain high contrast and resolution reconstructions. 

Mathematical models of neuronal networks play a crucial role in understanding sleep dynamics and associated disorders. However, validating these models through parameter estimation remains a significant challenge. In this work, we introduce an automated parameter estimation framework for sleep models that satisfy two key assumptions: (i) they consist of competing neuronal populations, each driving a distinct sleep stage (stage-promoting), and (ii) their dynamics evolve independently of weakly observed variables or external inputs (self-contained). We apply our method to a system of coupled nonlinear ordinary differential equations (ODEs) representing three interacting neuronal populations. Direct firing rates of these populations are typically unobservable, and hypnograms provide only the dominant sleep stage at each time point. Despite the limited information available in hypnograms, we successfully estimate ODE parameters for the underlying neuronal population model directly from hypnogram data. We use a smoothed winner-takes-all strategy within a constrained minimization framework, reformulate the problem in an unconstrained setting through the Lagrangian, and derive the corresponding optimality conditions from state and adjoint equations. A projected nonlinear conjugate gradient scheme is then used to estimate the parameters numerically. We validate our approach by accurately reconstructing 111 out of 139 hypnograms from the Sleep-EDF database. The inferred population-level parameters provide insights into sleep regulation by capturing interaction strengths, timescale constants and non-rapid eye movement-related variability. 

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.