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Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
We establish quantitative estimates for solutions $u(t,x)$ to the fractional nonlinear diffusion equation, $\partial_t u +(-\Delta)^s (u^m)=0$ in the whole range of exponents $m>0$, $0<s<1$. The equation is posed in the whole space…
This paper addresses Bayesian inference related to partial differential equations (PDEs), particularly nonparametric regression constrained by PDEs. To effectively encode prior information, we propose a novel framework that learns a…
In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and…
We propose a non uniform web spline based finite element analysis for elliptic partial differential equation with the gradient type nonlinearity in their principal coefficients like p-laplacian equation and Quasi-Newtonian fluid flow…
In this paper, we propose a globally convergent method for solving constrained nonlinear systems. The method combines an efficient Newton conditional gradient method with a derivative-free and nonmonotone linesearch strategy. The global…
Recent developments in generative modeling have utilized score-based methods coupled with stochastic differential equations to sample from complex probability distributions. However, these and other performant sampling methods generally…
We establish sharp geometric Holder regularity estimates for Gradient for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy. Such regularity estimates simplify and generalize, to some extent,…
We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…
The purpose of this research work is to employ the Optimal Auxiliary Function Method (OAFM) for obtaining numerical approximations of time-dependent nonlinear partial differential equations (PDEs) that arise in many disciplines of science…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…
In this work, we provide a comprehensive gradient regularity theory for a broad class of nonlinear kinetic Fokker-Planck equations. We achieve this by establishing precise pointwise estimates in terms of the data in the spirit of nonlinear…
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Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of…
This paper presents a comprehensive analysis of the well-known extragradient (EG) method for solving both equations and inclusions. First, we unify and generalize EG for [non]linear equations to a wider class of algorithms, encompassing…
We consider the defocusing nonlinear Schr{\"o}dinger equation with a gauge invariant power-like nonlinearity. We prove global dispersive estimates in a semi-classical scaling, after rescaling the solution thanks to a suitable distorsion of…
We propose an online parametric estimation method of stochastic differential equations with discrete observations and misspecified modelling based on online gradient descent. Our study provides uniform upper bounds for the risks of the…
We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain…
In this paper, different strands of literature are combined in order to obtain algorithms for semi-parametric estimation of discrete choice models that include the modelling of unobserved heterogeneity by using mixing distributions for the…