Related papers: Extremum Seeking for Stefan PDE with Moving Bounda…
We examine travelling wave solutions of the reaction-diffusion equation, $\partial_t u= R(u) + \partial_x \left[D(u) \partial_x u\right]$, with a Stefan-like condition at the edge of the moving front. With only a few assumptions on $R(u)$…
Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured…
This paper develops an input-to-state stability (ISS) analysis of the Stefan problem with respect to an unknown heat loss. The Stefan problem represents a liquid-solid phase change phenomenon which describes the time evolution of a…
In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design…
In this work, stabilization of an axonal growth in a neuron associated with the dynamics of tubulin concentration is proposed by designing a boundary control. The dynamics are given by a parabolic Partial Differential Equation (PDE) of the…
This paper presents a class of turbulence models written in terms of fractional partial differential equations (FPDEs) with stochastic loads. Every solution of these FPDE models is an incompressible velocity field and the distribution of…
This paper focuses on the further development of the Lie bracket approximation approach for extremum seeking systems. Classical results in this area provide extremum seeking algorithms with exponential convergence rates for quadratic-like…
In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…
For characterizing the Brownian motion in a bounded domain: $\Omega$, it is well-known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; on…
Modeling physical systems in a generative manner offers several advantages, including the ability to handle partial observations, generate diverse solutions, and address both forward and inverse problems. Recently, diffusion models have…
Score-based diffusion models have emerged as a powerful class of generative methods, achieving state-of-the-art performance across diverse domains. Despite their empirical success, the mathematical foundations of those models remain only…
The aim of this paper is to investigate extremum problems with pay-off being the total variational distance metric defined on the space of probability measures, subject to linear functional constraints on the space of probability measures,…
The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit…
Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the…
Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or…
The generalization of neural networks is a central challenge in machine learning, especially concerning the performance under distributions that differ from training ones. Current methods, mainly based on the data-driven paradigm such as…
This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L\'{e}vy-type…
In this paper, we develop an extremum seeking control method integrated with iterative learning control to track a time-varying optimizer within finite time. The behavior of the extremum seeking system is analyzed via an approximating…
In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity…
We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model…