On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral

2000 ◽  
pp. 254-271
Author(s):  
M. I. Zelikin
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Minimization Problem ◽  
Multiple Integral ◽  
Quadratic System ◽  
Partial Differential

scholarly journals Robust Low-Rank Discovery of Data-Driven Partial Differential Equations

2020 ◽  
Vol 34 (01) ◽  
pp. 767-774
Author(s):  
Jun Li ◽  
Gan Sun ◽  
Guoshuai Zhao ◽  
Li-wei H. Lehman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Minimization Problem ◽  
Real World ◽  
Low Rank ◽  
Dynamic Processes ◽  
Complex Data ◽  
Sparse Regression ◽  
Partial Differential ◽  
Regression Methods

Partial differential equations (PDEs) are essential foundations to model dynamic processes in natural sciences. Discovering the underlying PDEs of complex data collected from real world is key to understanding the dynamic processes of natural laws or behaviors. However, both the collected data and their partial derivatives are often corrupted by noise, especially from sparse outlying entries, due to measurement/process noise in the real-world applications. Our work is motivated by the observation that the underlying data modeled by PDEs are in fact often low rank. We thus develop a robust low-rank discovery framework to recover both the low-rank data and the sparse outlying entries by integrating double low-rank and sparse recoveries with a (group) sparse regression method, which is implemented as a minimization problem using mixed nuclear norms with ℓ1 and ℓ0 norms. We propose a low-rank sequential (grouped) threshold ridge regression algorithm to solve the minimization problem. Results from several experiments on seven canonical models (i.e., four PDEs and three parametric PDEs) verify that our framework outperforms the state-of-art sparse and group sparse regression methods. Code is available at https://github.com/junli2019/Robust-Discovery-of-PDEs

A New Image Demising Method Based on Partial Differential Equations

2013 ◽  
Vol 443 ◽  
pp. 22-26
Author(s):  
Yong Xing Lin ◽  
Xiao Yan Xu ◽  
Xian Dong Zhang
Keyword(s):  
Image Processing ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Objective Function ◽  
Thermal Diffusion ◽  
Minimization Problem ◽  
Iterative Process ◽  
Function Minimization ◽  
Partial Differential

In the paper, we discuss the image demising models, based on partial differential equations. It is through the use of the concept of variations in the calculus of the objective function minimization problem, defines the image processing tasks. The results show that the model expands 2d thermal diffusion equation. Therefore, it is easy to get solution is to use a simple iterative process.

scholarly journals Parameter Estimation for Partial Differential Equations by Collage-Based Numerical Approximation

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Xiaoyan Deng ◽  
Qingxi Liao
Keyword(s):  
Inverse Problem ◽  
Parameter Estimation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Minimization Problem ◽  
Engineering Practice ◽  
Unknown Parameters ◽  
Partial Differential ◽  
Excellent Starting Point ◽  
Starting Point

The inverse problem of using measurements to estimate unknown parameters of a system often arises in engineering practice and scientific research. This paper proposes a Collage-based parameter inversion framework for a class of partial differential equations. The Collage method is used to convert the parameter estimation inverse problem into a minimization problem of a function of several variables after the partial differential equation is approximated by a differential dynamical system. Then numerical schemes for solving this minimization problem are proposed, including grid approximation and ant colony optimization. The proposed schemes are applied to a parameter estimation problem for the Belousov-Zhabotinskii equation, and the results show that the proposed approximation method is efficient for both linear and nonlinear partial differential equations with respect to unknown parameters. At worst, the presented method provides an excellent starting point for traditional inversion methods that must first select a good starting point.

Partial Differential Equations

2020 ◽  
Author(s):  
A. K. Nandakumaran ◽  
P. S. Datti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential
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