scholarly journals A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems

2018 ◽  
Vol 52 (1) ◽  
pp. 123-145 ◽  
Author(s):  
Laurent Bourgeois ◽  
Arnaud Recoquillay
Keyword(s):  
Partial Differential Equations ◽  
Finite Elements ◽  
Differential Equations ◽  
Level Set Method ◽  
Level Set ◽  
Tikhonov Regularization ◽  
Obstacle Problem ◽  
Numerical Experiments ◽  
Mixed Formulation ◽  
Ill Posed

This paper is dedicated to a new way of presenting the Tikhonov regularization in the form of a mixed formulation. Such formulation is well adapted to the regularization of linear ill-posed partial differential equations because when it comes to discretization, the mixed formulation enables us to use some standard finite elements. As an application of our theory, we consider an inverse obstacle problem in an acoustic waveguide. In order to solve it we use the so-called “exterior approach”, which couples the mixed formulation of Tikhonov regularization and a level set method. Some 2d numerical experiments show the feasibility of our approach.

Numerical and Experimental Analysis of the Nonlinear Dynamics due to Impacts of a Continuous Overhung Rotor

1997 ◽  
Author(s):  
Mohammed F. Abdul Azeez ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Experiments ◽  
The Other ◽  
Experimental Setup ◽  
Partial Differential ◽  
Qualitative Comparison ◽  
Set Up ◽  
Theory And Experiments

Abstract This work is aimed at obtaining the transient response of an overhung rotor when there are impacts occurring in the system. An overhung rotor clamped on one end, with a flywheel on the other and impacts occurring in between, due to a bearing with clearance, is considered. The system is modeled as a continuous rotor system and the governing partial differential equations are set up and solved. The method of assumed modes is used to discretize the system in order to solve the partial differential equations. Using this method numerical experiments are run and a few of the results are presented. The different numerical issues involved are also discussed. An experimental setup was built to run experiments and validate the results. Preliminary experimental observations are presented to show qualitative comparison of theory and experiments.

Computation of geometric partial differential equations and mean curvature flow

Acta Numerica ◽  
2005 ◽  
Vol 14 ◽  
pp. 139-232 ◽  
Author(s):  
Klaus Deckelnick ◽  
Gerhard Dziuk ◽  
Charles M. Elliott
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Level Set ◽  
Phase Field ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Partial Differential ◽  
Geometric Partial Differential Equations ◽  
Geometric Pdes

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

SAINT-VENANT TYPE DECAY RESULTS FOR ILL-POSED ELLIPTIC PROBLEMS

2000 ◽  
Vol 10 (05) ◽  
pp. 771-783 ◽  
Author(s):  
KAREN A. AMES ◽  
LAWRENCE E. PAYNE
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Problems ◽  
Vital Role ◽  
Cylindrical Domain ◽  
Differential Inequalities ◽  
Partial Differential ◽  
First Order ◽  
Decay Of Solutions ◽  
Ill Posed

Differential inequalities play a vital role in the study of ordinary and partial differential equations. In this paper we make use of first-order differential inequalities to investigate the decay of solutions to two ill-posed elliptic problems in a semi-infinite cylindrical domain.

scholarly journals Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Nathan Bush ◽  
Baasansuren Jadamba ◽  
Akhtar A. Khan ◽  
Fabio Raciti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Optimization Problem ◽  
Short Note ◽  
Variable Parameter ◽  
Partial Differential ◽  
Convergence Results ◽  
Equation Error

AbstractThe objective of this short note is to employ an equation error approach to identify a variable parameter in fourth-order partial differential equations. Existence and convergence results are given for the optimization problem emerging from the equation error formulation. Finite element based numerical experiments show the effectiveness of the proposed framework.

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