A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian Equation

Alexandre Caboussat; Roland Glowinski; Dimitrios Gourzoulidis

doi:10.1007/s10915-022-01968-8

A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian Equation

Alexandre Caboussat^*
, Roland Glowinski
, Dimitrios Gourzoulidis

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

1 Citation (Scopus)

Abstract

We consider a least-squares/relaxation finite element method for the numerical solution of the prescribed Jacobian equation. We look for its solution via a least-squares approach. We introduce a relaxation algorithm that decouples this least-squares problem into a sequence of local nonlinear problems and variational linear problems. We develop dedicated solvers for the algebraic problems based on Newton’s method and we solve the differential problems using mixed low-order finite elements. Various numerical experiments demonstrate the accuracy, efficiency and the robustness of the proposed method, compared for instance to augmented Lagrangian approaches.

Original language	English
Article number	15
Number of pages	32
Journal	Journal of Scientific Computing
Volume	93
Issue number	1
DOIs	https://doi.org/10.1007/s10915-022-01968-8
Publication status	Published - 24 Aug 2022

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SDG 9 Industry, Innovation, and Infrastructure

User-Defined Keywords

Biharmonic regularization
Finite element method
Jacobian determinant
Least-squares method
Newton methods
Nonlinear constrained minimization

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10.1007/s10915-022-01968-8Licence: CC BY

Cite this

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title = "A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian Equation",

abstract = "We consider a least-squares/relaxation finite element method for the numerical solution of the prescribed Jacobian equation. We look for its solution via a least-squares approach. We introduce a relaxation algorithm that decouples this least-squares problem into a sequence of local nonlinear problems and variational linear problems. We develop dedicated solvers for the algebraic problems based on Newton{\textquoteright}s method and we solve the differential problems using mixed low-order finite elements. Various numerical experiments demonstrate the accuracy, efficiency and the robustness of the proposed method, compared for instance to augmented Lagrangian approaches.",

keywords = "Biharmonic regularization, Finite element method, Jacobian determinant, Least-squares method, Newton methods, Nonlinear constrained minimization",

author = "Alexandre Caboussat and Roland Glowinski and Dimitrios Gourzoulidis",

note = "Open access funding provided by University of Applied Sciences and Arts Western Switzerland (HES-SO) This work was supported by the Swiss National Science Foundation (Grant Number 165785). Roland Glowinski has received research support from the Hong Kong based Kennedy Wong Foundation. This work was partially funded by the Swiss National Science Foundation (Grant Number 165785), and by the Hong Kong based Kennedy Wong Foundation. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

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AU - Gourzoulidis, Dimitrios

N1 - Open access funding provided by University of Applied Sciences and Arts Western Switzerland (HES-SO) This work was supported by the Swiss National Science Foundation (Grant Number 165785). Roland Glowinski has received research support from the Hong Kong based Kennedy Wong Foundation. This work was partially funded by the Swiss National Science Foundation (Grant Number 165785), and by the Hong Kong based Kennedy Wong Foundation. Publisher Copyright: © 2022, The Author(s).

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N2 - We consider a least-squares/relaxation finite element method for the numerical solution of the prescribed Jacobian equation. We look for its solution via a least-squares approach. We introduce a relaxation algorithm that decouples this least-squares problem into a sequence of local nonlinear problems and variational linear problems. We develop dedicated solvers for the algebraic problems based on Newton’s method and we solve the differential problems using mixed low-order finite elements. Various numerical experiments demonstrate the accuracy, efficiency and the robustness of the proposed method, compared for instance to augmented Lagrangian approaches.

AB - We consider a least-squares/relaxation finite element method for the numerical solution of the prescribed Jacobian equation. We look for its solution via a least-squares approach. We introduce a relaxation algorithm that decouples this least-squares problem into a sequence of local nonlinear problems and variational linear problems. We develop dedicated solvers for the algebraic problems based on Newton’s method and we solve the differential problems using mixed low-order finite elements. Various numerical experiments demonstrate the accuracy, efficiency and the robustness of the proposed method, compared for instance to augmented Lagrangian approaches.

KW - Biharmonic regularization

KW - Finite element method

KW - Jacobian determinant

KW - Least-squares method

KW - Newton methods

KW - Nonlinear constrained minimization

UR - https://www.scopus.com/pages/publications/85137062243

UR - https://link.springer.com/article/10.1007/s10915-022-01968-8#article-info

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DO - 10.1007/s10915-022-01968-8

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JF - Journal of Scientific Computing

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M1 - 15

ER -

A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian Equation

Abstract

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