A NONLINEAR ELIMINATION PRECONDITIONED INEXACT NEWTON METHOD FOR HETEROGENEOUS HYPERELASTICITY
We propose and study a nonlinear elimination preconditioned inexact Newton
method for the numerical simulation of diseased human arteries with a heterogeneous hyperelastic model. We assume the artery is made of layers of distinct tissues and also contains plaque. Traditional Newton methods often work well for smooth and homogeneous arteries but suffer from slow or no convergence due to the heterogeneousness of diseased soft tissues when the material is quasiincompressible. The proposed nonlinear elimination method adaptively finds a small number of equations causing the nonlinear stagnation and then eliminates them from the global nonlinear system. By using the theory of affine invariance of Newton method, we provide insight into why the nonlinear elimination method can improve the convergence of Newton iterations. Our numerical results show that the combination of nonlinear elimination with an initial guess interpolated from a coarse level solution can lead to the uniform convergence of Newton method for this class of very difficult nonlinear problems.
 Award ID(s):
 1720366
 Publication Date:
 NSFPAR ID:
 10155912
 Journal Name:
 SIAM journal on scientific computing
 ISSN:
 10648275
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract This paper presents a method to derive the virtual fields for identifying constitutive model parameters using the Virtual Fields Method (VFM). The VFM is an approach to identify unknown constitutive parameters using deformation fields measured across a given volume of interest. The general principle for solving identification problems with the VFM is first to derive parametric stress field, where the stress components at any point depend on the unknown constitutive parameters, across the volume of interest from the measured deformation fields. Applying the principle of virtual work to the parametric stress fields, one can write scalar equations of themore »

This work concerns the local convergence theory of Newton and quasiNewton methods for convexcomposite optimization: where one minimizes an objective that can be written as the composition of a convex function with one that is continuiously differentiable. We focus on the case in which the convex function is a potentially infinitevalued piecewise linearquadratic function. Such problems include nonlinear programming, minimax optimization, and estimation of nonlinear dynamics with nonGaussian noise as well as many modern approaches to largescale data analysis and machine learning. Our approach embeds the optimality conditions for convexcomposite optimization problems into a generalized equation. We establish conditions formore »

Electrical Impedance Tomography (EIT) is a wellknown imaging technique for detecting the electrical properties of an object in order to detect anomalies, such as conductive or resistive targets. More specifically, EIT has many applications in medical imaging for the detection and location of bodily tumors since it is an affordable and noninvasive method, which aims to recover the internal conductivity of a body using voltage measurements resulting from applying low frequency current at electrodes placed at its surface. Mathematically, the reconstruction of the internal conductivity is a severely illposed inverse problem and yields a poor quality image reconstruction. To remedymore »

Abstract This paper considers an inverse problem for a reaction diffusion equation from overposed final time data. Specifically, we assume that the reaction term
is known but modified by a spacedependent coefficient to obtain . Thus the strength of the reaction can vary with location. The inverse problem is to recover this coefficient. Our technique is to use iterative Newtontype methods although we also use and analyse higher order schemes of Halley type. We show that such schemes are well defined and prove convergence results. Our assumption about the diffusion process is also more general in thatmore » 
Beattie, C.A. ; Benner, P. ; Embree, M. ; Gugercin, S. ; Lefteriu, S. (Ed.)This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a largescale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the firstorder global convergence property, under suitable assumptions. Thus evenmore »