Nonlinear optimization in Hilbert space using Sobolev gradients with applications

2011/12/14 by Parimah Kazemi, Kazemi, Parimah, Robert J. Renka +1
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Methods and Algorithms #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1112.3150

openalex publication_date 2011/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The problem of finding roots or solutions of a nonlinear partial differential equation may be formulated as the problem of minimizing a sum of squared residuals. One then defines an evolution equation so that in the asymptotic limit a minimizer, and often a solution of the PDE, is obtained. The corresponding discretized nonlinear least squares problem is an often met problem in the field of numerical optimization, and thus there exist a wide variety of methods for solving such problems. We review here Newton's method from nonlinear optimization both in a discrete and continuous setting and present results of a similar nature for the Levernberg-Marquardt method. We apply these results to the Ginzburg-Landau model of superconductivity.

Related

• A gradient method in a Hilbert space with an optimized inner product: achieving a Newton-like convergence
• Optimizing Schroedinger functionals using Sobolev gradients: Applications to Quantum Mechanics and Nonlinear Optics
• Combined LP and Quasi-Newton Methods for Nonlinear l₁ Optimization
• The q-Levenberg-Marquardt method for unconstrained nonlinear\n optimization
• Partial differential equation solver based on optimization methods