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A Globalized Inexact Semismooth Newton Method for Nonsmooth Fixed-point Equations involving Variational Inequalities

Amal Alphonse, Constantin Christof, Michael Hintermüller, Ioannis P. A. Papadopoulos

Abstract

We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure $q$-superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and $q$-superlinear convergence of the developed solution algorithm.

A Globalized Inexact Semismooth Newton Method for Nonsmooth Fixed-point Equations involving Variational Inequalities

Abstract

We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure -superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and -superlinear convergence of the developed solution algorithm.
Paper Structure (20 sections, 26 theorems, 66 equations, 4 figures, 3 tables, 2 algorithms)

This paper contains 20 sections, 26 theorems, 66 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation and basic concepts
  4. Semismooth Newton methods for fixed-point problems
  5. Vanilla inexact semismooth Newton method
  6. Global convergence for contractive equations
  7. Localization of the contraction assumptions and multiple solutions
  8. Composite fixed-point equations
  9. Elliptic obstacle-type QVIs
  10. The solution operator of the obstacle problem
  11. Solution operators of semilinear PDEs as obstacle maps
  12. An alternative application: semilinear VIs
  13. Numerical experiments for obstacle-type QVIs
  14. Implementation details
  15. Test 1: a one-dimensional QVI with a known solution
  16. ...and 5 more sections

Key Result

Theorem 2.1

Consider the fixed-point equation eq:general_FP_H_intro involving a Banach space $X$ and a map $H\colon X \to X$. Let $R\colon X \to X$, $R(x) := x - H(x)$, denote the residue function of eq:general_FP_H_intro. Suppose that the following holds: Then there exists $r >0$ such that the standard semismooth Newton method, i.e., algo:semiNewtonAbstractGeneral, satisfies the following for all $x_0 \in B

Figures (4)

  • Figure 1: (Test 1) Convergence behavior and results of the fixed-point method \ref{['item:C1']} and \ref{['algo:semiNewtonAbstract']} for the parameter choices (I) and (II) as well as \ref{['algo:semiNewtonAbstractGeneral']} and \ref{['item:C4']} (\ref{['algo:semiNewtonAbstractGeneral']} with a backtracking linesearch), for the parameter choice (II) with the initial guesses $u_0 = 0$ and $u_0 = (-\Delta)^{-1} f$ in the situation of \ref{['eq:ex_1_setup']}. The mesh size was chosen as $h= 5 \times 10^{-4}$. The numbers next to the graphs are the EOCs in \ref{['eq:EOC']}.
  • Figure 2: (Test 2) Divergence behavior of the fixed-point method \ref{['item:C1']}, \ref{['algo:semiNewtonAbstract']}, and \ref{['algo:semiNewtonAbstractGeneral']} with and without a backtracking linesearch \ref{['item:C4']} in the situation of the QVI \ref{['eq:ModelQVINum']} with setup \ref{['eq:ex_2_setup']} and $(\alpha_1,\alpha_2) = (10, 1)$. No projection is used and the initial guess is (the Lagrange interpolant of) $\bar{u}_2$. The mesh size is $h=5 \times 10^{-4}$. It can be checked that the pure fixed-point method actually converges to $\bar{u}_1$ although it is initialized as close to $\bar{u}_2$ as possible. \ref{['algo:semiNewtonAbstract']} and \ref{['algo:semiNewtonAbstractGeneral']} both with and without a backtracking linesearch stagnate/enter a cycle and are unable to reduce the residue below $10^{-6}$ in this test case---a bound much larger than observed in the other experiments; cf. \ref{['fig:tf-solutions']}. This indicates that, on the discrete level, there is no solution corresponding to $\bar{u}_2$.
  • Figure 3: (Test 3) Surface plots of the membrane (a) and corresponding mould (b) together with a slice plot at $x_2=1/2$ (c) for the thermoforming setting \ref{['eq:ex_3_setup']}. Figure (d) depicts the outer loop convergence of \ref{['algo:semiNewtonAbstract']}, the fixed-point method \ref{['item:C1']}, and the regularization method \ref{['item:C2']} for $h=0.02$ and various $\rho$. The residue depicted for \ref{['item:C2']} is that of the smoothed system. It can be seen that \ref{['algo:semiNewtonAbstract']} converges the fastest and that the convergence speed of the Moreau--Yosida-based method \ref{['item:C2']} degrades for $\rho \to 0$ due to ill-conditioning effects.
  • Figure 4: (Test 4) Solution of the semilinear VI \ref{['eq:semilinear_VI']} in the situation of \ref{['tab:test4-mesh-independence']}. The mesh size is $h=1.56 \times 10^{-4}$.

Theorems & Definitions (33)

  • Theorem 2.1: Local convergence of \ref{['algo:semiNewtonAbstractGeneral']}
  • Lemma 2.3: Unique solvability of \ref{['eq:general_FP_H_intro']}
  • Theorem 2.4: Finite and global $q$-superlinear convergence of \ref{['algo:semiNewtonAbstract']}
  • Remark 2.6
  • Lemma 2.7: Properties of $H_B$
  • Theorem 2.8: Convergence in the localized setting
  • Lemma 2.9: Projections onto closed balls
  • Remark 2.10
  • Corollary 2.11: Local convergence of \ref{['algo:semiNewtonAbstractGeneral']} for \ref{['eq:SPhi_FP']}
  • Corollary 2.12: Global convergence of \ref{['algo:semiNewtonAbstract']} for \ref{['eq:SPhi_FP']} with global contraction
  • ...and 23 more