Conjugate gradient algorithms in the solution of optimization problems for nonlinear elliptic partial differential equations

Several variants of the conjugate gradient algorithm are discussed with emphasis on determining the parameters without performing line searches and on using splitting techniques to accelerate convergence. The splittings used here are related to the nonlinear SSOR algorithm. The behavior of the methods is illustrated on a discretization of a nonlinear elliptic partial differential boundary value problem, the minimal surface equation. A conjugate gradient algorithm with splittings is also developed for constrained minimization with upper and lower bounds on the variables, and the method is applied to the obstacle problem for the minimal surface equation.ZusammenfassungWir besprechen mehrere Varianten des konjugierten Gradienten-Algorithmus unter Hervorhebung der Parameterbestimmung ohne Minimierung entlang von Linien und der Konvergenzbeschleunigung durch Zerlegung. Die hier verwendeten Zerlegungen sind dem nichtlinearen SSOR-Algorithmus verwandt. Das Verhalten der Methoden wird illustriert an der Diskretisierung eines nichtlinearen elliptischen partiellen Randwertproblems, nämlich der Minimalflächen-Gleichung. Wir entwickeln auch einen konjugierten Gradienten-Algorithmus mit Zerlegungen für Minimierung mit von oben und unten beschränkten Variabeln; ferner zeigen wir eine Anwendung der Methode auf das Hindernisproblem bei der Minimalflächen-Gleichung.

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