TY - GEN A1 - Weiser, Martin T1 - Linear convergence of an interior point method for linear control constrained optimal control problems N2 - The paper provides a detailed analysis of a short step interior point algorithm applied to linear control constrained optimal control problems. Using an affine invariant local norm and an inexact Newton corrector, the well-known convergence results from finite dimensional linear programming can be extended to the infinite dimensional setting of optimal control. The present work complements a recent paper of Weiser and Deuflhard, where convergence rates have not been derived. The choice of free parameters, i.e. the corrector accuracy and the number of corrector steps, is discussed. T3 - ZIB-Report - 02-13 KW - linear optimal control KW - infinite dimensional KW - interior point methods KW - inexact Newton methods Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6809 ER - TY - GEN A1 - Schiela, Anton A1 - Weiser, Martin T1 - Superlinear Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimization N2 - A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global linear convergence we show, that this method converges locally almost quadratically, if the optimal solution satisfies a function space analogue to a non-degeneracy condition. In numerical experiments we observe, that a prototype implementation of our method behaves in compliance with our theoretical results. T3 - ZIB-Report - 05-15 KW - interior point methods in function space KW - optimal control KW - superlinear convergence Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8490 ER - TY - GEN A1 - Weiser, Martin A1 - Schiela, Anton T1 - Function space interior point methods for PDE constrained optimization N2 - A primal-dual interior point method for optimal control problems with PDE constraints is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed. Numerical results from an inexact continuation method applied to a model problem are shown. T3 - ZIB-Report - 04-27 KW - interior point methods in function space KW - optimal control KW - complementarity functions Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8027 ER - TY - GEN A1 - Prüfert, Uwe A1 - Tröltzsch, Fredi A1 - Weiser, Martin T1 - The convergence of an interior point method for an elliptic control problem with mixed control-state constraints N2 - The paper addresses primal interior point method for state constrained PDE optimal control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central path are established, and linear convergence of a short-step pathfollowing method is shown. The behaviour of the regularizations are demonstrated by numerical examples. T3 - ZIB-Report - 04-47 KW - interior point methods in function space KW - optimal control KW - mixed control-state constraints KW - Lavrentiev regularization Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8223 ER - TY - GEN A1 - Weiser, Martin A1 - Gänzler, Tobias A1 - Schiela, Anton T1 - A Control Reduced Primal Interior Point Method for PDE Constrained Optimization N2 - A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparable coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. T3 - ZIB-Report - 04-38 KW - interior point methods in function space KW - optimal control KW - finite elements KW - discretization error Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8138 ER - TY - GEN A1 - Weiser, Martin T1 - Interior Point Methods in Function Space N2 - A primal-dual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a short step pathfollowing method is established. T3 - ZIB-Report - 03-35 KW - interior point methods in function space KW - optimal control KW - complementarity functions Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7578 ER - TY - GEN A1 - Weiser, Martin A1 - Schiela, Anton A1 - Deuflhard, Peter T1 - Asymptotic Mesh Independence of Newton's Method Revisited N2 - The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method in nonlinear PDEs. Compared to earlier attempts, the new approach is both much simpler and more natural from the algorithmic point of view. The theory is exemplified at collocation methods for ODE boundary value problems and at finite element methods for elliptic PDE problems. T3 - ZIB-Report - 03-13 KW - mesh independence KW - nonlinear partial differential equations KW - Newton method KW - finite element method KW - collocation method Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7352 ER - TY - GEN A1 - Weiser, Martin A1 - Deuflhard, Peter A1 - Erdmann, Bodo T1 - Affine conjugate adaptive Newton methods for nonlinear elastomechanics N2 - The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning. T3 - ZIB-Report - 04-01 KW - affine conjugate Newton methods KW - nonconvex minimization KW - nonlinear elastomechnics KW - cranio-maxillofacial surgery KW - soft tissue simulation KW - multilev Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7768 ER - TY - GEN A1 - Weiser, Martin T1 - On goal-oriented adaptivity for elliptic optimal control problems N2 - The paper proposes goal-oriented error estimation and mesh refinement for optimal control problems with elliptic PDE constraints using the value of the reduced cost functional as quantity of interest. Error representation, hierarchical error estimators, and greedy-style error indicators are derived and compared to their counterparts when using the all-at-once cost functional as quantity of interest. Finally, the efficiency of the error estimator and generated meshes are demonstrated on numerical examples. T3 - ZIB-Report - 09-08 KW - optimal control KW - error estimation KW - adaptive mesh refinement Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11192 SN - 1438-0064 ER - TY - GEN A1 - Weiser, Martin T1 - Optimization and Identification in Regional Hyperthermia N2 - Regional hyperthermia is a cancer therapy aiming at heating tumors using phased array applicators. This article provides an overview over current mathematical challenges of delivering individually optimal treatments. The focus is on therapy planning and identification of technical as well as physiological quantities from MR thermometry measurements. T3 - ZIB-Report - 08-40 KW - Hyperthermie-Therapieplanung KW - Identifizierung KW - MR-Thermometrie KW - Innere-Punkte-Methoden KW - hyperthermia treatment planning KW - identification KW - MR thermometry KW - interior point methods Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10916 SN - 1438-0064 ER -