700
eng
reportzib
0
2002-10-24
2002-10-24
--
Affine Invariant Adaptive Newton Codes for Discretized PDEs
The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and inexact Newton methods, where an inner iteration is used instead, are synoptically presented, both in affine invariant convergence theory and in numerical experiments. The three types of algorithms are: (a) affine covariant (formerly just called affine invariant) Newton algorithms, oriented toward the iterative errors, (b) affine contravariant Newton algorithms, based on iterative residual norms, and (c) affine conjugate Newton algorithms for convex optimization problems and discrete nonlinear elliptic PDEs.
02-33
701
urn:nbn:de:0297-zib-7005
Appeared in: P. Deuflhard: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer 2004. Series Computational Mathemtics, 35
Peter Deuflhard
Ulrich Nowak
Martin Weiser
ZIB-Report
02-33
eng
uncontrolled
Affine invariant Newton methods
eng
uncontrolled
global Newton methods
eng
uncontrolled
inexact Newton methods
eng
uncontrolled
adaptive trust region methods
eng
uncontrolled
nonlinear partial differential equa
Informatik, Informationswissenschaft, allgemeine Werke
Systems of equations
Global methods, including homotopy approaches [See also 58C30, 90C30]
Computational Medicine
Deuflhard, Peter
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/700/ZR-02-33.ps
https://opus4.kobv.de/opus4-zib/files/700/ZR-02-33.pdf
638
eng
reportzib
0
2001-06-07
2001-06-07
--
The Central Path towards the Numerical Solution of Optimal Control Problems
A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior-point concepts in function space --- realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path. Existence of the central path and its continuation towards the solution point is analyzed in some theoretical detail. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of a first version of our new type of algorithm is documented by the successful treatment of the well-known intricate windshear problem.
01-12
639
urn:nbn:de:0297-zib-6380
Appeared under the title " Inexact Central Path Following Algorithms for Optimal Control Problems" in: SIAM J. Contr. Opt. 46 (3) (2007) 792-815
Martin Weiser
Peter Deuflhard
ZIB-Report
01-12
eng
uncontrolled
optimal control
eng
uncontrolled
interior point methods
eng
uncontrolled
affine invariance
Informatik, Informationswissenschaft, allgemeine Werke
Newton-type methods
Optimization and variational techniques [See also 49Mxx, 93B40]
Programming in abstract spaces
Interior-point methods
Computational Medicine
Deuflhard, Peter
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/638/ZR-01-12.ps
https://opus4.kobv.de/opus4-zib/files/638/ZR-01-12.pdf
849
eng
reportzib
0
2005-02-09
2005-02-09
--
Superlinear Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimization
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.
05-15
849
urn:nbn:de:0297-zib-8490
Appeared in: Comp. Opt. and Appl. 39(3): 369-393, 2008
Anton Schiela
Martin Weiser
ZIB-Report
05-15
eng
uncontrolled
interior point methods in function space
eng
uncontrolled
optimal control
eng
uncontrolled
superlinear convergence
Informatik, Informationswissenschaft, allgemeine Werke
Newton-type methods
Interior-point methods
Numerical Mathematics
ZIB Allgemein
Computational Medicine
Weiser, Martin
MATHEON-A17
ZIB-Kaskade7
https://opus4.kobv.de/opus4-zib/files/849/ZR-05-15.pdf
https://opus4.kobv.de/opus4-zib/files/849/ZR-05-15.ps
802
eng
reportzib
0
2004-06-24
2004-06-24
--
Function space interior point methods for PDE constrained optimization
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.
04-27
803
urn:nbn:de:0297-zib-8027
Appeared in : PAMM 4 (1) 43-46 (2004)
Martin Weiser
Anton Schiela
ZIB-Report
04-27
eng
uncontrolled
interior point methods in function space
eng
uncontrolled
optimal control
eng
uncontrolled
complementarity functions
Informatik, Informationswissenschaft, allgemeine Werke
Newton-type methods
Programming in abstract spaces
Interior-point methods
Numerical Mathematics
Computational Medicine
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/802/ZR-04-27.ps
https://opus4.kobv.de/opus4-zib/files/802/ZR-04-27.pdf
822
eng
reportzib
0
2004-12-01
2004-12-01
--
The convergence of an interior point method for an elliptic control problem with mixed control-state constraints
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.
04-47
823
urn:nbn:de:0297-zib-8223
Appeared in: Computational Optimization and Applications 39, No. 2 (2008) 183-218
Uwe Prüfert
Fredi Tröltzsch
Martin Weiser
ZIB-Report
04-47
eng
uncontrolled
interior point methods in function space
eng
uncontrolled
optimal control
eng
uncontrolled
mixed control-state constraints
eng
uncontrolled
Lavrentiev regularization
Informatik, Informationswissenschaft, allgemeine Werke
Optimal control problems involving partial differential equations
Error bounds
Interior-point methods
Numerical Mathematics
ZIB Allgemein
Computational Medicine
Weiser, Martin
MATHEON-A17
https://opus4.kobv.de/opus4-zib/files/822/ZR-04-47.ps
https://opus4.kobv.de/opus4-zib/files/822/ZR-04-47.pdf
813
eng
reportzib
0
2004-09-07
2004-09-07
--
A Control Reduced Primal Interior Point Method for PDE Constrained Optimization
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.
04-38
814
urn:nbn:de:0297-zib-8138
Appeared under the title "Control Reduced Primal Interior Point Methods" in: Comp. Optim. and Appl. 41 (2008) pp. 127-145
Martin Weiser
Tobias Gänzler
Anton Schiela
ZIB-Report
04-38
eng
uncontrolled
interior point methods in function space
eng
uncontrolled
optimal control
eng
uncontrolled
finite elements
eng
uncontrolled
discretization error
Informatik, Informationswissenschaft, allgemeine Werke
Newton-type methods
Error bounds
Interior-point methods
Numerical Mathematics
Computational Medicine
Weiser, Martin
MATHEON-A17
ZIB-Kaskade7
https://opus4.kobv.de/opus4-zib/files/813/ZR-04-38.ps
https://opus4.kobv.de/opus4-zib/files/813/ZR-04-38.pdf
757
eng
reportzib
0
2003-11-03
2003-11-03
--
Interior Point Methods in Function Space
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.
03-35
758
urn:nbn:de:0297-zib-7578
Appeared in : SIAM J. Control Opt. 44(5),1766-1786 (2005)
Martin Weiser
ZIB-Report
03-35
eng
uncontrolled
interior point methods in function space
eng
uncontrolled
optimal control
eng
uncontrolled
complementarity functions
Informatik, Informationswissenschaft, allgemeine Werke
Newton-type methods
Programming in abstract spaces
Interior-point methods
Numerical Mathematics
Computational Medicine
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/757/ZR-03-35.ps
https://opus4.kobv.de/opus4-zib/files/757/ZR-03-35.pdf
735
eng
reportzib
0
2003-05-12
2003-05-12
--
Asymptotic Mesh Independence of Newton's Method Revisited
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.
03-13
736
urn:nbn:de:0297-zib-7352
Appeared in: SIAM Journal on Numerical Analysis Vol. 42, No. 5, pp. 1830-1845
Martin Weiser
Anton Schiela
Peter Deuflhard
ZIB-Report
03-13
eng
uncontrolled
mesh independence
eng
uncontrolled
nonlinear partial differential equations
eng
uncontrolled
Newton method
eng
uncontrolled
finite element method
eng
uncontrolled
collocation method
Informatik, Informationswissenschaft, allgemeine Werke
Equations with nonlinear operators (do not use 65Hxx)
Solution of discretized equations [See also 65Fxx, 65Hxx]
Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/735/ZR-03-13.ps
https://opus4.kobv.de/opus4-zib/files/735/ZR-03-13.pdf
776
eng
reportzib
0
2004-04-23
2004-04-23
--
Affine conjugate adaptive Newton methods for nonlinear elastomechanics
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.
04-01
777
urn:nbn:de:0297-zib-7768
Appeared in: Opt. Meth. Softw. 22(3): 413-431 (2007)
Martin Weiser
Peter Deuflhard
Bodo Erdmann
ZIB-Report
04-01
eng
uncontrolled
affine conjugate Newton methods
eng
uncontrolled
nonconvex minimization
eng
uncontrolled
nonlinear elastomechnics
eng
uncontrolled
cranio-maxillofacial surgery
eng
uncontrolled
soft tissue simulation
eng
uncontrolled
multilev
Informatik, Informationswissenschaft, allgemeine Werke
Newton-type methods
Optimization and variational techniques [See also 49Mxx, 93B40]
Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Nonlinear elasticity
Energy minimization
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
FacialSurgery
https://opus4.kobv.de/opus4-zib/files/776/ZR-04-01.ps
https://opus4.kobv.de/opus4-zib/files/776/ZR-04-01.pdf
1119
eng
reportzib
0
2009-02-24
2009-02-24
--
On goal-oriented adaptivity for elliptic optimal control problems
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.
09-08
1438-0064
1167
urn:nbn:de:0297-zib-11192
appeared in Opt. Meth. Softw. 28(13), 969-992, 2013
Martin Weiser
unknown unknown
ZIB-Report
09-08
eng
uncontrolled
optimal control
eng
uncontrolled
error estimation
eng
uncontrolled
adaptive mesh refinement
Mathematik
Optimization and variational techniques [See also 49Mxx, 93B40]
Mesh generation and refinement
Numerical Mathematics
Computational Medicine
Weiser, Martin
MATHEON-A17
ZIB-Kaskade7
https://opus4.kobv.de/opus4-zib/files/1119/ZR_09_08.pdf