378
eng
reportzib
0
1998-12-10
1998-12-10
--
A New Nonlinear Elliptic Multilevel FEM Applied to Regional Hyperthermia
In the clinical cancer therapy of regional hyperthermia nonlinear perfusion effects inside and outside the tumor seem to play a not negligible role. A stationary model of such effects leads to a nonlinear Helmholtz term within an elliptic boundary value problem. The present paper reports about the application of a recently designed adaptive multilevel FEM to this problem. For several 3D virtual patients, nonlinear versus linear model is studied. Moreover, the numerical efficiency of the new algorithm is compared with a former application of an adaptive FEM to the corresponding instationary model PDE.
SC-98-35
379
urn:nbn:de:0297-zib-3785
Appeared under the title: "A New Nonlinear Elliptic Multilevel FEM in Clinical Cancer Therapy Planning" in: Comput Visual Sc 3 (2000) 115-120
Peter Deuflhard
Martin Weiser
Martin Seebass
ZIB-Report
SC-98-35
eng
uncontrolled
hyperthermia
eng
uncontrolled
nonlinear elliptic
eng
uncontrolled
multilevel FEM
Informatik, Informationswissenschaft, allgemeine Werke
Nonlinear elliptic equations
Equations with nonlinear operators (do not use 65Hxx)
Medical applications (general)
ZIB Allgemein
Deuflhard, Peter
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/378/SC-98-35.ps
https://opus4.kobv.de/opus4-zib/files/378/SC-98-35.pdf
3996
2000
2000
eng
1
6
3
article
0
--
--
--
A New Nonlinear Elliptic Multilevel FEM Applied to Regional Hyperthermia
Comput. Visual. Sci.
10.1007/PL00013546
urn:nbn:de:0297-zib-3785
Peter Deuflhard
Martin Weiser
Martin Seebaß
Visual Data Analysis
Visual Data Analysis in Science and Engineering
Therapy Planning
Deuflhard, Peter
Weiser, Martin
Visual and Data-centric Computing
2104
2000
2000
eng
115
120
3
article
0
--
--
--
A new nonlinear elliptic multilevel FEM in clinical cancer therapy planning
Comput. Vis. Sci.
Peter Deuflhard
Martin Weiser
Martin Seebass
Numerical Mathematics
Deuflhard, Peter
Weiser, Martin
2106
2012
2012
eng
book
de Gruyter
Berlin
0
--
--
--
Adaptive numerical solution of PDEs
Peter Deuflhard
Martin Weiser
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
ZIB-Kaskade7
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
2459
2007
2007
eng
413
431
3
22
article
0
--
--
--
Affine conjugate adaptive Newton methods for nonlinear elastomechanics
Opt. Meth. Softw.
urn:nbn:de:0297-zib-7768
Martin Weiser
Peter Deuflhard
Bodo Erdmann
Numerical Mathematics
Deuflhard, Peter
Weiser, Martin
FacialSurgery
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
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
2456
2005
2005
eng
1830
1845
5
42
article
0
--
--
--
Asymptotic Mesh Independence of Newton’s Method Revisited
SIAM J. Num. Anal.
urn:nbn:de:0297-zib-7352
Martin Weiser
Anton Schiela
Peter Deuflhard
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
3697
2010
2010
eng
258
263
5
52
article
Oldenbourg Verlagsgruppe
0
--
--
--
Craniofacial Surgery Planning Based on Virtual Patient Models
it - Information Technology
10.1524/itit.2010.0600
yes
Martin Weiser
Stefan Zachow
Peter Deuflhard
Numerical Mathematics
Visual Data Analysis
Computational Medicine
Therapy Planning
Deuflhard, Peter
Weiser, Martin
Zachow, Stefan
FacialSurgery
Visual and Data-centric Computing
243
eng
reportzib
0
1996-10-14
1996-10-14
--
Global Inexact Multilevel FEM for Nonlinear Elliptic Problems
The paper deals with the multilevel solution of {\em elliptic} partial differential equations (PDEs) in a {\em finite element} setting: {\em uniform ellipticity} of the PDE then goes with {\em strict monotonicity} of the derivative of a nonlinear convex functional. A {\em Newton multigrid method} is advocated, wherein {\em linear residuals} are evaluated within the multigrid method for the computation of the Newton corrections. The globalization is performed by some {\em damping} of the ordinary Newton corrections. The convergence results and the algorithm may be regarded as an extension of those for local Newton methods presented recently by the authors. An {\em affine conjugate} global convergence theory is given, which covers both the {\em exact} Newton method (neglecting the occurrence of approximation errors) and {\em inexact} Newton--Galerkin methods addressing the crucial issue of accuracy matching between discretization and iteration errors. The obtained theoretical results are directly applied for the construction of adaptive algorithms. Finally, illustrative numerical experiments with a~{\sf NEWTON--KASKADE} code are documented.
SC-96-33
244
urn:nbn:de:0297-zib-2430
Appeared in: W. Hackbusch, G. Wittum (eds.) Multigrid Methods. Lecture Notes in Computational Science and Engineering 3. Springer, 1998, pp. 71-89
Peter Deuflhard
Martin Weiser
ZIB-Report
SC-96-33
Informatik, Informationswissenschaft, allgemeine Werke
ZIB Allgemein
Deuflhard, Peter
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/243/SC-96-33.ps
https://opus4.kobv.de/opus4-zib/files/243/SC-96-33.pdf
2103
1998
1998
eng
71
89
3
conferenceobject
Berlin: Springer
0
--
--
--
Global inexact Newton multilevel FEM for nonlinear elliptic problems
Multigrid methods V. proceedings of the 5th European multigrid conference, held in Stuttgart, Germany, October 1–4, 1996.
Lect. Notes Comput. Sci. Eng.
urn:nbn:de:0297-zib-2430
Peter Deuflhard
Wolfgang et al. Hackbusch
Martin Weiser
Numerical Mathematics
Deuflhard, Peter
Weiser, Martin
1909
2007
2007
eng
792
815
3
46
article
0
--
--
--
Inexact central path following algorithms for optimal control problems
SIAM J. Control Opt.
urn:nbn:de:0297-zib-6380
Martin Weiser
Peter Deuflhard
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
240
eng
reportzib
0
1996-09-25
1996-09-25
--
Local Inexact Newton Multilevel FEM for Nonlinear Elliptic Problems
The finite element setting for nonlinear elliptic PDEs directly leads to the minimization of convex functionals. Uniform ellipticity of the underlying PDE shows up as strict convexity of the arising nonlinear functional. The paper analyzes computational variants of Newton's method for convex optimization in an affine conjugate setting, which reflects the appropriate affine transformation behavior for this class of problems. First, an affine conjugate Newton--Mysovskikh type theorem on the local quadratic convergence of the exact Newton method in Hilbert spaces is given. It can be easily extended to inexact Newton methods, where the inner iteration is only approximately solved. For fixed finite dimension, a special implementation of a Newton--PCG algorithm is worked out. In this case, the suggested monitor for the inner iteration guarantees quadratic convergence of the outer iteration. In infinite dimensional problems, the PCG method may be just formally replaced by any Galerkin method such as FEM for linear elliptic problems. Instead of the algebraic inner iteration errors we now have to control the FE discretization errors, which is a standard task performed within any adaptive multilevel method. A careful study of the information gain per computational effort leads to the result that the quadratic convergence mode of the Newton--Galerkin algorithm is the best mode for the fixed dimensional case, whereas for an adaptive variable dimensional code a special linear convergence mode of the algorithm is definitely preferable. The theoretical results are then illustrated by numerical experiments with a {\sf NEWTON--KASKADE} algorithm.
SC-96-29
240
urn:nbn:de:0297-zib-2401
Appeared in: Computational Science for the 21st Century. M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.-L. Lions, J. Periaux, M. Wheeler (eds.) Wiley-Interscience-Europe, 1997, pp. 129-138
Peter Deuflhard
Martin Weiser
ZIB-Report
SC-96-29
Informatik, Informationswissenschaft, allgemeine Werke
ZIB Allgemein
Deuflhard, Peter
Weiser, Martin
https://opus4.kobv.de/opus4-zib/files/240/SC-96-29.ps
https://opus4.kobv.de/opus4-zib/files/240/SC-96-29.pdf
2102
1997
1997
eng
129
138
bookpart
Chichester: John Wiley & Sons.
0
--
--
--
Local inexact Newton multilevel FEM for nonlinear elliptic problems
Computational science for the 21st century
urn:nbn:de:0297-zib-2401
yes
accepted for publication
Peter Deuflhard
M.-O. et al. Bristeau
Martin Weiser
Numerical Mathematics
Deuflhard, Peter
Weiser, Martin
1420
eng
reportzib
0
2011-10-21
2011-10-21
--
Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia
This paper surveys the required mathematics for a typical challenging problem from computational medicine, the cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem gave rise to quite a number of subtle mathematical problems, part of which had been unsolved when the common project started. Efficiency of numerical algorithms, i.e. computational speed and monitored reliability, play a decisive role for the medical treatment. Off-the-shelf software had turned out to be not sufficient to meet the requirements of medicine. Rather, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, a virtual lab, including 3D geometry processing of individual virtual patients had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analyzed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the individual patients' treatment and on the construction of an improved hyperthermia applicator.
1438-0064
11-39
urn:nbn:de:0297-zib-14204
Appeared in: Acta Numerica 21: 307-378, 2012
Peter Deuflhard
Anton Schiela
Anton Schiela
Martin Weiser
ZIB-Report
11-39
eng
uncontrolled
hyperthermia therapy planning
eng
uncontrolled
mathematical modelling
eng
uncontrolled
Maxwell equations
eng
uncontrolled
state constraints
eng
uncontrolled
parameter identification
CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX]
Numerical methods [See also 90Cxx, 65Kxx]
NUMERICAL ANALYSIS
Partial differential equations, boundary value problems
Physiological, cellular and medical topics
Medical applications (general)
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
MATHEON-A17
ZIB-Kaskade7
https://opus4.kobv.de/opus4-zib/files/1420/ZR-11-39.pdf
4662
2012
eng
307
378
21
article
0
--
--
--
Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia
This paper surveys the required mathematics for a typical challenging problem from computational medicine, the cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem gave rise to quite a number of subtle mathematical problems, part of which had been unsolved when the common project started. Efficiency of numerical algorithms, i.e. computational speed and monitored reliability, play a decisive role for the medical treatment. Off-the-shelf software had turned out to be not sufficient to meet the requirements of medicine. Rather, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, a virtual lab, including 3D geometry processing of individual virtual patients had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analyzed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the individual patients' treatment and on the construction of an improved hyperthermia applicator.
Acta Numerica
yes
urn:nbn:de:0297-zib-14204
Peter Deuflhard
Martin Weiser
Anton Schiela
Martin Weiser
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
MATHEON-A17
ZIB-Kaskade7
5070
2014
eng
7
25
1
incollection
European Mathematical Society
0
--
--
--
Mathematics cures virtual patients
MATHEON-Mathematics for Key Technologies
Series in Industrial and Applied Mathematics
no
Peter Deuflhard
Martin Weiser
Peter Deuflhard
Ralf Kornhuber
Martin Grötschel
Dietmar Hömberg
Oliver Sander
Anton Schiela
Ulrich Horst
Martin Weiser
Jürg Kramer
Volker Mehrmann
Konrad Polthier
Frank Schmidt
Christof Schütte
Martin Skutella
Jürgen Sprekels
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
MATHEON-A17
ZIB-Kaskade7
2107
2006
2006
eng
1012
1016
9
53
article
0
--
--
--
Mathematics in Facial Surgery
AMS Notices
Peter Deuflhard
Martin Weiser
Stefan Zachow
Numerical Mathematics
Computational Medicine
Deuflhard, Peter
Weiser, Martin
Zachow, Stefan
FacialSurgery
2591
2008
2008
eng
140
156
bookpart
Health Academy
0
--
--
--
Modellgestützte Operationsplanung in der Kopfchirurgie
Modellgestützte Therapie
Stefan Zachow
Wolfgang Niederlag
Martin Weiser
Heinz Lemke
Peter Deuflhard
Jürgen Meixensberger
Michael Baumann
Numerical Mathematics
Computational Medicine
Therapy Planning
Deuflhard, Peter
Weiser, Martin
Zachow, Stefan
FacialSurgery