TY - GEN A1 - Deuflhard, Peter A1 - Weiser, Martin A1 - Seebass, Martin T1 - A New Nonlinear Elliptic Multilevel FEM Applied to Regional Hyperthermia N2 - 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. T3 - ZIB-Report - SC-98-35 KW - hyperthermia KW - nonlinear elliptic KW - multilevel FEM Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3785 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Weiser, Martin A1 - Seebaß, Martin T1 - A New Nonlinear Elliptic Multilevel FEM Applied to Regional Hyperthermia JF - Comput. Visual. Sci. Y1 - 2000 U6 - https://doi.org/10.1007/PL00013546 VL - 3 SP - 1 EP - 6 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Weiser, Martin A1 - Seebass, Martin T1 - A new nonlinear elliptic multilevel FEM in clinical cancer therapy planning JF - Comput. Vis. Sci. Y1 - 2000 VL - 3 SP - 115 EP - 120 ER - TY - BOOK A1 - Deuflhard, Peter A1 - Weiser, Martin T1 - Adaptive numerical solution of PDEs Y1 - 2012 PB - de Gruyter CY - Berlin 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 - JOUR A1 - Weiser, Martin A1 - Deuflhard, Peter A1 - Erdmann, Bodo T1 - Affine conjugate adaptive Newton methods for nonlinear elastomechanics JF - Opt. Meth. Softw. Y1 - 2007 VL - 22 IS - 3 SP - 413 EP - 431 ER - TY - GEN A1 - Deuflhard, Peter A1 - Nowak, Ulrich A1 - Weiser, Martin T1 - Affine Invariant Adaptive Newton Codes for Discretized PDEs N2 - 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. T3 - ZIB-Report - 02-33 KW - Affine invariant Newton methods KW - global Newton methods KW - inexact Newton methods KW - adaptive trust region methods KW - nonlinear partial differential equa Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7005 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 - JOUR A1 - Weiser, Martin A1 - Schiela, Anton A1 - Deuflhard, Peter T1 - Asymptotic Mesh Independence of Newton’s Method Revisited JF - SIAM J. Num. Anal. Y1 - 2005 VL - 42 IS - 5 SP - 1830 EP - 1845 ER - TY - JOUR A1 - Weiser, Martin A1 - Zachow, Stefan A1 - Deuflhard, Peter T1 - Craniofacial Surgery Planning Based on Virtual Patient Models JF - it - Information Technology Y1 - 2010 U6 - https://doi.org/10.1524/itit.2010.0600 VL - 52 IS - 5 SP - 258 EP - 263 PB - Oldenbourg Verlagsgruppe ER - TY - GEN A1 - Deuflhard, Peter A1 - Weiser, Martin T1 - Global Inexact Multilevel FEM for Nonlinear Elliptic Problems N2 - 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. T3 - ZIB-Report - SC-96-33 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2430 ER - TY - CHAP A1 - Deuflhard, Peter A1 - Weiser, Martin ED - et al. Hackbusch, Wolfgang T1 - Global inexact Newton multilevel FEM for nonlinear elliptic problems T2 - Multigrid methods V. proceedings of the 5th European multigrid conference, held in Stuttgart, Germany, October 1–4, 1996. Y1 - 1998 VL - 3 SP - 71 EP - 89 PB - Berlin: Springer ER - TY - JOUR A1 - Weiser, Martin A1 - Deuflhard, Peter T1 - Inexact central path following algorithms for optimal control problems JF - SIAM J. Control Opt. Y1 - 2007 VL - 46 IS - 3 SP - 792 EP - 815 ER - TY - GEN A1 - Deuflhard, Peter A1 - Weiser, Martin T1 - Local Inexact Newton Multilevel FEM for Nonlinear Elliptic Problems N2 - 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. T3 - ZIB-Report - SC-96-29 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2401 ER - TY - CHAP A1 - Deuflhard, Peter A1 - Weiser, Martin ED - et al. Bristeau, M.-O. T1 - Local inexact Newton multilevel FEM for nonlinear elliptic problems T2 - Computational science for the 21st century Y1 - 1997 SP - 129 EP - 138 PB - Chichester: John Wiley & Sons. ER - TY - GEN A1 - Deuflhard, Peter A1 - Schiela, Anton A1 - Weiser, Martin T1 - Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia N2 - 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. T3 - ZIB-Report - 11-39 KW - hyperthermia therapy planning KW - mathematical modelling KW - Maxwell equations KW - state constraints KW - parameter identification Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14204 SN - 1438-0064 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Schiela, Anton A1 - Weiser, Martin T1 - Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia JF - Acta Numerica N2 - 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. Y1 - 2012 VL - 21 SP - 307 EP - 378 ER - TY - GEN A1 - Deuflhard, Peter A1 - Kornhuber, Ralf A1 - Sander, Oliver A1 - Schiela, Anton A1 - Weiser, Martin ED - Deuflhard, Peter ED - Grötschel, Martin ED - Hömberg, Dietmar ED - Horst, Ulrich ED - Kramer, Jürg ED - Mehrmann, Volker ED - Polthier, Konrad ED - Schmidt, Frank ED - Schütte, Christof ED - Skutella, Martin ED - Sprekels, Jürgen T1 - Mathematics cures virtual patients T2 - MATHEON-Mathematics for Key Technologies Y1 - 2014 VL - 1 SP - 7 EP - 25 PB - European Mathematical Society ER - TY - JOUR A1 - Deuflhard, Peter A1 - Weiser, Martin A1 - Zachow, Stefan T1 - Mathematics in Facial Surgery JF - AMS Notices Y1 - 2006 VL - 53 IS - 9 SP - 1012 EP - 1016 ER - TY - CHAP A1 - Zachow, Stefan A1 - Weiser, Martin A1 - Deuflhard, Peter ED - Niederlag, Wolfgang ED - Lemke, Heinz ED - Meixensberger, Jürgen ED - Baumann, Michael T1 - Modellgestützte Operationsplanung in der Kopfchirurgie T2 - Modellgestützte Therapie Y1 - 2008 SP - 140 EP - 156 PB - Health Academy ER -