@article{GoetschelNagaiahKunischetal.2014, author = {G{\"o}tschel, Sebastian and Nagaiah, Chamakuri and Kunisch, Karl and Weiser, Martin}, title = {Lossy Compression in Optimal Control of Cardiac Defibrillation}, volume = {60}, journal = {J. Sci. Comput.}, number = {1}, doi = {10.1007/s10915-013-9785-x}, pages = {35 -- 59}, year = {2014}, abstract = {This paper presents efficient computational techniques for solving an optimization problem in cardiac defibrillation governed by the monodomain equations. Time-dependent electrical currents injected at different spatial positions act as the control. Inexact Newton-CG methods are used, with reduced gradient computation by adjoint solves. In order to reduce the computational complexity, adaptive mesh refinement for state and adjoint equations is performed. To reduce the high storage and bandwidth demand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compression technique for storing trajectory data is applied. An adaptive choice of quantization tolerance based on error estimates is developed in order to ensure convergence. The efficiency of the proposed approach is demonstrated on numerical examples.}, language = {en} } @article{GoetschelWeiserMaierhoferetal.2013, author = {G{\"o}tschel, Sebastian and Weiser, Martin and Maierhofer, Christiane and Richter, Regina and R{\"o}llig, Mathias}, title = {Fast Defect Shape Reconstruction Based on the Travel Time in Pulse Thermography}, volume = {6}, journal = {Nondestructive Testing of Materials and Structures}, pages = {83 -- 89}, year = {2013}, language = {en} } @article{GoetschelWeiser2015, author = {G{\"o}tschel, Sebastian and Weiser, Martin}, title = {Lossy Compression for PDE-constrained Optimization: Adaptive Error Control}, volume = {62}, journal = {Comput. Optim. Appl.}, number = {1}, publisher = {Springer}, pages = {131 -- 155}, year = {2015}, abstract = {For the solution of optimal control problems governed by nonlinear parabolic PDEs, methods working on the reduced objective functional are often employed to avoid a full spatio-temporal discretization of the problem. The evaluation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the ad-joint equation. The state enters into the adjoint equation, requiring the storage of a full 4D data set. If Newton-CG methods are used, two additional trajectories have to be stored. To get numerical results which are accurate enough, in many case very fine discretizations in time and space are necessary, which leads to a significant amount of data to be stored and transmitted to mass storage. Lossy compression methods were developed to overcome the storage problem by reducing the accuracy of the stored trajectories. The inexact data induces errors in the reduced gradient and reduced Hessian. In this paper, we analyze the influence of such a lossy trajectory compression method on Newton-CG methods for optimal control of parabolic PDEs and design an adaptive strategy for choosing appropriate quantization tolerances.}, language = {en} } @article{SchenklMuggenthalerHubigetal.2017, author = {Schenkl, Sebastian and Muggenthaler, Holger and Hubig, Michael and Erdmann, Bodo and Weiser, Martin and Zachow, Stefan and Heinrich, Andreas and G{\"u}ttler, Felix Victor and Teichgr{\"a}ber, Ulf and Mall, Gita}, title = {Automatic CT-based finite element model generation for temperature-based death time estimation: feasibility study and sensitivity analysis}, volume = {131}, journal = {International Journal of Legal Medicine}, number = {3}, doi = {doi:10.1007/s00414-016-1523-0}, pages = {699 -- 712}, year = {2017}, abstract = {Temperature based death time estimation is based either on simple phenomenological models of corpse cooling or on detailed physical heat transfer models. The latter are much more complex, but allow a higher accuracy of death time estimation as in principle all relevant cooling mechanisms can be taken into account. Here, a complete work flow for finite element based cooling simulation models is presented. The following steps are demonstrated on CT-phantoms: • CT-scan • Segmentation of the CT images for thermodynamically relevant features of individual geometries • Conversion of the segmentation result into a Finite Element (FE) simulation model • Computation of the model cooling curve • Calculation of the cooling time For the first time in FE-based cooling time estimation the steps from the CT image over segmentation to FE model generation are semi-automatically performed. The cooling time calculation results are compared to cooling measurements performed on the phantoms under controlled conditions. In this context, the method is validated using different CTphantoms. Some of the CT phantoms thermodynamic material parameters had to be experimentally determined via independent experiments. Moreover the impact of geometry and material parameter uncertainties on the estimated cooling time is investigated by a sensitivity analysis.}, language = {en} } @article{WeiserFreytagErdmannetal.2018, author = {Weiser, Martin and Freytag, Yvonne and Erdmann, Bodo and Hubig, Michael and Mall, Gita}, title = {Optimal Design of Experiments for Estimating the Time of Death in Forensic Medicine}, volume = {34}, journal = {Inverse Problems}, number = {12}, doi = {10.1088/1361-6420/aae7a5}, pages = {125005}, year = {2018}, abstract = {Estimation of time of death based on a single measurement of body core temperature is a standard procedure in forensic medicine. Mechanistic models using simulation of heat transport promise higher accuracy than established phenomenological models in particular in nonstandard situations, but involve many not exactly known physical parameters. Identifying both time of death and physical parameters from multiple temperature measurements is one possibility to reduce the uncertainty significantly. In this paper, we consider the inverse problem in a Bayesian setting and perform both local and sampling-based uncertainty quantification, where proper orthogonal decomposition is used as model reduction for fast solution of the forward model. Based on the local uncertainty quantification, optimal design of experiments is performed in order to minimize the uncertainty in the time of death estimate for a given number of measurements. For reasons of practicability, temperature acquisition points are selected from a set of candidates in different spatial and temporal locations. Applied to a real corpse model, a significant accuracy improvement is obtained already with a small number of measurements.}, language = {en} } @article{GoetschelSchielaWeiser2021, author = {G{\"o}tschel, Sebastian and Schiela, Anton and Weiser, Martin}, title = {Kaskade 7 - a Flexible Finite Element Toolbox}, volume = {81}, journal = {Computers and Mathematics with Applications}, doi = {10.1016/j.camwa.2020.02.011}, pages = {444 -- 458}, year = {2021}, abstract = {Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the \textsc{Dune} interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables. We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way.}, language = {en} } @inproceedings{KrauseWeiser2021, author = {Krause, Rolf and Weiser, Martin}, title = {Multilevel augmented Lagrangian solvers for overconstrained contact formulations}, volume = {71}, booktitle = {ESAIM: ProcS}, doi = {10.1051/proc/202171175}, pages = {175 -- 184}, year = {2021}, abstract = {Multigrid methods for two-body contact problems are mostly based on special mortar discretizations, nonlinear Gauss-Seidel solvers, and solution-adapted coarse grid spaces. Their high computational efficiency comes at the cost of a complex implementation and a nonsymmetric master-slave discretization of the nonpenetration condition. Here we investigate an alternative symmetric and overconstrained segment-to-segment contact formulation that allows for a simple implementation based on standard multigrid and a symmetric treatment of contact boundaries, but leads to nonunique multipliers. For the solution of the arising quadratic programs, we propose augmented Lagrangian multigrid with overlapping block Gauss-Seidel smoothers. Approximation and convergence properties are studied numerically at standard test problems.}, language = {en} } @article{AlhaddadFoerstnerGrothetal.2021, author = {Alhaddad, Samer and F{\"o}rstner, Jens and Groth, Stefan and Gr{\"u}newald, Daniel and Grynko, Yevgen and Hannig, Frank and Kenter, Tobias and Pfreundt, Franz-Josef and Plessl, Christian and Schotte, Merlind and Steinke, Thomas and Teich, J{\"u}rgen and Weiser, Martin and Wende, Florian}, title = {HighPerMeshes - A Domain-Specific Language for Numerical Algorithms on Unstructured Grids}, journal = {Euro-Par 2020: Parallel Processing Workshops.}, publisher = {Springer}, doi = {10.1007/978-3-030-71593-9_15}, pages = {185 -- 196}, year = {2021}, abstract = {Solving partial differential equations on unstructured grids is a cornerstone of engineering and scientific computing. Nowadays, heterogeneous parallel platforms with CPUs, GPUs, and FPGAs enable energy-efficient and computationally demanding simulations. We developed the HighPerMeshes C++-embedded Domain-Specific Language (DSL) for bridging the abstraction gap between the mathematical and algorithmic formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different parallel programming and runtime models on the other hand. Thus, the HighPerMeshes DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HighPer-Meshes DSL, and demonstrate its usage with three examples, a Poisson and monodomain problem, respectively, solved by the continuous finite element method, and the discontinuous Galerkin method for Maxwell's equation. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters is presented. Finally, the achievable performance and scalability are demonstrated for a typical example problem on a multi-core CPU cluster.}, language = {en} } @inproceedings{MuellerGoetschelWeiseretal.2017, author = {M{\"u}ller, Jan and G{\"o}tschel, Sebastian and Weiser, Martin and Maierhofer, Christiane}, title = {Thermografie mit optimierter Anregung f{\"u}r die quantitative Untersuchung von Delaminationen in kohlenstofffaserverst{\"a}rkten Kunststoffen}, booktitle = {NDT.net Proc. DGZfP 2017}, year = {2017}, abstract = {Da kohlenstofffaserverst{\"a}rkte Kunststoffe (CFK) in anspruchsvollen sicherheitsrelevanten Einsatzgebieten wie im Automobilbau und in der Luftfahrt eingesetzt werden, besteht ein zunehmender Bedarf an zerst{\"o}rungsfreien Pr{\"u}fmethoden. Ziel ist die Gew{\"a}hrleistung der Sicherheit und Zuverl{\"a}ssigkeit der eingesetzten Bauteile. Aktive Thermografieverfahren erm{\"o}glichen die effiziente Pr{\"u}fung großer Fl{\"a}chen mit hoher Aufl{\"o}sung in wenigen Arbeitsschritten. Ein wichtiges Teilgebiet der Pr{\"u}fungen ist die Ortung und Charakterisierung von Delaminationen, die sowohl bereits in der Fertigung als auch w{\"a}hrend der Nutzung eines Bauteils auftreten k{\"o}nnen, und dessen strukturelle Integrit{\"a}t schw{\"a}chen. ;In diesem Beitrag werden CFK-Strukturen mit k{\"u}nstlichen und nat{\"u}rlichen Delaminationen mit Hilfe unterschiedlich zeitlich modulierter Strahlungsquellen experimentell untersucht. Verwendet werden dabei Anregungen mit Blitzlampen und mit frequenzmodulierten Halogenlampen. Mittels Filterfunktionen im Zeit- und Frequenzbereich wird das Kontrast-zu-Rausch-Verh{\"a}ltnis (CNR) der detektierten Fehlstellen optimiert. Verglichen werden anschließend die Nachweisempfindlichkeit, das CNR und die Ortsaufl{\"o}sung der zu charakterisierenden Delaminationen f{\"u}r die unterschiedlichen Anregungs- und Auswertungstechniken. Erg{\"a}nzt werden die Experimente durch numerische Simulationen des dreidimensionalen W{\"a}rmetransportes.}, language = {de} } @misc{WeiserFreytagErdmannetal.2018, author = {Weiser, Martin and Freytag, Yvonne and Erdmann, Bodo and Hubig, Michael and Mall, Gita}, title = {Optimal Design of Experiments for Estimating the Time of Death in Forensic Medicine}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-67247}, year = {2018}, abstract = {Estimation of time of death based on a single measurement of body core temperature is a standard procedure in forensic medicine. Mechanistic models using simulation of heat transport promise higher accuracy than established phenomenological models in particular in nonstandard situations, but involve many not exactly known physical parameters. Identifying both time of death and physical parameters from multiple temperature measurements is one possibility to reduce the uncertainty significantly. In this paper, we consider the inverse problem in a Bayesian setting and perform both local and sampling-based uncertainty quantification, where proper orthogonal decomposition is used as model reduction for fast solution of the forward model. Based on the local uncertainty quantification, optimal design of experiments is performed in order to minimize the uncertainty in the time of death estimate for a given number of measurements. For reasons of practicability, temperature acquisition points are selected from a set of candidates in different spatial and temporal locations. Applied to a real corpse model, a significant accuracy improvement is obtained already with a small number of measurements.}, language = {en} } @article{MoldenhauerWeiserZachow2017, author = {Moldenhauer, Marian and Weiser, Martin and Zachow, Stefan}, title = {Adaptive Algorithms for Optimal Hip Implant Positioning}, volume = {17}, journal = {PAMM}, number = {1}, doi = {10.1002/pamm.201710071}, pages = {203 -- 204}, year = {2017}, abstract = {In an aging society where the number of joint replacements rises, it is important to also increase the longevity of implants. In particular hip implants have a lifetime of at most 15 years. This derives primarily from pain due to implant migration, wear, inflammation, and dislocation, which is affected by the positioning of the implant during the surgery. Current joint replacement practice uses 2D software tools and relies on the experience of surgeons. Especially the 2D tools fail to take the patients' natural range of motion as well as stress distribution in the 3D joint induced by different daily motions into account. Optimizing the hip joint implant position for all possible parametrized motions under the constraint of a contact problem is prohibitively expensive as there are too many motions and every position change demands a recalculation of the contact problem. For the reduction of the computational effort, we use adaptive refinement on the parameter domain coupled with the interpolation method of Kriging. A coarse initial grid is to be locally refined using goal-oriented error estimation, reducing locally high variances. This approach will be combined with multi-grid optimization such that numerical errors are reduced.}, language = {en} } @article{TierneyMiraReinholdetal.2018, author = {Tierney, Nicholas J. and Mira, Antonietta and Reinhold, J. Jost and Weiser, Martin and Burkart, Roman and Benvenuti, Claudio and Auricchio, Angelo}, title = {Novel relocation methods for automatic external defibrillator improve out-of-hospital cardiac arrest coverage under limited resources}, journal = {Resuscitation}, number = {125}, doi = {10.1016/j.resuscitation.2018.01.055}, pages = {83 -- 89}, year = {2018}, abstract = {Background Mathematical optimisation models have recently been applied to identify ideal Automatic External Defibrillator (AED) locations that maximise coverage of Out of Hospital Cardiac Arrest (OHCA). However, these fixed location models cannot relocate existing AEDs in a flexible way, and have nearly exclusively been applied to urban regions. We developed a flexible location model for AEDs, compared its performance to existing fixed location and population models, and explored how these perform across urban and rural regions. Methods Optimisation techniques were applied to AED deployment and OHCA coverage was assessed. A total of 2802 geolocated OHCAs occurred in Canton Ticino, Switzerland, from January 1st 2005 to December 31st 2015. Results There were 719 AEDs in Canton Ticino. 635 (23\%) OHCA events occurred within 100m of an AED, with 306 (31\%) in urban, and 329 (18\%) in rural areas. Median distance from OHCA events to the nearest AED was 224m (168m urban vs. 269m rural). Flexible location models performed better than fixed location and population models, with the cost to deploy 20 new AEDs instead relocating 171 existing AEDs to new locations, improving OHCA coverage to 38\%, compared to 26\% using fixed models, and 24\% with the population based model. Conclusions Optimisation models for AEDs placement are superior to population models and should be strongly considered by communities when selecting areas for AED deployment. Compared to other models, flexible location models increase overall OHCA coverage, and decreases the distance to nearby AEDs, even in rural areas, while saving significant financial resources.}, language = {en} } @article{WeiserErdmannSchenkletal.2018, author = {Weiser, Martin and Erdmann, Bodo and Schenkl, Sebastian and Muggenthaler, Holger and Hubig, Michael and Mall, Gita and Zachow, Stefan}, title = {Uncertainty in Temperature-Based Determination of Time of Death}, volume = {54}, journal = {Heat and Mass Transfer}, number = {9}, publisher = {Springer}, doi = {10.1007/s00231-018-2324-4}, pages = {2815 -- 2826}, year = {2018}, abstract = {Temperature-based estimation of time of death (ToD) can be per- formed either with the help of simple phenomenological models of corpse cooling or with detailed mechanistic (thermodynamic) heat transfer mod- els. The latter are much more complex, but allow a higher accuracy of ToD estimation as in principle all relevant cooling mechanisms can be taken into account. The potentially higher accuracy depends on the accuracy of tissue and environmental parameters as well as on the geometric resolution. We in- vestigate the impact of parameter variations and geometry representation on the estimated ToD based on a highly detailed 3D corpse model, that has been segmented and geometrically reconstructed from a computed to- mography (CT) data set, differentiating various organs and tissue types.}, language = {en} } @misc{OezelKulkarniHasanetal.2019, author = {{\"O}zel, M. Neset and Kulkarni, Abhishek and Hasan, Amr and Brummer, Josephine and Moldenhauer, Marian and Daumann, Ilsa-Maria and Wolfenberg, Heike and Dercksen, Vincent J. and Kiral, F. Ridvan and Weiser, Martin and Prohaska, Steffen and von Kleist, Max and Hiesinger, Peter Robin}, title = {Serial synapse formation through filopodial competition for synaptic seeding factors}, issn = {1438-0064}, doi = {10.1016/j.devcel.2019.06.014}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74397}, year = {2019}, abstract = {Following axon pathfinding, growth cones transition from stochastic filopodial exploration to the formation of a limited number of synapses. How the interplay of filopodia and synapse assembly ensures robust connectivity in the brain has remained a challenging problem. Here, we developed a new 4D analysis method for filopodial dynamics and a data-driven computational model of synapse formation for R7 photoreceptor axons in developing Drosophila brains. Our live data support a 'serial synapse formation' model, where at any time point only a single 'synaptogenic' filopodium suppresses the synaptic competence of other filopodia through competition for synaptic seeding factors. Loss of the synaptic seeding factors Syd-1 and Liprin-α leads to a loss of this suppression, filopodial destabilization and reduced synapse formation, which is sufficient to cause the destabilization of entire axon terminals. Our model provides a filopodial 'winner-takes-all' mechanism that ensures the formation of an appropriate number of synapses.}, language = {en} } @misc{GoetschelWeiser2019, author = {G{\"o}tschel, Sebastian and Weiser, Martin}, title = {Lossy Compression for Large Scale PDE Problems}, issn = {1438-0064}, doi = {10.1101/506378}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-73817}, year = {2019}, abstract = {Solvers for partial differential equations (PDE) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that needs to be stored and transmitted on all levels of the memory hierarchy. Often, bandwidth is the limiting factor due to relatively small arithmetic intensity, and increasingly so due to the growing disparity between computing power and bandwidth. Consequently, data compression techniques have been investigated and tailored towards the specific requirements of PDE solvers during the last decades. This paper surveys data compression challenges and corresponding solution approaches for PDE problems, covering all levels of the memory hierarchy from mass storage up to main memory. Exemplarily, we illustrate concepts at particular methods, and give references to alternatives.}, language = {en} } @misc{GoetschelSchielaWeiser2019, author = {G{\"o}tschel, Sebastian and Schiela, Anton and Weiser, Martin}, title = {Kaskade 7 -- a Flexible Finite Element Toolbox}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74616}, year = {2019}, abstract = {Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables. We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way.}, language = {en} } @article{SchneckWeiserWende2021, author = {Schneck, Jakob and Weiser, Martin and Wende, Florian}, title = {Impact of mixed precision and storage layout on additive Schwarz smoothers}, volume = {28}, journal = {Numerical Linear Algebra with Applications}, number = {4}, doi = {10.1002/nla.2366}, year = {2021}, abstract = {The growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth-bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage.}, language = {en} } @misc{DeuflhardWeiser1996, author = {Deuflhard, Peter and Weiser, Martin}, title = {Local Inexact Newton Multilevel FEM for Nonlinear Elliptic Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2401}, number = {SC-96-29}, year = {1996}, abstract = {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.}, language = {en} } @misc{DeuflhardWeiser1996, author = {Deuflhard, Peter and Weiser, Martin}, title = {Global Inexact Multilevel FEM for Nonlinear Elliptic Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2430}, number = {SC-96-33}, year = {1996}, abstract = {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.}, language = {en} } @misc{Weiser2003, author = {Weiser, Martin}, title = {Interior Point Methods in Function Space}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7578}, number = {03-35}, year = {2003}, abstract = {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.}, language = {en} } @misc{WeiserSchielaDeuflhard2003, author = {Weiser, Martin and Schiela, Anton and Deuflhard, Peter}, title = {Asymptotic Mesh Independence of Newton's Method Revisited}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7352}, number = {03-13}, year = {2003}, abstract = {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.}, language = {en} } @misc{HorenkoWeiser2002, author = {Horenko, Illia and Weiser, Martin}, title = {Adaptive Integration of Multidimensional Molecular Dynamics with Quantum Initial Conditions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6967}, number = {02-29}, year = {2002}, abstract = {The paper presents a particle method framework for resolving molecular dynamics. Error estimators for both the temporal and spatial discretization are advocated and facilitate a fully adaptive propagation. For time integration, the implicit trapezoidal rule is employed, where an explicit predictor enables large time steps. The framework is developed and exemplified in the context of the classical Liouville equation, where Gaussian phase-space packets are used as particles. Simplified variants are discussed shortly, which should prove to be easily implementable in common molecular dynamics codes. A concept is illustrated by numerical examples for one-dimensional dynamics in double well potential.}, language = {en} } @misc{DeuflhardNowakWeiser2002, author = {Deuflhard, Peter and Nowak, Ulrich and Weiser, Martin}, title = {Affine Invariant Adaptive Newton Codes for Discretized PDEs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7005}, number = {02-33}, year = {2002}, abstract = {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.}, language = {en} }