Refine
Year of publication
Document Type
- Article (67)
- ZIB-Report (52)
- In Proceedings (22)
- In Collection (9)
- Book (4)
- Book chapter (3)
- Software (2)
- Report (1)
Is part of the Bibliography
- no (160)
Keywords
- optimal control (18)
- interior point methods in function space (5)
- interior point methods (4)
- trajectory storage (4)
- discretization error (3)
- finite elements (3)
- flight planning (3)
- free flight (3)
- lossy compression (3)
- shortest path (3)
Institute
- Numerical Mathematics (134)
- Computational Medicine (103)
- Modeling and Simulation of Complex Processes (40)
- Visual and Data-centric Computing (13)
- Visual Data Analysis (12)
- Mathematics for Life and Materials Science (8)
- Network Optimization (8)
- Therapy Planning (8)
- ZIB Allgemein (7)
- Distributed Algorithms and Supercomputing (3)
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.
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.
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.
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.
Reasons for the failure of adaptive methods to deliver improved efficiency when integrating monodomain models for myocardiac excitation are discussed. Two closely related techniques for reducing the computational complexity of linearly implicit integrators, deliberate sparsing and splitting, are investigated with respect to their impact on computing time and accuracy.
Pulse thermography of concrete structures is used in civil engineering for detecting voids, honeycombing and delamination. The physical situation is readily modeled by Fourier's law. Despite the simplicity of the PDE structure, quantitatively realistic numerical 3D simulation faces two major obstacles. First, the short heating pulse induces a thin boundary layer at the heated surface which encapsulates all information and therefore has to be resolved faithfully. Even with adaptive mesh refinement techniques, obtaining useful accuracies requires an unsatisfactorily fine discretization. Second, bulk material parameters and boundary conditions are barely known exactly. We address both issues by a semi-analytic reformulation of the heat transport problem and by parameter identification. Numerical results are compared with measurements of test specimens.
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.
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.
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.
An affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods is presented. The theory is used for the construction of an accuracy matching between iteration errors and truncation errors, which arise from the inexact linear system solves. The theoretical investigations are illustrated numerically by an optimal control problem for the Burgers equation.