Deuflhard, Peter
Refine
Year of publication
Document Type
- ZIB-Report (95)
- In Proceedings (92)
- Article (91)
- Book chapter (25)
- Book (18)
- Other (12)
- In Collection (5)
- Doctoral Thesis (3)
- Proceedings (1)
- Report (1)
Is part of the Bibliography
- no (343)
Keywords
- hyperthermia (5)
- cluster analysis (4)
- metastability (3)
- Dynamical contact problems (2)
- Hamiltonian dynamics (2)
- Markov chain (2)
- Maxwell's equations (2)
- Newmark method (2)
- Self-Organizing Maps (2)
- Signorini condition (2)
Institute
- Numerical Mathematics (170)
- ZIB Allgemein (73)
- Visual Data Analysis (49)
- Visual and Data-centric Computing (49)
- Therapy Planning (48)
- Computational Medicine (39)
- Visual Data Analysis in Science and Engineering (15)
- Computational Systems Biology (12)
- Computational Molecular Design (7)
- Computational Nano Optics (7)
We consider periodic orbits of autonomous parameter dependent ODE's. Starting from a shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincar\'e-section we develop a pathfollowing algorithm for periodic solutions based on a tangential continuation method with implicit reparametrization. For ODE's equivariant w.r.t. a finite group we show that spatial as well as spatio-temporal symmetries of periodic orbits can be exploited within the (multiple) shooting context. We describe how turning points, period doubling bifurcations and Hopf points along the branch of periodic solutions can be handled. Furthermore equivariant Hopf points and generic secondary bifurcations of periodic orbits with $ Z_m$-symmetry are treated. We tested the code with standard examples, e.g., the period doubling cascade in the Lorenz equations. To show the efficiency of the described methods we also used the program for an application from electronics, a ring oscillator with $n $ inverters. In this example the exploitation of symmetry reduces the amount of work for the continuation of periodic orbits from ${\cal O}(n^2)$ to ${\cal O}(n)$
In the field of deep regional hyperthermia, one of the most widely used devices is the BSD--2000 Hyperthermia System which employs the Sigma 60 applicator. The Sigma 60 consists of four independent sources, giving it the potential to control the energy pattern within the patient. The independent amplitudes and phases, as well as frequency selection and applicator position, present a large number of parameters for the operator to determine. Computer simulation has long been recognized as an attractive approach to optimizing these parameters. A treatment planning program was used in clinical practice at Stanford University Medical Center for two years. It demonstrated the feasibility of computer simulation for deep regional hyperthermia in a clinical situation. However, several parts of this system were written in a language specific to one workstation, which severely restricted the wider distribution of the program to other users of the Sigma 60. A new treatment planning system for the BSD 2000 has been developed and put into clinical practice at the Rudolf Virchow Clinic of the Free University of Berlin. The new method, which we will refer to as the Berlin system, has a simpler model construction program and a considerably better graphics capability. However, the most important feature is that all programs are written in FORTRAN, C, or the X Window graphics system. Therefore, the entire treatment planning system is completely portable to other workstations.
The paper presents a construction scheme of deriving transparent , i. e. reflection-free, boundary conditions for the numerical solution of Fresnel's equation (being formally equivalent to Schrödinger's equation). These boundary conditions appear to be of a nonlocal Cauchy type. As it turns out, each kind of linear implicit discretization induces its own discrete transparent boundary conditions.
Cascadic conjugate gradient methods for the numerical solution of elliptic partial differential equations consists of Galerkin finite element methods as outer iteration and (possibly preconditioned) conjugate gradient methods as inner iteration. Both iterations are known to minimize the energy norm of the arising iterations errors. A simple but efficient strategy to control the discretization errors versus the PCG iteration errors in terms of energy error norms is derived and worked out in algorithmic detail. In a unified setting, the relative merits of different preconditioners versus the case of no preconditioning is compared. Surprisingly, it appears that the cascadic conjugate gradient method without any preconditioning is not only simplest but also fastest. The numerical results seem to indicate that the cascade principle in itself already realizes some kind of preconditioning. A theoretical explanation of these observations will be given in Part II of this paper.
In this paper, a rather recent algorithmic approac to the numerical simulation of macromolecula processes is surveyed. It avoids the numerical stiff integration o thousands up to millions of ODE's by constructing a scale of discret Hilbert spaces, especially weighted sequence spaces, and establishing corresponding Galerkin method. Examples including polyreactions o industrial relevance and ecological waste management by biochemica recycling illustrate the importance and efficiency of the algorithm.
The Lanczos iteration for symmetric indefinite linear systems seems to be well--known for quite a while. However, in order to modify it with the aim of improved performance, the present paper studies certain aspects in terms of an adjoint scalar three--term recurrence. Thus, at least a different view is opened. Moreover, an alternative $3n$--implementation in terms of the Euclidean orthogonal basis has been found that easily permits generalizations. The study is understood as a start--off for further numerical investigations and experiments.
The numerical solution of optimal control problems by indirect methods (such as multiple shooting or collocation) requires a considerable amount of analytic calculation to establish a numerically tractable system. These analytic calculations, though being rather tedious in realistic examples, are nowadays mostly still done by hand-and thus prone to calculation errors. The paper aims at automating this analytic processing to a reasonable extent by means of a modern symbolic manipulation language (here: REDUCE). In its present stage of development the package OCCAL (mnemotechnically for \underline{O}ptimal \underline{C}ontrol \underline{CAL}culator) permits an interactive use, covering tasks like automatic determination of control and, in case of a singular control, of its order. In simpler problems, the present version of OCCAL automatically produces the full subroutine input for a MULtiple shooting code (MULCON) with adaptive numerical CONtinuation. In more complicate problems where singular sub-arcs may occur or where the sequence of sub-arcs of the optimal trajectory is unclear OCCAL is a significant help in reducing analytic pre-processing. Numerical examples illustrate the performance of OCCAL/MULCON.
The rapidly increasing number of AIDS cases requires a realistic estimation of the future development of the HIV/AIDS disease. For that purpose we develop a large system of coupled nonlinear differential equations describing simultaneously the dynamics of the development of the disease, the population size, the gender and age structure. A set of 1650 coupled equations are linked by balanced parameters. The balancing procedure is described by a set of (formally) 2,178,000 additional algebraic conditions. As the considered system is stiff, it requires new special extrapolation methods combined with techniquees of dynamical sparsing for the solution of sparsely filled systems. According to our simulations we expect 19,0,000 deaths caused by AIDS in the Federal Republic of Germany (former territories) in the year 2000. Such an epidemical spread would tie up about 4-7 percent of the actual health care workers.
The present paper contains a generalization of a refinement of the Newton- Mysovskii theorem, recently obtained by the authors, to the case of Gauss-Newton procedures for solving nonlinear least-squares problems with full Jacobians. Invariant sufficient conditions are given that ensure the convergence of the Gauss-Newton iterates towards a solution of the problem, as well as the uniqueness of that solution in an explicitely defined neighborhood. It is shown by a counter- example that the results do not carry over to the rank deficient case.