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A 3-D Tensor FDTD-Formulation for Treatment of Sloped Interfaces in Electrically Inhomogeneous Media
(2003)
This article studies the relation of the two scientific languages Chemistry and Mathematics via three selected comparisons: (a) QSSA versus dynamic ILDM in reaction kinetics, (b) lumping versus discrete Galerkin methods in polymer chemistry, and (c) geometrical conformations versus metastable conformations in drug design. The common clear message from these comparisons is that chemical intuition may pave the way for mathematical concepts just as chemical concepts may gain from mathematical precisioning. Along this line, significant improvements in chemical research and engineering have already been possible -- and can be further expected in the future from the dialogue between the two scientific languages.
This paper aims at presenting the complex coupled network of the human menstrual cycle to the interested community. Beyond the presently popular smaller models, where important network components arise only as extremely simplified source terms, we add: the GnRH pulse generator in the hypothalamus, receptor binding, and the biosynthesis in the ovaries. Simulation and parameter identification are left to a forthcoming paper.
The numerical integration of dynamical contact problems often leads to instabilities at contact boundaries caused by the non-penetration condition between bodies in contact. Even a recent energy dissipative modification due to Kane et al. (1999), which discretizes the non-penetration constraints implicitly, is not able to circumvent artificial oscillations. For this reason, the present paper suggests a contact stabilization which avoids artificial oscillations at contact interfaces and is also energy dissipative. The key idea of this contact stabilization is an additional $L^2$-projection at contact interfaces, which can easily be added to any existing time integration scheme. In case of a lumped mass matrix, this projection can be carried out completely locally, thus creating only negligible additional numerical cost. For the new scheme, an elementary analysis is given, which is confirmed by numerical findings in an illustrative test example (Hertzian two body contact).
Recently, a novel concept for the computation of essential features of the dynamics of Hamiltonian systems (such as molecular dynamics) has been proposed. The realization of this concept had been based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework. This approach allows to define the phrase ``conformation'' in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator $T$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of $T$ via specific hybrid Monte Carlo techniques is shown to lead to a stochastic matrix $P$. With these theoretical preparations, an identification algorithm for conformations is applicable. It is demonstrated that the discretization of $T$ can be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked. Numerical results for the n-pentane molecule and the triribonucleotide adenylyl\emph{(3'-5')}cytidylyl\emph{(3'-5')}cytidin are given and interpreted.
Recently, a novel concept for the computation of essential features of Hamiltonian systems (such as those arising in molecular dynamics) has been proposed. The realization of that concept was based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept based on statistical mechanics, which allows to attack realistic molecular systems. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator $T$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of $T$ via hybrid Monte Carlo techniques (based on short term subtrajectories only) is shown to lead to a stochastic matrix $P$. With these theoretical preparations, an identification algorithm for conformations is applicable (to be presented elsewhere). Numerical results for the n-pentane molecule are given and interpreted.
This study presents a differential equation model for the feedback mechanisms between Gonadotropin-releasing Hormone (GnRH), Follicle-Stimulating Hormone (FSH), Luteinizing Hormone (LH), development of follicles and corpus luteum, and the production of estradiol (E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual cycle.
In contrast to other models, this model does not involve delay differential equations and is based on deterministic modelling of the GnRH pulse pattern, which allows for faster simulation times and efficient parameter identification.
These steps were essential to tackle the task of developing a mathematical model for the administration of GnRH analogues.
The focus of this paper is on model development for GnRH receptor binding and the integration of a pharmacokinetic/pharmacodynamic model for the GnRH agonist Nafarelin and the GnRH antagonist Cetrorelix into the menstrual cycle model.
The final mathematical model describes the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle in 12 healthy women.
Moreover, it correctly predicts the changes in the cycle following single and multiple dose administration of Nafarelin or Cetrorelix at different stages in the cycle.
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 is motivated by the need for a fast robust adaptive multigrid method to solve complex Helmholtz eigenvalue problems arising from the design of optical chips. A nonlinear multigrid method is developed, which can be regarded as an extension of a previous adaptive Rayleigh quotient minimization method for selfadjoint Helmholtz eigenproblems. Since the complex Helmholtz operator is just a compact nonselfadjoint perturbation of a selfadjoint operator, linear algebra techniques like Schur decomposition can be extended from the finite dimensional case. The efficiency of the derived adaptive nonlinear multigrid method is illustrated by computations for a technologically relevant integrated optics component containing Multi Quantum Well Layers.
A slight modification of the extended Stoermer discretization for non self-adjoint second order ODE systems is derived on the basis of a simple stability analysis. This discretization easily extends to implicit ODE systems, which are known to arise e.g. in mechanical engineering. In addition, a special variant of semi-implicit Euler discretization is proposed, which essentially treats the state variables explicitly, but their derivatives implicitly. Numerical tests over critical parameter values of the van der Pol oscillator illustrate the domain of efficiency of the suggested discretizations.
This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.
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.
Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included are only known qualitatively as stimulatory or inhibitory effects, the model surprisingly well features quantitative observations made in reality. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences.
A Stepsize Control for Continuation Methods with Special Application to Multiple Shooting Techniques
(1976)
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.
A cascadic multigrid (CMG) method for elliptic problems with strong material jumps is proposed and analyzed. Non--matching grids at interfaces between subdomains are allowed and treated by mortar elements. The arising saddle point problems are solved by a subspace confined conjugate gradient method as smoother for the CMG. Details of algorithmic realization including adaptivity are elaborated. Numerical results illustrate the efficiency of this CMG algorithm.