A Comparison of Related Concepts in Computational Chemistry and Mathematics

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 precising. 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

Parameter Identification in a Tuberculosis Model for Cameroon

A deterministic model of tuberculosis in Cameroon is designed and analyzed with respect to its transmission dynamics. The model includes lack of access to treatment and weak diagnosis capacity as well as both frequency-and density-dependent transmissions. It is shown that the model is mathematically well-posed and epidemiologically reasonable. Solutions are non-negative and bounded whenever the initial values are non-negative. A sensitivity analysis of model parameters is performed and the most sensitive ones are identified by means of a state-of-the-art Gauss-Newton method. In particular, parameters representing the proportion of individuals having access to medical facilities are seen to have a large impact on the dynamics of the disease. The model predicts that a gradual increase of these parameters could significantly reduce the disease burden on the population within the next 15 years.IMU Berlin Einstein Foundation Progra

Computer Assisted Planning in Cranio-Maxillofacial Surgery

In cranio-maxillofacial surgery physicians are often faced with the reconstruction of massively destroyed or radically resected tissue structures caused by trauma or tumours. Also corrections of dislocated bone fragments up to the complete modeling of facial regions in cases of complex congenital malformations are common tasks of plastic and reconstructive surgeons. With regard to the individual anatomy and physiology, such procedures have to be planned and executed thoroughly in order to achieve the best functional as well as an optimal aesthetic rehabilitation. On this account a computer assisted modeling, planning and simulation approach is presented that allows for preoperative assessment of different therapeutic strategies on basis of three-dimensional patient models. Bone structures can be mobilized and relocated under consideration of anatomical and functional constraints. The resulting facial appearance is simulated via finite-element methods on basis of a biomechanical tissue model, and visualized using high quality rendering techniques. Such an approach is not only important for preoperative mental preparation, but also for vivid patient information, documentation, quality assurance as well as for surgical education and training

Stable computation of probability densities for metastable dynamical systems

Whenever the invariant stationary density of metastable dynamical systems decomposes into almost invariant partial densities, its computation as eigenvector of some transition probability matrix is an ill-conditioned problem. In order to avoid this computational difficulty, we suggest applying an aggregation/disaggregation method which addresses only well-conditioned subproblems aud thus results in a stable algorithm. In contrast to existing methods, the aggregation step is done via a sampling algorithm which covers only small patches of the sampling space. Finally, the theoretical analysis is illustrated by two biomolecular examples

Parametric estimation of the driving L\'evy process of multivariate CARMA processes from discrete observations

We consider the parametric estimation of the driving L\'evy process of a multivariate continuous-time autoregressive moving average (MCARMA) process, which is observed on the discrete time grid $(0,h,2h,...)$. Beginning with a new state space representation, we develop a method to recover the driving L\'evy process exactly from a continuous record of the observed MCARMA process. We use tools from numerical analysis and the theory of infinitely divisible distributions to extend this result to allow for the approximate recovery of unit increments of the driving L\'evy process from discrete-time observations of the MCARMA process. We show that, if the sampling interval $h=h_N$ is chosen dependent on $N$, the length of the observation horizon, such that $N h_N$ converges to zero as $N$ tends to infinity, then any suitable generalized method of moments estimator based on this reconstructed sample of unit increments has the same asymptotic distribution as the one based on the true increments, and is, in particular, asymptotically normally distributed.Comment: 38 pages, four figures; to appear in Journal of Multivariate Analysi

Domain Decomposition with Subdomain CCG for Material Jump Elliptic Problems

A combination of the cascadic conjugate gradient (CCG) method for homogeneous problems with a non-overlapping domain decomposition (DD) method is studied. Mortar finite elements on interfaces are applied to permit nonmatching grids in neighboring subdomains. For material jump problems, the method is designed as an alternative to the cascadic methods. 1 Introduction In this paper we consider linear elliptic problems #(a#u) + cu = f on general domains with space dimension p equal to 2 or 3, where typically the coe#cient a is strongly varying. This type of problems arises whenever di#erent materials are combined. Standard (multiplicative) multigrid methods [13] or additive multilevel methods such as KASKADE with BPX preconditioners [12, 20, 9] deal quite e#ciently with such a situation - apart from certain pathological examples in 3D. However, the recently developed cascadic multigrid methods such as CCG [11, 8, 7, 6], which are extremely fast for homogeneous problems, tend to exhi..

A Complex Mathematical Model of the Human Menstrual Cycle

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