Prof. Dr. Carsten Hartmann

Lehrstuhl für Wahrscheinlichkeitsrechnung und Statistik

Institut für Mathematik
BTU Cottbus-Senftenberg
03046 Cottbus
Website


Research focus

Computational Stochastics, Optimal Control, Model Reduction, Molecular Dynamics

Projects as a project leader

  • CH4

    Optimal control of chemical reaction systems and application to drug resistance mitigating therapy

    Prof. Dr. Carsten Hartmann / Dr. Max von Kleist / PD Dr. Marcus Weber

    Project heads: Prof. Dr. Carsten Hartmann / Dr. Max von Kleist / PD Dr. Marcus Weber
    Project members: Wei Zhang
    Duration: -
    Status: completed
    Located at: Freie Universität Berlin

    Description

    Development and spread of drug resistant microorganisms is a major health issue which, accompanied by an attrition in drug development, is expected to worsen in the near future. The source of drug resistance development is the inadequate use of antimicrobials: Inadequate therapies insufficiently suppress susceptible strains, which may give rise to a drug resistant type. At the same time, inadequate therapy exerts enough selective pressure to provide the newly emerged resistant strain with a selective advantage that allows it to become fixed in the population. In recent years, we have elaborated the idea, that an optimal switching between existing antimicrobial drugs may mitigate drug resistance development in the individual. Drug resistance development is an intrinsically stochastic process. This process can be accurately described by the chemical master equation (CME). A major mathematical drawback is the fact that the CME cannot be solved directly due to its numerical complexity. Therefore, computation of an optimal control/therapy based on a direct numerical solution of the CME is usually not feasible. The aim of the proposed project is to mathematically characterize and develop optimal control policies derived from approximations of the CME, and to use the developed methods to suggest drug mitigating therapies to clinical partners in the field of HIV-1 and antibiotic resistance.

    http://systems-pharmacology.de/?page_id=621
  • CH-AP8

    Probing scales in equilibrated systems by optimal nonequilibrium forcing

    Prof. Dr. Carsten Hartmann / Prof. Dr. Christof Schütte / PD Dr. Marcus Weber

    Project heads: Prof. Dr. Carsten Hartmann / Prof. Dr. Christof Schütte / PD Dr. Marcus Weber
    Project members: -
    Duration: 01.10.2014 - 30.06.2018
    Status: running
    Located at: Freie Universität Berlin

  • CH-AP9

    Origin of the scaling cascades in protein dynamics

    Prof. Dr. Carsten Hartmann

    Project heads: Prof. Dr. Carsten Hartmann
    Project members: -
    Duration: 01.10.2014 - 30.06.2018
    Status: running
    Located at: Freie Universität Berlin

    Description

    The molecular dynamics of proteins and peptides is a hierarchical process which in­volves characteristic time scales ranging from 10-12 seconds to 100 seconds. Although the physical models of the local intramolecular interactions are relatively well devel­oped, and molecular dynamics simulations have proven successful in recovering the dynamics of large-scale biomolecular systems, a mathematical understanding of how local interactions in the molecular root model give rise to a cascade of processes on different time scales is still lacking.
    In this project we will investigate how these scaling cascades arise from the physical models of molecular dynamics and develop mathematical tools for their analysis. Our root model is a diffusion in a high-dimensional potential energy landscape that mod­els the local interactions between atoms or groups of atoms. The local interactions in the molecular force .eld (i.e., the gradient of the potential energy) then induce long-range effects and may give rise to the observed long time scales on the order of seconds. Yet the predictability of molecular dynamics with respect to variations in the physical parameters (e.g., force .eld parameters) or boundary conditions (e.g., temperature) is re­markably poor, the reason being the nonlinearity, the large dimensionality of the models and noise present in the systems, which altogether promote large-scale effects induced by small noise or slow collective motions of atoms or groups of atoms.
    For molecular systems with reversible dynamics, the relevant so-called implied time scales are related to the dominant eigenvalues of the underlying Markov generator. These eigenvalues can be estimated from molecular dynamics simulations and serve as approximations of experimentally measurable quantities. In molecular dynamics simu­lations it is possible to selectively tune the strength of a speci.c physical interaction (e.g., strength of long-range forces between different amino acids) or boundary conditions (e.g., temperature or pH), rendering them an ideal tool for analyzing the connection between root model and observed time scales. To investigate how the cascades of time scales arise in molecular dynamics we will extend numerical continuation methods for dynamical systems to stochastic molecular systems in order to study the changes in the implied time scales under variation of force .eld parameters or boundary conditions. We will compare analytical results to results from numerical simulations (classical and ab-initio molecular dynamics) and to results from infrared (IR) spectroscopy. Despite its popularity in the protein folding community, implied time scales are only one possible way to quantify molecular dynamics time scales. For instance, the exponen­tial convergence rate towards the thermodynamic equilibrium state is closely linked to experimentally measurable quantities. A second focus of the project is therefore to com­pare quantities which represent these relaxation time scales. To this end we will extend the numerical continuation approach to other observables, such as entropy production rates that, in certain cases, can be related to the shape of the molecular potential or Han­kel singular values that characterize the response of the system to the environmental noise and can be related to the typical residence time of a conformation.
    The understanding how scaling cascades in protein dynamics originate from the known hierarchy of physical interactions will be crucial for the development of multi-scale models, which consistently capture time scales on any desired level of coarseness. Moreover it will yield insight into biological phenomena such as allosteric regulation mechanisms or pathological misfolding events caused by single-point mutations.

    http://sfb1114.imp.fu-berlin.de/research/index.php?option=com_projectlog&view=project&id=9
  • SE11

    Model order reduction for light-controlled nanocatalysis

    Prof. Dr. Carsten Hartmann

    Project heads: Prof. Dr. Carsten Hartmann
    Project members: PD Dr. Burkhard Schmidt
    Duration: -
    Status: completed
    Located at: Freie Universität Berlin

    Description

    Photocatalysis is a key application in the field of femtochemistry where chemical reaction dynamics is controlled by temporally shaped femtosecond laser pulses, with the target to promote specific product channels while suppressing competing undesired channels, e.g. pollutants. The optimal shaping of the laser pulse requires a detailed insight into the underlying reaction mechanisms at the atomic or molec- ular level that can often only be obtained by theoretical modelling and computer simulations of the quantum mechanical equations of motion. For catalytic system, this boils down to the iterated integration of the dissipative Liouville–von–Neumann (LvN) equation for reduced quantum mechanical density matrices, which represents the computational bottleneck for theoretical modelling, as the size of the matrices grows quadratically with the number of quantum states involved. The aim of this project is to study model order reduction (MOR) of LvN-based models to beat the curse of dimensionality in the simulation and (optimal) control of photocatalytic processes. In the setting of first-order perturbation theory, the laser field in these models is linearly coupled to the density matrix, which leads to a time- inhomogeneous bilinear system of equations of motion. MOR of bilinear systems has recently been a field of intense research. The downside of many available methods is their lack of structure preservation, most importantly, asymptotic stability of fixed points. An alternative that is in the focus of this project is MOR based on balancing the controllable and observable subspace of the system. Even though the identification of the essential subspace requires the solution of large-scale Lyapunov equations, which limits the applicability of the method to systems of moderate size (up to 100,000 DOFs), it has proven powerful for linear control systems in terms of computable error bounds and structure preservation. Whether these results carry over to the bilinear case is still open.

    https://sites.google.com/site/ecmathse11/