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    Estimation of the infinitesimal generator by square-root approximation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Donati, Luca; Heida, Martin; Weber, Marcus; Keller, Bettina
    For the analysis of molecular processes, the estimation of time-scales, i.e., tran- sition rates, is very important. Estimating the transition rates between molecular conformations is - from a mathematical point of view - an invariant subspace projec- tion problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is ap- proximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the ge- ometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.
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    Consistency and order 1 convergence of cell-centered finite volume discretizations of degenerate elliptic problems in any space dimension
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Heida, Martin; Sikorski, Alexander; Weber, Marcus
    We study consistency of cell-centered finite difference methods for elliptic equations with degenerate coefficients in any space dimension $dgeq2$. This results in order of convergence estimates in the natural weighted energy norm and in the weighted discrete $L^2$-norm on admissible meshes. The cells of meshes under consideration may be very irregular in size. We particularly allow the size of certain cells to remain bounded from below even in the asymptotic limit. For uniform meshes we show that the order of convergence is at least 1 in the energy semi-norm, provided the discrete and continuous solutions exist and the continuous solution has $H^2$ regularity.