TY - GEN A1 - Hammerschmidt, Martin A1 - Döpking, Sandra A1 - Burger, Sven A1 - Matera, Sebastian T1 - Field Heterogeneities and their Impact on Photocatalysis: Combining optical and kinetic Monte Carlo Simulations on the Nanoscale N2 - Gaining insights into the working principles of photocatalysts on an atomic scale is a challenging task. The obviously high complexity of the reaction mechanism involving photo-excited electrons and holes is one reason. Another complicating aspect is that the electromagnetic field, driving photocatalysis, is not homogeneous on a nanoscale level for particle based catalysts as it is influenced by the particle’s shape and size. We present a simple model, inspired by the CO2 reduction on titania anatase, which addresses the impact of these heterogeneities on the photocatalytic kinetics by combining kinetic Monte Carlo with electromagnetic wave simulations. We find that average activity and especially efficiency might differ significantly between different particles. Moreover, we find sizable variation of the catalytic activity on a single facet of a nanocrystal. Besides this quantitative heterogeneity, the coverage situation in general changes laterally on this facet and we observe a concomitant change of the rate-determining steps. This heterogeneity on all levels of photocatalytic activity is masked in experimental studies, where only the spatially averaged activity can be addressed. Microkinetic models based on experimental findings might therefore not represent the true micro- scopic behavior, and mechanistic conclusion drawn from these need to be handled with care. T3 - ZIB-Report - 17-20 Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-63690 SN - 1438-0064 ER - TY - JOUR A1 - Hammerschmidt, Martin A1 - Döpking, Sandra A1 - Burger, Sven A1 - Matera, Sebastian T1 - Field Heterogeneities and Their Impact on Photocatalysis: Combining Optical and Kinetic Monte Carlo Simulations on the Nanoscale JF - J. Phys. Chem. C Y1 - 2020 U6 - https://doi.org/10.1021/acs.jpcc.9b11469 VL - 124 SP - 3177 ER - TY - GEN A1 - Erdmann, Bodo A1 - Lang, Jens A1 - Matera, Sebastian A1 - Wilmanski, Krzysztof T1 - Adaptive Linearly Implicit Methods for Linear Poroelastic Equations N2 - Adaptive numerical methods in time and space are introduced and studied for linear poroelastic models in two and three space dimensions. We present equivalent models for linear poroelasticity and choose both the {\em displacement--pressure} and the {\em stress--pressure} formulation for our computations. Their discretizations are provided by means of linearly implicit schemes in time and linear finite elements in space. Our concept of adaptivity opens a way to a fast and reliable simulation of different loading cases defined by corresponding boundary conditions. We present some examples using our code {\sf Kardos} and show that the method works efficiently. In particular, it could be used in the simulation of some bone healing models. T3 - ZIB-Report - 06-37 KW - Poroelasticity KW - Biot's model KW - bone healing KW - adaptive finite elements KW - adaptive time integration KW - Rosenbrock methods Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9312 ER - TY - JOUR A1 - Gelß, Patrick A1 - Matera, Sebastian A1 - Schütte, Christof T1 - Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model JF - Journal of Computational Physics N2 - In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased Y1 - 2016 U6 - https://doi.org/10.1016/j.jcp.2016.03.025 VL - 314 SP - 489 EP - 502 ER - TY - GEN A1 - Gelß, Patrick A1 - Matera, Sebastian A1 - Schütte, Christof T1 - Solving the master equation without kinetic Monte Carlo: tensor train approximations for a CO oxidation model N2 - In multiscale models of heterogeneous catalysis, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. This usually is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO_2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness. T3 - ZIB-Report - 15-41 KW - heterogeneous catalysis KW - master equation KW - kinetic Monte Carlo KW - tensor decompositions KW - tensor train format KW - alternating linear scheme Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-55743 SN - 1438-0064 ER - TY - JOUR A1 - Gelß, Patrick A1 - Klein, Rupert A1 - Matera, Sebastian A1 - Schmidt, Burkhard T1 - Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains JF - The Journal of Chemical Physics N2 - We demonstrate how to apply the tensor-train format to solve the time-independent Schrödinger equation for quasi-one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fröhlich–Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low-rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach that directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in the case of homogeneous systems, the tensor-train ranks of state vectors only marginally depend on the chain length, which results in a linear growth of the storage consumption. However, the central processing unit time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton–phonon coupling strength, although only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich–Holstein type Hamiltonians. Y1 - 2022 U6 - https://doi.org/10.1063/5.0074948 VL - 156 IS - 2 SP - 024109 ER -