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- ItemComposition of titania coatings deposited by different techniques(Offenbach : Verlag der Deutschen Glastechnischen Gesellschaft, 1994) Laube, Michael; Wagner, Wolfgang; Rauch, Friedrich; Ottermann, Clemens; Bange, Klaus; Niederwald, HansjörgQuantitative dement concentrations in titania films produced by different deposition techniques (evaporation, sputtering, ion-assisted deposition, ion plating and dip coating) have been determined by means of Rutherford Backscattering Spectrometry and Nuclear Reaction Analysis with the reaction ¹H(¹⁵N, αγ)¹²C. Large differences of the hydrogen content are found for the various production techniques and the related deposition parameters, which correlate with the refractive index of the respective film. In dependence on the deposition conditions the oxygen/titanium ratio of the investigated titania films varies between 1.95 and 2.09. The impurities detected in the films (tantalum, molybdenum, silicon, argon, carbon, sodium) can be related to specific deposition conditions. Three multilayer interference systems containing TiO₂ and SiO₂ show large variations in hydrogen content resembling those found for single TiO₂ films.
- ItemA class of probabilistic models for the Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Wagner, WolfgangA class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.
- ItemCell size error in stochastic particle methods for coagulation equations with advection(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Patterson, Robert I.A.; Wagner, WolfgangThe paper studies the approximation error in stochastic particle methods for spatially inhomogeneous population balance equations. The model includes advection, coagulation and inception. Sufficient conditions for second order approximation with respect to the spatial discretization parameter (cell size) are provided. Examples are given, where only first order approximation is observed.
- ItemA kinetic equation for the distribution of interaction clusters in rarefied gases(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Patterson, Robert I.A.; Simonella, Sergio; Wagner, WolfgangWe consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.
- ItemRandom and deterministic fragmentation models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Wagner, WolfgangRandom and deterministic fragmentation models are considered. Their relationship is studied by deriving different forms of the kinetic fragmentation equation from the corresponding stochastic models. Results related to the problem of non-conservation of mass (phase transition into dust) are discussed. Illustrative examples are given and some open problems are mentioned.
- ItemPost-gelation behavior of a spatial coagulation model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Wagner, WolfgangA coagulation model on a finite spatial grid is considered. Particles of discrete masses jump randomly between sites and, while located at the same site, stick together according to some coagulation kernel. The asymptotic behavior (for increasing particle numbers) of this model is studied in the situation, when the coagulation kernel grows sufficiently fast so that the phenomenon of gelation is observed. Weak accumulation points of an appropriate sequence of measure-valued processes are characterized in terms of solutions of a nonlinear equation. A natural description of the behavior of the gel is obtained by using the one-point compactification of the size space. Two aspects of the limiting equation are of special interest. First, the formal extension of Smoluchowski's coagulation equation to the spatially inhomogeneous case has to be modified for a certain class of coagulation kernels. Second, due to spatial inhomogeneity, an equation for the time evolution of the gel mass density has to be added. The jump rates are assumed to vanish with increasing particle masses so that the gel is immobile. Two different gel growth mechanisms (active and passive gel) are found depending on the type of the coagulation kernel.
- ItemA random cloud model for the Wigner equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Wagner, WolfgangA probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.
- ItemA stochastic algorithm without time discretization error for the Wigner equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Muscato, Orazio; Wagner, WolfgangStochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wavevector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.
- ItemA class of stochastic algorithms for the Wigner equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Muscato, Orazio; Wagner, WolfgangA class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed in a benchmark test case, where certain advantages of the new class of algorithms are demonstrated.
- ItemStochastic weighted particle methods for population balance equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Patterson, Robert I.A.; Kraft, Markus; Wagner, WolfgangA class of stochastic algorithms for the numerical treatment of population balance equations is introduced. The algorithms are based on systems of weighted particles, in which coagulation events are modelled by a weight transfer that keeps the number of computational particles constant. The weighting mechanisms are designed in such a way that physical processes changing individual particles (such as growth, or other surface reactions) can be conveniently treated by the algorithms. Numerical experiments are performed for complex laminar premixed flame systems. Two members of the class of stochastic weighted particle methods are compared to each other and to a direct simulation algorithm. One weighted algorithm is shown to be consistently better than the other with respect to the statistical noise generated. Finally, run times to achieve fixed error tolerances for a real flame system are measured and the better weighted algorithm is found to be up to three times faster than the direct simulation algorithm.