Browsing by Author "Wagner, Wolfgang"

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    Cell 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, Wolfgang
    The 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.
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    Chemical properties of hydrolyzed surface layers on SiO2-BaO-B2O3
    (Offenbach : Verlag der Deutschen Glastechnischen Gesellschaft, 1990) Wagner, Wolfgang; Rauch, Friedrich; Bach, Hans
    The leaching process of SiO2—BaO—B2O3 glass in distilled water and the subsequent processes during storage in air were studied to complete the recent investigations on the interaction of SiO2—BaO—B2O3 glass with polishing slurries. The in-depth distributions of hydrogen and glass constituents existing in the leached surface layers were measured using nuclear reaction analysis and Rutherford backscattering spectrometry. Hydrogen uptake, barium loss and redeposition of dissolved zirconium-containing material on the surface occurred during the interaction with water. Changes of the in-depth distributions caused by reactions between the molecular water from the air, the materials of the leached surface layers and the glass components during a subsequent storage in air at room temperature were also observed. The analyses show unequivocally that the chemical driving forces of these reactions exhibit differences which depend exclusively on the different chemical properties of the leached surface-layer materials. These properties are on their part determined by different leaching agents and parameters. Consequences for the control of chemical interactions of silicate glass surfaces during the fabrication of optical surfaces are considered.
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    A class of probabilistic models for the Schrödinger equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Wagner, Wolfgang
    A 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.
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    A class of stochastic algorithms for the Wigner equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Muscato, Orazio; Wagner, Wolfgang
    A 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.
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    Coagulation and Fragmentation Models
    (Zürich : EMS Publ. House, 2007) Norris, James; Wagner, Wolfgang
    Analysis of coagulation and fragmentation is crucial to understanding many processes of scientific and industrial importance. In recent years this has led to intensified research activities in the areas of differential equations, probability theory, and combinatorics. The purpose of the workshop was to bring together people from these different areas working on various aspects of coagulation and fragmentation. We believe that the insights resulting from the interactions which have been stimulated that week should lead to further advances both in the development of mathematical techniques and in new applications.
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    Composition 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, Hansjörg
    Quantitative 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.
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    Deviational particle Monte Carlo for the Boltzmann equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Wagner, Wolfgang
    The paper describes the deviational particle Monte Carlo method for the Boltzmann equation. The approach is an application of the general ``control variates'' variance reduction technique to the problem of solving a nonlinear equation. The deviation of the solution from a reference Maxwellian is approximated by a system of positive and negative particles. Previous results from the literature are modified and extended. New algorithms are proposed that cover the nonlinear Boltzmann equation (instead of a linearized version) with a general interaction model (instead of hard spheres). The algorithms are obtained as procedures for generating trajectories of Markov jump processes. This provides the framework for deriving the limiting equations, when the number of particles tends to infinity. These equations reflect the influence of various numerical approximation parameters. Detailed simulation schemes are provided for the variable hard sphere interaction model.
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    A 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, Wolfgang
    We 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.
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    Mini-Workshop: Numerics for Kinetic Equations
    (Zürich : EMS Publ. House, 2008) Rjasanow, Sergej; Wagner, Wolfgang
    [no abstract available]
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    Mini-Workshop: Stochastic Models for Coagulation Processes
    (Oberwolfach-Walke : Mathematisches Forschungsinstitut Oberwolfach, 2001) Wagner, Wolfgang
    [no abstract available]
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    Numerical study of the systematic error in Monte Carlo schemes for semiconductors
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Muscato, Orazio; Di Stefano, Vincenza; Wagner, Wolfgang
    The paper studies the convergence behavior of Monte Carlo schemes for semiconductors. A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field calculations, vanishes sufficiently fast. The error due to the approximation of the trajectories of particles depends on the ODE solver used in the algorithm. It is negligible compared to the other sources of time step error, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanism is the most significant source of error with respect to the time step.
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    Post-gelation behavior of a spatial coagulation model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Wagner, Wolfgang
    A 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.
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    Properties of the steady state distribution of electrons in semiconductors
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Muscato, Orazio; Wagner, Wolfgang; Di Stefano, Vincenza
    This paper studies a Boltzmann transport equation with several electron-phonon scattering mechanisms, which describes the charge transport in semiconductors. The electric field is coupled to the electron distribution function via Poisson's equation. Both the parabolic and the quasi-parabolic band approximations are considered. The steady state behaviour of the electron distribution function is investigated by a Monte Carlo algorithm. More precisely, several nonlinear functionals of the solution are calculated that quantify the deviation of the steady state from a Maxwellian distribution with respect to the wave-vector. On the one hand, the numerical results illustrate known theoretical statements about the steady state and indicate possible directions for future studies. On the other hand, the nonlinear functionals provide tools that can be used in the framework of Monte Carlo algorithms for detecting regions in which the steady state distribution has a relatively simple structure, thus providing a basis for domain decomposition methods
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    Random and deterministic fragmentation models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Wagner, Wolfgang
    Random 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.
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    A random cloud model for the Schrödinger equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Wagner, Wolfgang
    The paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offsprings. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.
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    A random cloud model for the Wigner equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Wagner, Wolfgang
    A 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.
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    A random walk model for the Schrödinger equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Wagner, Wolfgang
    A random walk model for the spatially discretized time-dependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complex-valued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation.
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    Some properties of the kinetic equation for electron transport in semiconductors
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Wagner, Wolfgang
    The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.
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    A stochastic algorithm without time discretization error for the Wigner equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Muscato, Orazio; Wagner, Wolfgang
    Stochastic 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.
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    Stochastic models in kinetic theory
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Wagner, Wolfgang
    The paper is concerned with some aspects of stochastic modelling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (mono-molecular processes), or via binary interactions (bi-molecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent