# Zur Analyse großer strukturierter chemischer Reaktionssysteme mit Waveform-Iterationsverfahren.

*Authors*

- Borchardt, Jürgen
- Bremer, Ingo

*2010 Mathematics Subject Classification*

- 65L05 65Y05 80A30

*Keywords*

- Kinetik chemischer Reaktionssysteme, Anfangswertproblem fiir gewohnliche Differentialgleichungen, Waveform-Iteration, strukturierte Analyse, parallele Algorithmen

*DOI*

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# Regularity of solutions for some problems of mathematical physics.

*Authors*

- Koshelev, A.

*DOI*

*Appeared in*

- Math. Nachr., 162 (1993) pp. 59--88.

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# Stability of numerical schemes for stochastic differential equations with multiplicative noise.

*Authors*

- Hofmann, N.

*2010 Mathematics Subject Classification*

- 60H10

*Keywords*

- Numerical stability, stochastic differential equations, numerical simulations, implicit schemes

*DOI*

*Abstract*

A notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise for which there is a connection between the parameters in the drift and diffusion coefficient. By means of the Euler scheme and two different implicit Euler schemes a method to find the regions of stability is also examined.

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# Asymptotical mean stability of numerical solutions with multiplicative noise.

*Authors*

- Schurz, Henri

*2010 Mathematics Subject Classification*

- 65C20 65U05 60H10 65L20

*Keywords*

- Stochastic differential equations, numerical methods, weak and strong convergence, asymptotically absolute stability, p-th and weak mean stability, stability function and domain, simulation experiments

*DOI*

*Abstract*

As an extent of asymptotically absolute stability of numerical methods in deterministic situation, in this report the asymptotically absolute mean stability of the null solution for stochastic differential equations with respect to different criterions will be examined, both for the exact solution and for its numerical approximations. Among the considered criterions the mean square stability plays the main role in the examinations. For the class of scalar, bilinear, complex-valued stochastic differential equations, comparison studies for different numerical schemes are provided and show their different stability features. However the balanced implicit methods have proved to be rich enough to possess appropriately large stability domains. Finally, experiments for the Kubo oscillator indicate how efficient the asymptotical mean stability examinations could be for the reality.

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# Super-Brownian motions in higher dimensions with absolutely continuous measure states.

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus

*2010 Mathematics Subject Classification*

- 60J80 60J65 60G57

*Keywords*

- Absolutely continuous states, fractal catalytic medium, fundamental solutions, super-Brownian motion, superprocess, branching rate functional, measure-valued branching, additive functional approach, critical branching, collision local time

*DOI*

*Abstract*

Continuous super-Brownian motions in two and higher dimensions are known to have singular measure states. However, by weakening the branching mechanism in an irregular way they can be forced to have absolutely continuous states. The sufficient conditions we impose are identified in a couple of examples with irregularities in only one coordinate. This includes the case of branching restricted to some densely situated ensemble of hyperplanes.

*Appeared in*

- J. Theoret. Probab., 8 (1995), pp. 179--206

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# Attractors of non invertible maps.

*Authors*

- Bothe, Hans-Guenter

*2010 Mathematics Subject Classification*

- 58F12 58F15

*Keywords*

- Attractors, non invertible maps, solenoids

*DOI*

*Abstract*

For mappings ƒ : S^{1} x ℝ^{n} → S^{1} x ℝ^{n} (n≥2) of the form ƒ ( t,*x* ) = (Θt,λ*x*+*v*(t)), where Θ ∈ ℤ, Θ ≥ 2, λ ∈ (0, 1), *v* ∈ C^{1}(S^{1},ℝ^{n}) we consider the open subset S_{n,Θ,λ} of C^{1}(S^{1},ℝ^{n}) which consist of all *v* for which the restriction of ƒ to its attractor is injective. It is shown that for λ < min(½,Θ^{-2/(n-1)}) this set S_{n,Θ,λ} is dense in C^{1}(S^{1},ℝ^{n}) and that for each odd *n* it is not dense provided λ ≥ 64Θ^{-2/(n-1)}.

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# The Hausdorff dimension of certain attractors.

*Authors*

- Bothe, Hans Guenter

*2010 Mathematics Subject Classification*

- 58F12 28A78

*Keywords*

- Hyperbolic attractors, Hausdorff dimension, pressure

*DOI*

*Abstract*

For the solid torus V = S^{1} x D^{2} and a C^{1} embedding ƒ : V → V given by ƒ(t,*x*_{1},*x*_{2}) = (φ(t),λ_{1}(t)·*x*_{1}+*z*_{1}(t), λ_{2}(t)·*x*_{2}+*z*_{2}(t)) with dφ⁄_{dt} > 1, 0 < λ_{i}(t) < 1 the attractor Λ= ∩^{∞}_{i=0} ƒ^{i} (V) is a solenoid, and for each disk D(t) = {t} x D^{2} (t∈S^{1}) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which the Hausdorff dimension of Λ(t) is independent of t and determined by dim_{H} Λ (t) = max(*p*_{1},*p*_{2}) (0.1) where the real numbers *p*_{i} are characterized by the condition that the pressure of the function log λ_{i}^{pi}: S^{1} → ℝ with respect to the expanding mapping φ : S^{1} → S^{1} becomes zero. (There are two further characterizations of these numbers.) It is proved that (0.1) holds provided λ_{1}, λ_{2} are sufficiently small and Λ satisfies a condition called intrinsic transverseness. Then it is shown that in the space of all embeddings ƒ with sup λ_{i} < Θ^{-2} (Θ the mapping degree of φ) the subset of those ƒ which have an intrinsically transverse attractor Λ is open and dense with respect to the C^{1} topology.

*Appeared in*

- Ergodic Theory Dynam. Systems 15 (1995), no. 3, pp. 449--474.

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# Iterative Verfahren für lineare Gleichungssysterne auf Distributed Memory Systemen

*Authors*

- Schlundt, Rainer

ORCID: 0000-0002-4424-4301

*2010 Mathematics Subject Classification*

- 65F10 65F50 65Y05 68Q22

*Keywords*

- große lineare Systeme, iterative Verfahren, Krylov-Unterraum-Methoden, GMRES-Algorithmus, QMR-Verfahren, Distributed Memory Systeme

*DOI*

*Abstract*

Für die Lösung großer linearer Gleichungssysteme mit schwach bzw. voll besetzten Koeffizientenmatrizen für Distributed Memory Systeme werden das GMRES-Verfahren und die QMR-Methode vorgestellt. Beide iterativen Verfahren basieren auf Krylov-Unterraum-Methoden. Die Weiterentwicklung dieser beiden Verfahren für Distributed Memory Systeme erfolgt in zwei Richtungen. Die erste Variante beruht auf einer Parallelisierung der Matrix* Vektor-Operation. Die zweite Richtung beinhaltet die Aufspaltung der Gesamtaufgabe in disjunkte bzw. sich überlappende Teilprobleme.

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# Singularity of super-Brownian local time at a point catalyst.

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus
- Li, Yi
- Müller, Carl

*2010 Mathematics Subject Classification*

- 60J80 60J65 60G57

*Keywords*

- Point catalytic medium, critical branching, super-Brownian local time, occupation time, occupation density, measure-valued branching, superprocess

*DOI*

*Abstract*

In a one-dimensional single point-catalytic continuous super-Brownian motion studied by Dawson and Fleischmann (1993), the occupation density measure *λ*^{c} at the catalyst's position *c* is shown to be a singular (diffuse) random measure. The source of this qualitative new effect is the irregularity of the varying medium δ_{c} describing the point catalyst. The proof is based on a probabilistic characterization of the law of the Palm canonical clusters χ appearing in the Lévy-Khinchin representation of *λ*^{c} in a historical process setting and the fact that these χ have infinite left upper density (with respect to Lebesgue measure) at the Palm time point.

*Appeared in*

- The Annals of Probability 23(1)(1995), pp. 37-55

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# Automatic bandwidth choice and confidence intervals in nonparametric regression.

*Authors*

- Neumann, Michael H.

*2010 Mathematics Subject Classification*

- 62G15 62G07 62G20

*Keywords*

- Nonparametric regression, bandwidth choice, confidence intervals, Edgeworth expansions

*DOI*

*Abstract*

In the present paper we combine the issues of bandwidth choice and construction of confidence intervals in nonparametric regression. We modify the √*n*-consistent bandwidth selector of Härdle, Hall and Marron (1991) such that it is appropriate for heteroscedastic data and show how one can adapt the bandwidth g of the pilot estimator *m*̂_{g} in a reasonable data-dependent way. Then we compare the coverage accuracy of classical confidence intervals based on kernel estimators with data-driven bandwidths. We propose a method to put undersmoothing with a data-driven bandwidth into practice and show that this procedure outperforms explicit bias correction.

*Appeared in*

- Ann. Statist., 23 (1996), pp. 1937--1959

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# Finite element approximation of transport of reactive solutes in porous media. Part 2: Error estimates for equilibrium adsorption processes.

*Authors*

- Barrett, John W.
- Knabner, Peter

*2010 Mathematics Subject Classification*

- 65M15 65M60 35K65 35R35 35K55 76S05

*Keywords*

- Finite element approximation, error estimates, degenerate parabolic equation, energy norm estimates, flow in porous media

*DOI*

*Abstract*

In this paper we analyse a fully practical piecewise linear finite element approximation; involving numerical integration, backward Euler time discretisation and possibly regularization and relaxation; of the following degenerate parabolic equation arising in a model of reactive solute transport in porous media: Find u(x,t) such that

∂_{t}u + ∂_{t} [φ(u)] - Δu = f in Ω x (0,T]

u = 0 on ∂Ω x (0,T] u(•,0) = g (•) in Ω

for known data Ω ⊂ ℝ^{d}, 1 ≤ d ≤ 3, f, g and a monotonically increasing φ ∈ C^{0}(ℝ) ∩C^{1}(-∞,0] ∪ (0,∞) satisfying φ(0) = 0, which is only locally Hölder continuous, with exponent p ∈ (0,1), at at the origin; e.g. φ(s) ≡ [s]^{p}_{+}. This lack of Lipschitz continuity at the origin limits the regularity of the unique solution u and leads to difficulties in the finite element error analysis.

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# Finite element approximation of transport of reactive solutes in porous media. Part I: Error estimates for non-equilibrium adsorption processes.

*Authors*

- Barrett, John W.
- Knabner, Peter

*2010 Mathematics Subject Classification*

- 65M15 65M60 35K65 35R35 35K55 76S05

*Keywords*

- Finite element approximation, error estimates, degenerate parabolic equation, energy norm estimates, flow in porous media

*DOI*

*Abstract*

In this paper we analyse a fully practical piecewise linear finite element approximation; involving regularization, numerical integration and backward Euler time discretisation; of the following degenerate parabolic system arising in a model of reactive solute transport in porous media: Find {u(x,t),v(x,t)} such that

∂_{t}u + ∂_{t}v - Δu = f in Ω x (0,T] u=0 on ∂ Ω x (0,T]

∂_{t}v = k (φ(u)-v) in Ω x (0,T]

u(•,0) = g_{1} (•) v(•,0) = g _{2 } (•) in Ω ⊂ ℝ^{d}, 1 ≤ d ≤ 3

^{+}, f, g

_{1}, g

_{2}and a monotonically increasing φ ∈ C

^{0}(ℝ) ∩C

^{1}(-∞,0]∪(0,∞) satisfying φ(0) = 0; which is only locally Hölder continuous, with exponent p ∈ (0,1), at the origin; e.g. φ (s) ≡ [s]

^{p}

_{+}. This lack of Lipschitz continuity at the origin limits the regularity of the unique solution {u,v} and leads to difficulties in the finite element error analysis. Nevertheless we arrive at error bounds which in some cases exhibit the full approximation power of the trial space.

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# Nyström's method and iterative solvers for the solution of the double layer potential equation over polyhedral boundaries.

*Authors*

- Kleemann, Bernd
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 45L10 65R20 65F10

*Keywords*

- potential equation, Nyström's method, two-grid iteration

*DOI*

*Abstract*

In this paper we consider a quadrature method for the solution of the double layer potential equation corresponding to Laplace's equation in a three-dimensional polyhedron. We prove the stability for our method in case of special triangulations over the boundary of the polyhedron. For the solution of the corresponding system of linear equations, we consider a two-grid iteration and a further simple iteration procedure. Finally, we establish the rates of convergence and complexity and discuss the effect of mesh refinement near the corners and edges of the polyhedron.

*Appeared in*

- SIAM J. Numer. Anal., 32 (1995), pp. 924--951

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# Travelling wave behaviour of crystal dissolution in porous media flow.

*Authors*

- Knabner, Peter
- van Duijn, Cornelius J.

*2010 Mathematics Subject Classification*

- 35K65 35K57 35R35 35B30 76S05

*Keywords*

- Degenerate parabolic equation, flow in porous media, travelling wave solutions, asymptotic analysis

*DOI*

*Abstract*

In this paper we present a model for crystal dissolution in porous media and we analyse travelling wave solutions of the ensuing equations for a one-dimensional flow situation. We demonstrate the structure of the waves and we prove existence and uniqueness. The travelling wave description is characterized by a rate parameter *k* and a diffusion/dispersion parameter D. We investigate the limit processes as *k* → ∞ and

D → 0 and we obtain expressions for the rate of convergence. We also present some numerical results.

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# Most β shifts have bad ergodic properties.

*Authors*

- Schmeling, Jörg

ORCID: 0000-0001-6956-9463

*DOI*

*Abstract*

More than 30 years ago Rényi [1] introduced the representations of real numbers with an arbitrary base β > 1 as a generalization of the p-adic representations. One of the most studied problems in this field is the link between expansions to base β and ergodic properties of the corresponding β-shift. In this paper we will follow the bibliography of F. Blanchard [2] and give an affirmative answer to a question on the size of the set of real numbers β having the worst ergodic properties of their β-shifts.

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# Typical dimension of the graph of certain functions.

*Authors*

- Schmeling, Jörg

ORCID: 0000-0001-6956-9463 - Winkler, Reinhard

*DOI*

*Abstract*

Most functions from the unit interval to itself have a graph with. Hausdorff and lower entropy dimension 1 and upper entropy dimension 2. The same holds for several other Baire spaces of functions. In this paper it will be proved that this is the case also in the spaces of all mappings that are Lebesque measurable, Borel measurable, integrable in the Riemann sense, continuous, uniform distribution preserving (and continuous).

*Appeared in*

- Monatsh. Math., 119 (1995), pp. 303--320

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# Numerical methods for stochastic differential equations.

*Authors*

- Kloeden, Peter E.
- Platen, Eckhard

*2010 Mathematics Subject Classification*

- 60H10

*Keywords*

- Stochastic differential equations, numerical stimulations

*DOI*

*Abstract*

Numerical methods for stochastic differential equations, including Taylor expansion approximations, Runge-Kutta like methods and implicit methods, are summarized. Important differences between simulation techniques with respect to the strong (pathwise) and the weak (distributional) approximation criteria are discussed. Applications to the visualization of nonlinear stochastic dynamics. the computation of Lyapunov exponents and stochastic bifurcations are also presented.

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# Dissipative quantum mechanics using GENERIC

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34D20 37N20 47N50 80A99 81Q05 81V19

*Keywords*

- Quantum mechanics, density matrices, Hamiltonian systems, gradient systems, Onsager systems, GENERIC, von Neumann entropy, canonical correlation

*DOI*

*Abstract*

Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo-Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare of our gradient flow formulation for quantum systems with the Wasserstein gradient flow formulation for the Fokker-Planck equation and the entropy gradient flow formulation for reversible Markov chains.

*Appeared in*

- Recent Trends in Dynamical Systems, Proceedings of a Conference in Honor of Jürgen Scheurle, A. Johann, H.-P. Kruse, S. Schmitz, eds., vol. 35 of Proceedings in Mathematics and Statistics, Springer, Heidelberg, 2013, pp. 555--585

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# Mean square stability for discrete linear stochastic systems.

*Authors*

- Schurz, Henri

*2010 Mathematics Subject Classification*

- 60H10 65C20 65L20 65U05

*Keywords*

- Stochastic differential equations, numerical methods, mean square stability of the null solution

*DOI*

*Abstract*

Several results concerning asymptotical mean square stability of the null solution of specific linear stochastic systems are presented and proven. It is shown that the mean square stability of the implicit Euler method, taken from the monography of Kloeden and Platen (1992) and applied to linear stochastic differential equations, is necessary for the mean square stability of the corresponding implicit Milstein method (using the same implicitness parameter). Furthermore, a sufficient condition for the mean square stability of the implicit Euler method can be varified for autonomous systems. Additionally, the principle of 'monotonous inclusion' of the sequel of mean square stability domains holds for linear systems. The paper generalizes the results due to Schurz (1993) where one-dimensional linear complex systems with respect to asymptotical p-th mean stability have been investigated. Finally, a simple example confirms these assertions. The results can also be used to deduce recommendations for the practical implementation of numerical methods solving nonlinear systems by orienting on their linearization.

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# Ladungstransport und Oberflächenpotentialkinetik in ungeordneten dünnen Schichten.

*Authors*

- Brehmer, L.
- Liemant , Alfred
- Müller, I.

*Keywords*

- Charge carrier transport, hopping model, localized states, energetic states, distribution, electrical conductivity, surface potential kinetics, transit time

*DOI*

*Abstract*

Starting from a microscopic model of the solid a macroscopic nonlinear partial differential equation describes the space charge evolution process in dependence of the energetic distribution of the localized states, electric field strenght a.s.o. Using definite boundary and initial conditions the transport equation can be solved numerically and allows the calculation of important observabeles. A very useful observable both from the scientific and technological point of view is the surface potential and his kinetic. Therefore, the surface potential kinetics is discussed thoroughly in dependence on material parameters, film thickness, and surface charge. Furthermore, it could be calculated the space charge evolution process, the conductivity, transit time, and the concentration of localized states. The interpretation and the numerical solution of the presented transport equation give a new insight in fundamental problems of charge carrier transport in disordered materials

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# About loss of regularity and "blow up" of solutions for quasilinear parabolic systems.

*Authors*

- Gajewski, Herbert
- Jäger, Willi
- Koshelev, Alexander

*2010 Mathematics Subject Classification*

- 35B65 35D10 35K40 35K57

*Keywords*

- Parabolic systems, weak solutions, loss of regularity, chemotaxis system, semiconductor equations, numerical evidence

*DOI*

*Abstract*

Starting from sufficient conditions for regularity of weak solutions to quasilinear parabolic systems, uecessary conditions for loss of regularity are formulated. It is shown numerically that in some situations loss of regularity ("blow up") really happens accordingly to these conditions.

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# On the regularization of the ill-posed logarithmic kernel integral equation of the first kind.

*Authors*

- Bruckner, Gottfried

*2010 Mathematics Subject Classification*

- 65R30 65T05 65T10

*Keywords*

- decomposition, ill-posed logarithmic kernel integral equations, ill-posed problem, singular value decomposition, parameter choice, regularization method, optimal convergence

*DOI*

*Abstract*

The logarithmic kernel integral equation of the first kind is investigated as improperly posed problem considering its right-hand side as observed quantity in a suitable space with a weaker norm. The improperly posed problem is decomposed into a well-posed one, extensively studied in the literature (cf. e.g. [11], [13], [14]), and an ill-posed imbedding problem. For the ill-posed part a modified truncated singular value decomposition regularization method is proposed that allows an easily performable a-posteriori parameter choice. The whole problem is then solved by combining the regularization method with a numerical procedure from [13] for the well-posed part. Finally, an error estimate is given revealing the influence of the observation error on the approximation error of the numerical procedure. For a specification of the discretization parameter as a known function of the noise level only, the optimal convergence order is achieved.

*Appeared in*

- Inverse Problems, 11 (1995), pp. 65--77

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# The long term behavior of a Stochastic PDE.

*Authors*

- Tribe, Roger

*DOI*

*Abstract*

The one-dimensional heat equation driven by a particular white noise term is studied. From initial conditions with compact support, solutions retain this compact support and die out in finite time. The long term behavior of solutions from certain initial conditions can be described by a system of wavefronts whose positions move approximately as Brownian motions and such that two wavefronts annihilate when they collide.

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# Stability of weak numerical schemes for stochastic differential equations.

*Authors*

- Hofmann, N.
- Platen, Eckhard

*2010 Mathematics Subject Classification*

- 60H10

*Keywords*

- Numerical stability, stochastic differential equations, weak numerical schemes, implicit schemes, regions of stability

*DOI*

*Abstract*

The paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes the regions of stability are also examined.

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# Pricing via anticipative stochastic calculus.

*Authors*

- Platen, Eckhard
- Rebolledo, Rolando

*2010 Mathematics Subject Classification*

- 90A09 60G35 60H10

*Keywords*

- Pricing, derivative securities, bonds, anticipative linear stochastic equations

*DOI*

*Abstract*

The paper proposes a general model for pricing of derivative securities with different maturity. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are studied.

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# An approach to bond pricing.

*Authors*

- Platen, Eckhard

*2010 Mathematics Subject Classification*

- 60H10

*Keywords*

- Bond pricing, stochastic differential equations, nonlinear partial differential equations, term structure of interest rates

*DOI*

*Abstract*

The paper proposes a simple arbitrage free approach for modelling bond prices. A natural structure of the volatility and expected return premium of bond price processes is directly obtained.

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# On the functional equations of dynamical theta functions I.

*Authors*

- Juhl, Andreas

ORCID: 0000-0002-0097-9760

*2010 Mathematics Subject Classification*

- 11F72 58F06 58F17 58F18 58F20

*Keywords*

- Selberg zeta function, functional equation, dynamical theta function, duality

*DOI*

*Abstract*

The twisted geodesic flow of compact locally symmetric spaces of rank one gives rise to a series of meromorphic functions on the complex plane satisfying simple functional equations. These results are discussed as part of geometric quantization and index theory.

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# On bond price dynamics.

*Authors*

- Platen, Eckhard
- Rebolledo, R.

*2010 Mathematics Subject Classification*

- 90A09 60G35 60H10

*Keywords*

- Pricing, bonds, stochastic differential equations, martingales

*DOI*

*Abstract*

This article proposes a new approach to bond price dynamics. By means of exponential formulae and a notion of forward derivatives we construct a general theoretical framework, which allows to include most of known bond price properties. In particular, we perform a new analysis of no arbitrage conditions together with their consequences on the corresponding return premium. An expression for the general bond price is obtained which also turns out to be computationally convenient. Finally, we specify our result in a general multifactor bond pricing model.

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# Wavelet approximation methods for pseudodifferential equations II: matrix compression and fast solution

*Authors*

- Dahmen, W.
- Prössdorf, Siegfried
- Schneider, R.

*2010 Mathematics Subject Classification*

- 65F35 65J10 65N30 65N35 65R20 47A20 47G30 45P05 41A25

*Keywords*

- Periodic pseudodifferential equations, pre-wavelets, biorthogonal wavelets, generalized Petrov-Galerkin schemes, wavelet representation, atomic decomposition, Calderón-Zygmund operators, matrix compression, error analysis

*DOI*

*Abstract*

This is the second part of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in ℝ^{n}. This setting covers classical Galerkin methods, collocation, and quasiinterpolation. The numerical methods are based on a general framework of multiresolution analysis, i.e., of sequences of nested spaces which are generated by refinable functions. In this part we analyse compression techniques for the resulting stiffness matrices relative to wavelet type bases. We will show that, although these stiffness matrices are generally not sparse, the order of the overall computational work which is needed to realize a certain accuracy is of the form O(N(logN)^{b}) where N is the number of unknowns and b ≥ 1 is some real number.

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# On the smoothness of the solution to a boundary value problem for a differential-difference equation.

*Authors*

- Ivanova, I. P.
- Kamenskij, G. A.

*DOI*

*Abstract*

This paper deals with the first boundary value problem (BVP) for equations which are a differential with respect to one variable (*t*) and difference with respect to the other variable (*s*) in a bounded domain. The initial value problem for differential-difference equations of this type was studied in [1], [2]. The theory of the BVP under investigation is connected with the theory of the BVP for strongly elliptic differential-difference equations which are difference and differential with respect to the same variable (see [3]). Some questions of this work were considered earlier in the papers [4], [5], [8]. Section 1 considers the solvability of the BVP for differential-difference equations. In contrast to differential equations the smoothness of the generalized solutions can be broken in the domain Q and is preserved only in some subdomains Q_{r} ⊂ Q where (∪_{r}Q̅_{r} = Q̅). In section 2 we construct such a set of Q_{r}. Section 3 deals with the smoothness of generalized solutions in the subdomains Q_{r}. Section 4 considers the conditions under which the smoothness is preserved when passing the boundaries between neighboring subdomains Q_{r}.

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# An optimal order collocation method for first kind boundary integral equations on polygons.

*Authors*

- Elschner, Johannes
- Graham, I. G.

*2010 Mathematics Subject Classification*

- 65R20 45B05

*Keywords*

- First kind boundary integral equations, collocation, nonlinear parametrization, Mellin convolution method

*DOI*

*Abstract*

This paper discusses the convergence of the collocation method using splines of any order *k* for first kind integral equations with logarithmic kernels on closed polygonal boundaries in ℝ^{2}. Before discretization the equation is transformed to an equivalent equation over [-π,π] using a nonlinear parametrization of the polygon which varies more slowly than arc-length near each corner. This has the effect of producing a transformed equation with a solution which is smooth on [-π,π]. This latter integral equation is shown to be well-posed in appropriate Sobolev spaces. The structure of the integral operator is described in detail, and can be written in terms of certain non-standard Mellin convolution operators. Using this information we are able to show that the collocation method using splines of order *k* (degree *k*-1) converges with optimal order O(h^{k}). (The collocation points are the midpoints of subintervals when *k* is odd and the break-points when *k* is even, and stability is shown under the assumption that the method may be modified slightly.) Using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm. Finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method. The convergence results are illustrated with some numerical examples.

*Appeared in*

- Numer. Math., 70 (1995), pp. 1--31

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# Self normal numbers.

*Authors*

- Schmeling, Jörg

ORCID: 0000-0001-6956-9463

*DOI*

*Abstract*

This paper is a continuation of [1] where we considered the link between expansions to a real base β and ergodic properties of the corresponding β-shift. We found there a certain gap in the hierarchy of the sizes in the classification proposed by F. Blanchard. The aim of this paper is to fill in this gap. We will follow the notations in [1].

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# A multiscale method for the double layer potential equation on a polyhedron.

*Authors*

- Dahmen, Wolfgang
- Kleemann, Bernd
- Prößdorf, Siegfried
- Schneider, Reinhold

*2010 Mathematics Subject Classification*

- 65N38 65N35 65N30 31B10 35J05

*Keywords*

- boundary element method, boundary integral equations, Laplace equation, exterior domain problem, unbounded domains, double layer potential equation, polyhedra; collocation, multiscale decompositions, linear finite element spaces, sparse matrices, algorithm, numerical experiments

*DOI*

*Abstract*

This paper is concerned with the numerical solution of the double layer potential equation on polyhedra. Specifically, we consider collocation schemes based on multiscale decompositions of piecewise linear finite element spaces defined on polyhedra. An essential difficulty is that the resulting linear systems are not sparse. However, for uniform grids and periodic problems one can show that the use of multiscale bases gives rise to matrices that can be well approximated by sparse matrices in such a way that the solutions to the perturbed equations exhibits still sufficient accuracy. Our objective is to explore to what extent the presence of corners and edges in the domain as well as the lack of uniform discretizations affects the performance of such schemes. Here we propose a concrete algorithm, describe its ingredients, discuss some consequences, future perspectives, and open questions, and present the results of numerical experiments for several test domains including non-convex domains.

*Appeared in*

- Advances in Computational Mathematics (H.P. Dikshit and C.A. Micchelli, eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, pp. 15--57, 1994

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# Rothe's method for equations modelling transport of dopants in semiconductors.

*Authors*

- Glitzky, Annegret

ORCID: 0000-0003-1995-5491 - Gröger, Konrad
- Hünlich, Rolf

*2010 Mathematics Subject Classification*

- 35A40 35B40 35B45 35D05 35D10 35K57 65M99

*Keywords*

- Transport of dopants in semiconductors, reaction-diffusion equations, Lyapunov function, a-priori estimates, global existence, asymptotic behaviour, discrete-time problems, convergence of Rothe's method, implicite and semi-implicite scheme

*DOI*

*Abstract*

This paper is devoted to the investigation of some nonlinear reaction-diffusion system modelling the transport of dopants in semiconductors and arising in semiconductor technology. Besides of results on existence and qualitative properties of the solution to the problem itself we are interested in the investigation of corresponding discrete-time problems. Using Rothe's method in a fully implicite and a semi-implicite version, respectively, we get analogous results on existence and qualitative behaviour of solutions to the discrete-time equations. Moreover, convergence in some strong sense will be proved. Essential tools are estimates of the energy functional, L^{∞}-estimates obtained by De Giorgi's method, L^{q} (S,W^{1,P} )-estimates for the continuous problem as well as a discrete version of Gronwall's lemma.

*Appeared in*

- Nonlinear analysis, 28 (1997), pp. 463-487

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# Autoregression approximation of a nonparametric diffusion model

*Authors*

- Milstein, Grigori N.
- Nussbaum, Michael

*2010 Mathematics Subject Classification*

- 62M10 62G07 62B15 60J60 60H10

*Keywords*

- Nonparametric experiments, deficiency distance, likelihood ratio process, stochastic differential equation, autoregression, diffusion sampling, asymptotic sufficiency

*DOI*

*Abstract*

We consider a model of small diffusion type where the function which governs the drift term varies in a nonparametric set. We investigate discrete versions of this continuous model with respect to statistical equivalence, in the sense of the asymptotic theory of experiments. It is shown that an Euler difference scheme as a discrete version of the stochastic differential equation is asymptotically equivalent in the sense of Le Cam's deficiency distance, when the discretization step decreases with the noise intensity ∈. We thus obtain a nonparametric version of diffusion limit results for autoregression. It follows that in the continuous diffusion model, discrete sampling on a uniform grid is asymptotically sufficient. The key technical step utilizes the notion of Hellinger process from semimartingale theory.

*Appeared in*

- Prob. Th. rel. Fields vol. 112 (1998) no. 4 pp. 535-543 under new title: Diffusion approximation for nonparametric autoregression.

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# Option pricing under incompleteness and stochastic volatility.

*Authors*

- Hofmann, N.
- Platen, Eckhard
- Schweizer, M.

*2010 Mathematics Subject Classification*

- 60H10

*Keywords*

- stochastic differential equations, option pricing, incompleteness, stochastic volatility, numerical solution

*DOI*

*Abstract*

We consider a very general diffusion model for asset prices which allows the description of stochastic and past-dependent volatilities. Since this model typically yields an incomplete market, we show that for the purpose of pricing options, a small investor should use the minimal equivalent martingale measure associated to the underlying stock price process. Then we present stochastic numerical methods permitting the explicit computation of option prices and hedging strategies, and we illustrate our approach by specific examples.

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# Numerical solution of hierarchically structured systems of algebraic-differential equations.

*Authors*

- Grund, Friedrich

*2010 Mathematics Subject Classification*

- 65Y05 65L05 65H10 65F50 68Q35

*Keywords*

- Algebraic-differential equations, systems of structured nonlinear equations, systems of linear sparse equations, parallelization of numerkal methods

*DOI*

*Abstract*

We consider hierarchically structured systems of algebraic-differential equations. Numerical methods for their solution are described. Parall methods are discussed.

*Appeared in*

- R. Bank, R. Burlisch, H. Gajewski and K. Merten, editors: Mathematical Modelling and Simulation of Electrical Circuits and Semiconductor Devices, volume 111 of International Series of Numerical Mathematics, pages 17-31. Birkhaeuser Verlag Basel, 1994.

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# A travelling wave solution to the Kolmogorov equation with noise.

*Authors*

- Tribe, Roger

*Keywords*

- Stochastic PDE, Kolmogorov equation, travelling wave

*DOI*

*Abstract*

We consider the one-dimensional Kolmogorov equation driven by a particular space-time white noise term and show that there exist stochastic wavelike solutions which travel with a linear limiting speed.

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# On the computation of hyperbolic sets and their invariant manifolds.

*Authors*

- Homburg, Ale Jan

*DOI*

*Abstract*

We describe a method for finding periodic orbits contained in a hyperbolic invariant set and of constructing their local stable and unstable manifolds, suitable to implement in a computer.

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# A rigorous renormalization group method for interfaces in random media.

*Authors*

- Bovier, Anton
- Külske, Christof

*2010 Mathematics Subject Classification*

- 60K40 82B44 82C22 60K35

*Keywords*

- Disordered systems, interfaces, SOS-model, renormalization group, contour models

*DOI*

*Abstract*

We prove the existence Gibbs states describing rigid interfaces in a disordered solid-on-solid (SOS) for low temperatures and for weak disorder in dimension D ≥ 4. This extends earlier results for hierarchical models to the more realistic models and proves a long-standing conjecture. The proof is based on the renormalization group method of Bricmont and Kupiainen originally developed for the analysis of low-temperature phases of the random field Ising model. In a broader context, we generalize this method to a class of systems with non-compact single-site state space.

*Appeared in*

- Rev. Math. Phys., 6 (1994), pp. 413--496

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# On Stationary Schrödinger-Poisson Equations

*Authors*

- Kaiser, Hans-Christoph
- Rehberg, Joachim
- Albinus, Günther

*2010 Mathematics Subject Classification*

- 35J05 35P15 47H05

*Keywords*

- Stationäry Schrödinger-Poisson system, monotone potential operators, iterative methods, discretization of the Schrödinger-Poisson system, electron gas with reduced dimension, nanoelectronics

*DOI*

*Abstract*

We regard the Schrödinger-Poisson system arising from the modelling of an electron gas with reduced dimension in a bounded up to three-dimensional domain and establish the method of steepest descent. The electrostatic potentials of the iteration scheme will converge uniformly on the spatial domain. To get this result we investigate the Schrödinger operator, the Fermi level and the quantum mechanical electron density operator for square integrable electrostatic potentials. On bounded sets of potentials the Fermi level is continuous and boundeq, and the electron density operator is monotone and Lipschitz continuous. - As a tool we develop a Riesz-Dunford functional calculus for semibounded self-adjoint operators using paths of integration which enclose a real half axis.

*Appeared in*

- Math. Methods Appl. Sci., 20 (1997), pp. 1283-1312

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# Rigorous results on the Hopfield model of neural networks.

*Authors*

- Bovier, Anton
- Gayrard, Véronique

*2010 Mathematics Subject Classification*

- 92B20 82C32

*Keywords*

- disordered systems, memory capacity, thermodynamic properties, Hopfield model, Curie-Weiss model, large deviation techniques, random interactions, thermodynamic limit, almost sure convergence of the free energy, local minima, Hamiltonian, dilute random graphs

*DOI*

*Abstract*

We review some recent rigorous results in the theory of neural networks, and in particular on the thermodynamic properties of the Hopfield model. In this context, the model is treated as a Curie-Weiss model with random interactions and large deviation techniques are applied. The tractability of the random interactions depends strongly on how the number, M, of stored patterns scales with the size, N, of the system. We present an exact analysis of the thermodynamic limit under the sole condition that M / N ↓ 0, as N ↑ ∞, i.e. we prove the almost sure convergence of the free energy to a non-random limit and the a.s. convergence of the measures induced on the overlap parameters. We also present results on the structure of local minima of the Hopfield Hamiltonian, originally derived by Newman. All these results are extended to the Hopfield model defined on dilute random graphs.

*Appeared in*

- Resenhas do IME-USP, 1 (1994), pp. 161--172

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# Convergence of particle schemes for the Boltzmann equation.

*Authors*

- Pulvirenti, M.
- Wagner, Wolfgang
- Zavelani Rossi, M.B.

*2010 Mathematics Subject Classification*

- 60K35 76P05 82C40

*Keywords*

- Stochastic particle schemes, Boltzmann equation, rate of convergence

*DOI*

*Abstract*

We show the convergence of a certain family of Markov chains, defined on the state space of a N-particle system (as the Bird's method), to the solutions of the (regularized) Boltzmann equation.

*Appeared in*

- Eur. J. Mech. B/Fluids, 13 (1994), pp. 339--351

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# Zur mathematischen Modellierung eines Halbleiterinjektionslasers mit Hilfe der Maxwellschen Gleichungen bei gegebener Stromverteilung.

*Authors*

- Förste, Joachim

*2010 Mathematics Subject Classification*

- 35Q60 78A60

*DOI*

*Abstract*

Zur mathematischen Beschreibung eines Halbleiterinjektionslasers wird ein System partieller Differentialgleichungen vorgeschlagen, das aus den Maxwellschen Gleichungen und der Ladungstragerbilanz in der aktiven Zone besteht. Es zeigt sich, daß man für dieses System ein vernünftig gestelltes Rand-Anfangswertproblem formulieren kann.

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