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- ItemTetGen: A quality tetrahedral mesh generator and a 3D Delaunay triangulator (Version 1.5 — User’s Manual)(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Si, HangTetGen is a software for tetrahedral mesh generation. Its goal is to generate good quality tetrahedral meshes suitable for numerical methods and scientific computing. It can be used as either a standalone program or a library component integrated in other software. The purpose of this document is to give a brief explanation of the kind of tetrahedralizations and meshing problems handled by TetGen and to give a fairly detailed documentation about the usage of the program. Readers will learn how to create tetrahedral meshes using input files from the command line. Furthermore, the programming interface for calling TetGen from other programs is explained.
- ItemAnalysis and simulation of multifrequency induction hardening(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hömberg, Dietmar; Petzold, Thomas; Rocca, ElisabettaWe study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwells equations coupled with an internal energy balance and an ODE for the volume fraction of austenite, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable a-priori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system.
- ItemMesh smoothing: An MMPDE approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Huang, Weizhang; Kamenski, Lennard; Si, HangWe study a mesh smoothing algorithm based on the moving mesh PDE (MMPDE) method. For the MMPDE itself, we employ a simple and efficient direct geometric discretization of the underlying meshing functional on simplicial meshes. The nodal mesh velocities can be expressed in a simple, analytical matrix form, which makes the implementation of the method relatively easy and simple. Numerical examples are provided.
- ItemOn the geometry of regular maps from a quasi-projective surface to a curve(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Parameswaran, A.J.; Tibar, M.We explore consequences of the triviality of the monodromy group, using the condition of purity of the mixed Hodge structure on the cohomology of the surface X.
- ItemObstacle mean-field game problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Gomes, Diogo A.; Patrizi, StefaniaIn this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.
- ItemFrom large deviations to Wasserstein gradient flows in multiple dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Erbar, Matthias; Maas, Jan; Renger, D.R. MichielWe study the large deviation rate functional for the empirical measure of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of -convergence) to the JordanKinderlehrerOtto functional arising in the Wasserstein gradient flow structure of the FokkerPlanck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.
- ItemA functional limit theorem for limit order books(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Bayer, Christian; Horst, Ulrich; Qiu, JinniaoWe consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tend to zero. The limits of the standing buy and sell volume densities are described by two linear stochastic partial differential equations, which are coupled with a two-dimensional reflected Brownian motion that is the limit of the best bid and ask price processes.
- ItemAdaptive behaviour in a predator-prey model leads to multiple equilibrium states(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Pimenov, Alexander; Korobeinikov, Andrei; Rachinskii, DmitriiThere is evidence that multiple stable equilibrium states are possible in real-life ecological systems. In order to verify a hypothesis that such a multitude of equilibrium states can be caused by adapting of animal behaviour to changes of environmental conditions, we consider a simple predator-prey model where prey changes a mode of behaviour in response to the pressure of predation. This model exhibits two stable coexisting equilibrium states with basins of attraction separated by a separatrix of a saddle point.
- ItemHölder-estimates for non-autonomous parabolic problems with rough data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Meinlschmidt, Hannes; Rehberg, JoachimIn this paper we establish Hölder estimates for solutions to non-autonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations.
- ItemBootstrap confidence sets under a model misspecification(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Spokoiny, Vladimir; Zhilova, MayyaA multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and possible model misspecification. Theoretical results justify the bootstrap consistency for small or moderate sample size and allow to control the impact of the parameter dimension: the bootstrap approximation works if the ratio of cube of the parameter dimension to the sample size is small. The main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called Small Modeling Bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modeling bias. We illustrate the results with numerical examples of misspecified constant and logistic regressions.