Density Matrix Renormalization for Model Reduction in Nonlinear Dynamics

We present a novel approach for model reduction of nonlinear dynamical systems based on proper orthogonal decomposition (POD). Our method, derived from Density Matrix Renormalization Group (DMRG), provides a significant reduction in computational effort for the calculation of the reduced system, compared to a POD. The efficiency of the algorithm is tested on the one dimensional Burgers equations and a one dimensional equation of the Fisher type as nonlinear model systems.Comment: 12 pages, 12 figure

Polynomial-Chaos-based Kriging

Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. To cope with demanding analysis such as optimization and reliability, surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade. Polynomial Chaos Expansions (PCE) and Kriging are two popular non-intrusive meta-modelling techniques. PCE surrogates the computational model with a series of orthonormal polynomials in the input variables where polynomials are chosen in coherency with the probability distributions of those input variables. On the other hand, Kriging assumes that the computer model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer runs, i.e. input vectors and response values. These two techniques have been developed more or less in parallel so far with little interaction between the researchers in the two fields. In this paper, PC-Kriging is derived as a new non-intrusive meta-modeling approach combining PCE and Kriging. A sparse set of orthonormal polynomials (PCE) approximates the global behavior of the computational model whereas Kriging manages the local variability of the model output. An adaptive algorithm similar to the least angle regression algorithm determines the optimal sparse set of polynomials. PC-Kriging is validated on various benchmark analytical functions which are easy to sample for reference results. From the numerical investigations it is concluded that PC-Kriging performs better than or at least as good as the two distinct meta-modeling techniques. A larger gain in accuracy is obtained when the experimental design has a limited size, which is an asset when dealing with demanding computational models

Statistical Inference for Time-changed Brownian Motion Credit Risk Models

We consider structural credit modeling in the important special case where the log-leverage ratio of the firm is a time-changed Brownian motion (TCBM) with the time-change taken to be an independent increasing process. Following the approach of Black and Cox, one defines the time of default to be the first passage time for the log-leverage ratio to cross the level zero. Rather than adopt the classical notion of first passage, with its associated numerical challenges, we accept an alternative notion applicable for TCBMs called "first passage of the second kind". We demonstrate how statistical inference can be efficiently implemented in this new class of models. This allows us to compare the performance of two versions of TCBMs, the variance gamma (VG) model and the exponential jump model (EXP), to the Black-Cox model. When applied to a 4.5 year long data set of weekly credit default swap (CDS) quotes for Ford Motor Co, the conclusion is that the two TCBM models, with essentially one extra parameter, can significantly outperform the classic Black-Cox model.Comment: 21 pages, 3 figures, 2 table

Infrared properties of propagators in Landau-gauge pure Yang-Mills theory at finite temperature

The finite-temperature behavior of gluon and of Faddeev-Popov-ghost propagators is investigated for pure SU(2) Yang-Mills theory in Landau gauge. We present nonperturbative results, obtained using lattice simulations and Dyson-Schwinger equations. Possible limitations of these two approaches, such as finite-volume effects and truncation artifacts, are extensively discussed. Both methods suggest a very different temperature dependence for the magnetic sector when compared to the electric one. In particular, a clear thermodynamic transition seems to affect only the electric sector. These results imply in particular the confinement of transverse gluons at all temperatures and they can be understood inside the framework of the so-called Gribov-Zwanziger scenario of confinement.Comment: 25 pages, 14 figures, 2 tables, minor changes of typographical and design character, some minor errors corrected, version to appear in PR

New, efficient, and accurate high order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions

We construct new, efficient, and accurate high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators' spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions.Comment: 16 pages, 9 figures. The files with the coefficients for the derivative and dissipation operators can be accessed by downloading the source code for the document. The files are located in the "coeffs" subdirector