A Chebychev propagator for inhomogeneous Schr\"odinger equations

We present a propagation scheme for time-dependent inhomogeneous Schr\"odinger equations which occur for example in optimal control theory or in reactive scattering calculations. A formal solution based on a polynomial expansion of the inhomogeneous term is derived. It is subjected to an approximation in terms of Chebychev polynomials. Different variants for the inhomogeneous propagator are demonstrated and applied to two examples from optimal control theory. Convergence behavior and numerical efficiency are analyzed.Comment: explicit description of algorithm and two appendices added version accepted by J Chem Phy

Signal and System Approximation from General Measurements

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley-Wiener space PW_\pi^1. We consider approximation processes, where the input signal is not directly used to generate the system output, but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.Comment: This paper will be published as part of the book "New Perspectives on Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's 85th birthday" in the Applied and Numerical Harmonic Analysis Series, Birkhauser (Springer-Verlag). Parts of this work have been presented at the IEEE International Conference on Acoustics, Speech, and Signal Processing 2014 (ICASSP 2014

A posteriori noise estimation in variable data sets

Most physical data sets contain a stochastic contribution produced by measurement noise or other random sources along with the signal. Usually, neither the signal nor the noise are accurately known prior to the measurement so that both have to be estimated a posteriori. We have studied a procedure to estimate the standard deviation of the stochastic contribution assuming normality and independence, requiring a sufficiently well-sampled data set to yield reliable results. This procedure is based on estimating the standard deviation in a sample of weighted sums of arbitrarily sampled data points and is identical to the so-called DER_SNR algorithm for specific parameter settings. To demonstrate the applicability of our procedure, we present applications to synthetic data, high-resolution spectra, and a large sample of space-based light curves and, finally, give guidelines to apply the procedure in situation not explicitly considered here to promote its adoption in data analysis.Comment: Accepted for publication in A&

Global smoothness estimation of a Gaussian process from regular sequence designs

We consider a real Gaussian process $X$ having a global unknown smoothness $(r_{\scriptscriptstyle 0},\beta_{\scriptscriptstyle 0})$, r_{\scriptscriptstyle 0}\in \mathds{N}_0 and $\beta_{\scriptscriptstyle 0} \in]0,1[$, with $X^{(r_{\scriptscriptstyle 0})}$ (the mean-square derivative of $X$ if $r_{\scriptscriptstyle 0}\ge 1$) supposed to be locally stationary with index $\beta_{\scriptscriptstyle 0}$. From the behavior of quadratic variations built on divided differences of $X$, we derive an estimator of $(r_{\scriptscriptstyle 0},\beta_{\scriptscriptstyle 0})$ based on - not necessarily equally spaced - observations of $X$. Various numerical studies of these estimators exhibit their properties for finite sample size and different types of processes, and are also completed by two examples of application to real data.Comment: 28 page

Nonparametric inference on L\'evy measures and copulas

In this paper nonparametric methods to assess the multivariate L\'{e}vy measure are introduced. Starting from high-frequency observations of a L\'{e}vy process $\mathbf{X}$, we construct estimators for its tail integrals and the Pareto-L\'{e}vy copula and prove weak convergence of these estimators in certain function spaces. Given n observations of increments over intervals of length $\Delta_n$, the rate of convergence is $k_n^{-1/2}$ for $k_n=n\Delta_n$ which is natural concerning inference on the L\'{e}vy measure. Besides extensions to nonequidistant sampling schemes analytic properties of the Pareto-L\'{e}vy copula which, to the best of our knowledge, have not been mentioned before in the literature are provided as well. We conclude with a short simulation study on the performance of our estimators and apply them to real data.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1116 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian non-linear large scale structure inference of the Sloan Digital Sky Survey data release 7

In this work we present the first non-linear, non-Gaussian full Bayesian large scale structure analysis of the cosmic density field conducted so far. The density inference is based on the Sloan Digital Sky Survey data release 7, which covers the northern galactic cap. We employ a novel Bayesian sampling algorithm, which enables us to explore the extremely high dimensional non-Gaussian, non-linear log-normal Poissonian posterior of the three dimensional density field conditional on the data. These techniques are efficiently implemented in the HADES computer algorithm and permit the precise recovery of poorly sampled objects and non-linear density fields. The non-linear density inference is performed on a 750 Mpc cube with roughly 3 Mpc grid-resolution, while accounting for systematic effects, introduced by survey geometry and selection function of the SDSS, and the correct treatment of a Poissonian shot noise contribution. Our high resolution results represent remarkably well the cosmic web structure of the cosmic density field. Filaments, voids and clusters are clearly visible. Further, we also conduct a dynamical web classification, and estimated the web type posterior distribution conditional on the SDSS data.Comment: 18 pages, 11 figure