The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

/pdf/the-ensemble-kalman-filter-theoretical-formulation-and-2xx6out3z9.pdf

The Ensemble Kalman Filter: theoretical formulation and practical implementation

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

/pdf/the-askey-scheme-of-hypergeometric-orthogonal-polynomials-46afheftah.pdf

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

http://cds.cern.ch/record/298578/files/SCAN-9603159.pdf

Review of mathematics, numerical factors, and corrections for dark matter experiments based on elastic nuclear recoil

In this paper, we examine the pricing of European call options on stocks that have vari? ance rates that change randomly. We study continuous time diffusion processes for the stock return and the standard deviation parameter, and we find that one must use the stock and two options to form a riskless hedge. The riskless hedge does not lead to a unique option pricing function because the random standard deviation is not a traded security. One must appeal to an equilibrium asset pricing model to derive a unique option pricing function. In general, the option price depends on the risk premium associated with the random standard deviation. We find that the problem can be simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with the stock return. The resulting solution is an integral ofthe Black-Scholes formula and the distribution function for the variance of the stock price. We show that accurate option prices can be computed via Monte Carlo simulations and we apply the model to a set of actual prices.

/pdf/option-pricing-when-the-variance-changes-randomly-theory-1000m1wpic.pdf

I. S Gradshteĭn

Papers

Table of Integrals, Series and Products, Corrected and Enlarged Edition