Stochastic collocation on unstructured multivariate meshes

Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification (UQ) community. Techniques for least-squares regularization, compressive sampling recovery, and interpolatory reconstruction are becoming standard tools used in a variety of applications. Selection of a collocation mesh is frequently a challenge, but methods that construct geometrically "unstructured" collocation meshes have shown great potential due to attractive theoretical properties and direct, simple generation and implementation. We investigate properties of these meshes, presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure

A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy-Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix

Hermite Methods for Aeroacoustics: Recent Progress

We present recent developments on Hermite methods for aeroacoustic simulations including time-stepping methods, hybridization with discontinuous Galerkin methods for handling of boundary conditions and adaptive implementations. By scaling studies reported below we show that the features unique to Hermite methods have promise to enable efficient exploitation of modern petascale architectures. We also present preliminary computations of turbulent jet noise obtained with the current implementation of our compressible Navier-Stokes solver

High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures

The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl

Function approximation for option pricing and risk management Methods, theory and applications.

PhD Thesis.This thesis investigates the application of function approximation techniques for computationally demanding problems in nance. We focus on the use of Chebyshev interpolation and its multivariate extensions. The main contribution of this thesis is the development of a new pricing method for path-dependent options. In each step of the dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. A key advantage of this approach is that it allows us to shift all modeldependent computations into a pre-computation step. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. We provide a theoretical error analysis and nd conditions that imply explicit error bounds. Numerical experiments con rm the fast convergence of prices and sensitivities. We use the new method to calculate credit exposures of European and path-dependent options for pricing and risk management. The simple structure of the Chebyshev interpolation allows for a highly e cient evaluation of the exposures. We validate the accuracy of the computed exposure pro les numerically for di erent equity products and a Bermudan swaption. Benchmarking against the least-squares Monte Carlo approach shows that our method delivers a higher accuracy in a faster runtime. We extend the method to e ciently price early-exercise options depending on several risk-factors. As an example, we consider the pricing of callable bonds in a hybrid twofactor model. We develop an e cient and stable calibration routine for the model based on our new pricing method. Moreover, we consider the pricing of early-exercise basket options in a multivariate Black-Scholes model. We propose a numerical smoothing in the dynamic programming time-stepping using the smoothing property of a Gaussian kernel. An extensive numerical convergence analysis con rms the e ciency

A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties

This work proposes a domain adaptive stochastic collocation approach for uncertainty quantification, suitable for effective handling of discontinuities or sharp variations in the random domain. The basic idea of the proposed methodology is to adaptively decompose the random domain into subdomains. Within each subdomain, a sparse grid interpolant is constructed using the classical Smolyak construction [S. Smolyak, Quadrature and interpo- lation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4 (1963) 240–243], to approximate the stochastic solution locally. The adaptive strategy is governed by the hierarchical surpluses, which are computed as part of the interpolation procedure. These hierarchical surpluses then serve as an error indicator for each subdo- main, and lead to subdivision whenever it becomes greater than a threshold value. The hierarchical surpluses also provide information about the more important dimensions, and accordingly the random elements can be split along those dimensions. The proposed adaptive approach is employed to quantify the effect of uncertainty in input parameters on the performance of micro-electromechanical systems (MEMS). Specifically, we study the effect of uncertain material properties and geometrical parameters on the pull-in behavior and actuation properties of a MEMS switch. Using the adaptive approach, we resolve the pull-in instability in MEMS switches. The results from the proposed approach are verified using Monte Carlo simulations and it is demonstrated that it computes the required statistics effectively