Estimation with Numerical Integration on Sparse Grids

For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions - a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques.Estimation; Quadrature; Simulation; Mixed Logit

Estimation with Numerical Integration on Sparse Grids

For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions - a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques

Quadrature filters for one-step randomly delayed measurements

In this paper, two existing quadrature filters, viz., the Gauss–Hermite filter (GHF) and the sparse-grid Gauss–Hermite filter (SGHF) are extended to solve nonlinear filtering problems with one step randomly delayed measurements. The developed filters are applied to solve a maneuvering target tracking problem with one step randomly delayed measurements. Simulation results demonstrate the enhanced accuracy of the proposed delayed filters compared to the delayed cubature Kalman filter and delayed unscented Kalman filter

Non-intrusive uncertainty quantification using reduced cubature rules

For the purpose of uncertainty quantification with collocation, a method is proposed for generating families of one-dimensional nested quadrature rules with positive weights and symmetric nodes. This is achieved through a reduction procedure: we start with a high-degree quadrature rule with positive weights and remove nodes while preserving symmetry and positivity. This is shown to be always possible, by a lemma depending primarily on Carathéodory's theorem. The resulting one-dimensional rules can be used within a Smolyak procedure to produce sparse multi-dimensional rules, but weight positivity is lost then. As a remedy, the reduction procedure is directly applied to multi-dimensional tensor-product cubature rules. This allows to produce a family of sparse cubature rules with positive weights, competitive with Smolyak rules. Finally the positivity constraint is relaxed to allow more flexibility in the removal of nodes. This gives a second family of sparse cubature rules, in which iteratively as many nodes as possible are removed. The new quadrature and cubature rules are applied to test problems from mathematics and fluid dynamics. Their performance is compared with that of the tensor-product and standard Clenshaw–Curtis Smolyak cubature rule

A quadrature algorithm for wavelet Galerkin methods

We consider the wavelet Galerkin method for the solution of boundary integral equations of the first and second kind including integral operators of order r less than zero. This is supposed to be based on an abstract wavelet basis which spans piecewise polynomials of order dT. For example, the bases can be chosen as the basis of tensor product interval wavelets defined over a set of parametrization patches. We define and analyze a quadrature algorithm for the wavelet Galerkin method which utilizes Smolyak quadrature rules of finite order. In particular, we prove that quadrature rules of an order larger than 2dT - r are sufficient to compose a quadrature algorithm for the wavelet Galerkin scheme such that the compressed and quadrature approximated method converges with the maximal order 2dT - r and such that the number of necessary arithmetic operations is less than 풪(N log N) with N the number of degrees of freedom. For the estimates, a degree of smoothness greater or equal to 2[2dT - r]+1 is needed

On Approximate Nonlinear Gaussian Message Passing On Factor Graphs

Factor graphs have recently gained increasing attention as a unified framework for representing and constructing algorithms for signal processing, estimation, and control. One capability that does not seem to be well explored within the factor graph tool kit is the ability to handle deterministic nonlinear transformations, such as those occurring in nonlinear filtering and smoothing problems, using tabulated message passing rules. In this contribution, we provide general forward (filtering) and backward (smoothing) approximate Gaussian message passing rules for deterministic nonlinear transformation nodes in arbitrary factor graphs fulfilling a Markov property, based on numerical quadrature procedures for the forward pass and a Rauch-Tung-Striebel-type approximation of the backward pass. These message passing rules can be employed for deriving many algorithms for solving nonlinear problems using factor graphs, as is illustrated by the proposition of a nonlinear modified Bryson-Frazier (MBF) smoother based on the presented message passing rules

Smolyak Quadrature

This thesis is an introduction to the theoretical foundation and practical usage of the Smolyak quadrature rule, which is used to evaluate high-dimensional integrals over regions of Euclidean spaces. Given a sequence of univariate quadrature rules, the Smolyak construction is defined in terms of tensor products taken over the univariate rules' consecutive differences. The evaluation points of the resulting multivariate quadrature rule are distributed more sparsely than those of e.g. tensor product quadrature. It can be shown that a multivariate quadrature rule formulated in this way inherits many useful properties of the underlying sequence of univariate quadrature rules, such as the polynomial exactness. The original formulation of the Smolyak rule is prone to a copious amount of cancellation of terms in practice. This issue can be circumvented by isolating the occurrence of duplicates to a separate term, which can be computed a priori. The resulting combination method forms the basis for a numerical implementation of the Smolyak quadrature rule, which we have provided using the MATLAB scripting language. Our tests suggest that the Smolyak rule provides a competitive alternative in the realm of multidimensional integration routines saturated by the stochastic Monte Carlo method and the deterministic Quasi-Monte Carlo method. This statement is especially valid in the case of smooth integrands and it is backed by the error analysis developed in the second chapter of this thesis. The classical convergence rate is also derived for integrands of sufficient smoothness in the case of a bounded integration region. The third chapter serves as a qualitative approach to generalized sparse grid quadrature. Especially of interest is the dimension-adaptive construction. While it lacks the theoretical foundation of the Smolyak quadrature rule, it has the added benefit of adapting to the spatial structure of the integrand. A MATLAB implementation of this routine is presented vis-à-vis the Smolyak quadrature rule

Collision and re-entry analysis under aleatory and epistemic uncertainty

This paper presents an approach to the design of optimal collision avoidance and re-entry maneuvers considering different types of uncertainty in initial conditions and model parameters. The uncertainty is propagated through the dynamics, with a non-intrusive approach, based on multivariate Tchebycheff series, to form a polynomial representation of the final states. The collision probability, in the cases of precise and imprecise probability measures, is computed considering the intersection between the uncertainty region of the end states of the spacecraft and a reference sphere. The re-entry probability, instead, is computed considering the intersection between the uncertainty region of the end states of the spacecraft and the atmosphere