Solving a class of multivariate integration problems via Laplace techniques

Lasserre, Jean-Bernard; Zeron, Eduardo S.

doi:10.4064/am28-4-2

Cited by 23 publications

(25 citation statements)

References 7 publications

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“…It is the same speed with which the truncated (Gamma) integral y 0 exp(−z)z (n+p)/d−1 dz converges to Γ((n + p)/d). (b) We also provide an alternative and simple proof based on Laplace transform (in the spirit of Lasserre and Zeron [13] to provide 1 The Gamma function Γ : R → R is defined by a → Γ(a) := ∞ 0 t a−1 exp(−t)dt, for every a > −1.…”

Section: Introductionmentioning

confidence: 99%

Level Sets and NonGaussian Integrals of Positively Homogeneous Functions

Lasserre

2015

Int. Game Theory Rev.

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We investigate various properties of the sublevel set {x : g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions. For instance, the latter integral reduces to integrating h exp(−g) on the whole space R n (a non Gaussian integral) and when g is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of g. In fact, whenever h is nonnegative, the functional φ(g(x))h(x)dx is a convex function of g for a large class of functions φ : R+ → R. We also provide a numerical approximation scheme to compute the volume or integrate h (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set {x : g(x) ≤ 1} of minimum volume that contains some given subset K is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.1991 Mathematics Subject Classification. 26B15 65K10 90C22 90C25.

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Section: Introductionmentioning

confidence: 99%

Level Sets and NonGaussian Integrals of Positively Homogeneous Functions

Lasserre

2015

Int. Game Theory Rev.

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“…Observe that since g d ≤ λ min for all d, As ε > 0 was arbitrary and G d ⊂ P, we obtain the desired result (12). The proof of (13) is similar.…”

Section: Proof Of Theoremmentioning

confidence: 54%

Inner Approximations for Polynomial Matrix Inequalities and Robust Stability Regions

Henrion

Lasserre

2012

IEEE Trans. Automat. Contr.

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Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modelled by a single polynomial superlevel set. Those inner approximations converge in a well-defined analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations.

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“…• It has the form of a product between an exponential term and a convergent power series with positive terms. The series is obtained by use of Laplace transform techniques [11]. The explicit form of the linear recurrence satisfied by the coefficients of the series is derived using properties of D-finite functions i.e., solutions of linear ordinary differential equations with polynomial coefficients [12,13].…”

Section: Contributionmentioning

confidence: 99%

Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

Serra

Arzelier

Joldeş

et al. 2016

Journal of Guidance, Control, and Dynamics

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International audienceThis article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature

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Solving a class of multivariate integration problems via Laplace techniques

Cited by 23 publications

References 7 publications

Level Sets and NonGaussian Integrals of Positively Homogeneous Functions

Level Sets and NonGaussian Integrals of Positively Homogeneous Functions

Inner Approximations for Polynomial Matrix Inequalities and Robust Stability Regions

Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

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