We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at min...

In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced concept of matroids. It is known that both matroid can be considered as special cases Kazhdan-Lusztig-Stanley locally finite posets. framework polynomials, we study inverse functions define We also compute these boolean matroids uniform As an unexpected application obtain a ne...

Let p(x) be a hyperbolic polynomial–like function of the form p(x) = (x − r1)1 · · · (x − rN )N , where m1, . . . ,mN are given positive real numbers and r1 < r2 < · · · < rN . Let x1 < x2 < · · · < xN−1 be the N − 1 critical points of p lying in Ik = (rk, rk+1), k = 1, 2, . . . , N−1. Define the ratios σk = xk−rk rk+1−rk , k = 1, 2, . . . , N−1. We prove that mk mk+···+mN < σk < m1+···+mk m1+·...