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the previous summation expressions become algorithms. As discussed in sec- tion 4.4.1 we only consider A ∼ Beta(a, b, [l , r]). For A ∼ Beta(a, b, [0, 1]), the values μk are known analytically [Zwillinger et al., 1996]: b+k k b+j μk−1 = ∏ , j=1 a+b+j+1 μˆj ≡μj(A)whereA∼Beta(a,b,[0,1]). For A ∼ Beta(a, b, [l , r]), ((r−l)τ+l)kρ(0,1) Beta(a,b) μ0 = 1, μk = k ≥ 1. μk(A) = μ(0,k) j j−i μ(i,j) = ∑( m=i m−i 18 The implementation is not straightforward, though it is cor- rect. It follows from organizing the moments into a matrix ⎡⎢ μ(0,0) μ(0,1) ... μ(0,k) ⎤⎥ ⎢ μ(1,1) ... μ(1,k) ⎥ ⎢ ⋱⋮⎥ ⎣ μ(k,k) ⎦ and filling in the entries μ(0,1), . . . , μ(0,k) from the ini- tially specified diagonal. At every step in the implementation, we compute a new diagonal. ζkρ(l,r) (ζ)dζ = r 1 E[Ak]= = ∑ ( ) μˆ j ( r − l ) l Beta(a,b) l0 kk jk−j j=0 j (4.28) and we can compute the moments of A ∼ Beta(a, b, [l , r]) by scaling and shifting those of A ∼ Beta(a, b, [0, 1]). Program 6 gives a simple implementa- tion of the path damping algorithms and an implementation of the recursion (4.29) )μˆm(r−l)m−ilj−m =(r−l)μ(i,j−1) +lμ(i+1,j) to compute the moments μ (A).18 k 4.6.4 Gaussian quadrature Integration and interpolation are the fundamental concepts behind the class of uncertainty quantification techniques known as stochastic colloca- tion [Xiu and Hesthaven, 2005], which were originally developed in the con- text of partial differential equation models with stochastic inputs. Much of the work in this context has focused on the problem of high-dimensional pa- rameter spaces [Nobile et al., 2008], where multi-dimensional interpolation and integration can have a computational cost that increases exponentially with the dimension of the parameter space; one aspect of the so-called curse of dimensionality. 4.6 ⋅ algorithms 83 a+b+k+1 To handle the general case, define (4.26) (4.27) (τ)dτPDF Image | Instagram Cheat Sheet
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