Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of the continuous-time optimal control problem. Us...

This paper presents some explicit results concerning an extension of the mechanical quadrature technique, namely, the Gauss-Jacobi numerical integration scheme, to the class of integrals whose kernels exhibit second order of singularity (i.e., one degree more singular than Cauchy). In order to ascribe numerical values to these integrals they must be understood in Hadamard's finite-part sense. T...

Algorithms that combat the curse of dimensionality take advantage of nonuniformity properties of the underlying functions, which may be rotational (e.g., grid alignment) or translational (e.g., near-singularities localized at certain points of the domain). The significance of such effects is explored for four different classes of algorithms: low-rank compression, quasi-Monte Carlo integration, ...

This paper presents a new univariate decomposition method for design sensitivity analysis and reliabilitybased design optimization of mechanical systems subject to uncertain performance functions in constraints. The method involves a novel univariate approximation of a general multivariate function in the rotated Gaussian space for reliability analysis, analytical sensitivity of failure probabi...

This paper presents a new and alternative univariate method for predicting component reliability of mechanical systems subject to random loads, material properties, and geometry. The method involves novel function decomposition at a most probable point that facilitates the univariate approximation of a general multivariate function in the rotated Gaussian space and one-dimensional integrations ...