Numerical Methods for Integration and Search for Minima

Abstract

Two topics are addressed. The first refers to the numerical computation of integrals and expected values of functions that may depend on a large number of random variables. Of course, integration includes the computation of the expected values of functions dependent on random variables. However, the latter shows peculiar nontrivial aspects that the former does not have. In case of a large number of random variables, the use of regular grids implies the risk of incurring the curse of dimensionality. Then, suitable sampling methods are taken into account to reduce such risk. In particular, Monte Carlo and quasi-Monte Carlo sequences are addressed. The second topic refers to the solution of the nonlinear programming problems obtained from the approximation of infinite-dimensional optimization problems by the Extended Ritz Method. We mention a few well-known direct techniques and gradient-based descent algorithms. In the case of nonlinear programming problems stated in stochastic frameworks, the stochastic approximation approach deserves attention and thus it is considered in some detail. Within this context, we describe the stochastic gradient algorithm enabling one to avoid the computation of integrals, hence, the computation of expected values of functions dependent on random variables. Convergence properties of that algorithm are reported.