Programme

I.11 -- Invited, IWTCP-2

Date: Thursday, 31-07-2014

Time: 08h30 - 09h05

Handling accuracy in Bayesian automatic adaptive quadrature

Gheorghe Adam

Joint Institute for Nuclear Research, Dubna, Russian Federation

The standard automatic adaptive quadrature of a Riemann integral [1, 2] was documented to fail (see e.g., [3, 4]) under circumstances like, for instance, the occurrence of inner abscissas of the integration domain at which the integrand or its first order derivative are either singular, or present finite discontinuities, or else slow convergence under subrange subdivision entails spurious activation of a procedure for convergence acceleration.The Bayesian automatic adaptive quadrature (Baaq) was proposed by us ([5, 6] and references therein) as a means to get either reliable outputs in difficult cases, or to identify the classes of integrals the solution of which cannot be obtained by the use of quadrature sums under floating point computations. A central issue enabling the success of the Baaq approach is the accuracy handling of the floating point computations involving differences of computed integrand values. In the present report we address a number of critical issues enabling the increase of the resolving power of the quadrature sums and strengthening the reliability of the inferences following from the Bayesian analysis: (i) implementation of local quadrature rules using redundancy; (ii) fast algorithm implementation of the Clenshaw-Curtis quadrature sums by using binary tree structures relating subsets of quadrature coefficients; (iii) refinement of the classification of the monotonicity intervals resolved within the generated integrand profile; (iv) resolving the endpoints of the distinct monotonicity intervals to machine accuracy; (v) adaptive subrange subdivision at discretization abscissas defined by the endpoints of the resolved subranges during the Bayesian analysis.

Presenter: Gheorghe Adam