Computing derivative-based global sensitivity measures using polynomial chaos expansions

In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol' indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The {\em derivative-based global sensitivity measures} (DGSM) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

We consider time series data modeled by ordinary differential equations (ODEs), widespread models in physics, chemistry, biology and science in general. The sensitivity analysis of such dynamical systems usually requires calculation of various derivatives with respect to the model parameters. We employ the adjoint state method (ASM) for efficient computation of the first and the second derivatives of likelihood functionals constrained by ODEs with respect to the parameters of the underlying ODE model. Essentially, the gradient can be computed with a cost (measured by model evaluations) that is independent of the number of the ODE model parameters and the Hessian with a linear cost in the number of the parameters instead of the quadratic one. The sensitivity analysis becomes feasible even if the parametric space is high-dimensional. The main contributions are derivation and rigorous analysis of the ASM in the statistical context, when the discrete data are coupled with the continuous ODE model. Further, we present a highly optimized implementation of the results and its benchmarks on a number of problems. The results are directly applicable in (e.g.) maximum-likelihood estimation or Bayesian sampling of ODE based statistical models, allowing for faster, more stable estimation of parameters of the underlying ODE model.Comment: 5 figure

Derivative based global sensitivity measures

The method of derivative based global sensitivity measures (DGSM) has recently become popular among practitioners. It has a strong link with the Morris screening method and Sobol' sensitivity indices and has several advantages over them. DGSM are very easy to implement and evaluate numerically. The computational time required for numerical evaluation of DGSM is generally much lower than that for estimation of Sobol' sensitivity indices. This paper presents a survey of recent advances in DGSM concerning lower and upper bounds on the values of Sobol' total sensitivity indices $S\_{i}^{tot}$. Using these bounds it is possible in most cases to get a good practical estimation of the values of $S\_{i}^{tot}$. Several examples are used to illustrate an application of DGSM

Monte Carlo evaluation of sensitivities in computational finance

In computational finance, Monte Carlo simulation is used to compute the correct prices for financial options. More important, however, is the ability to compute the so-called "Greeks'', the first and second order derivatives of the prices with respect to input parameters such as the current asset price, interest rate and level of volatility.\ud \ud This paper discusses the three main approaches to computing Greeks: finite difference, likelihood ratio method (LRM) and pathwise sensitivity calculation. The last of these has an adjoint implementation with a computational cost which is independent of the number of first derivatives to be calculated. We explain how the practical development of adjoint codes is greatly assisted by using Algorithmic Differentiation, and in particular discuss the performance achieved by the FADBAD++ software package which is based on templates and operator overloading within C++.\ud \ud The pathwise approach is not applicable when the financial payoff function is not differentiable, and even when the payoff is differentiable, the use of scripting in real-world implementations means it can be very difficult in practice to evaluate the derivative of very complex financial products. A new idea is presented to address these limitations by combining the adjoint pathwise approach for the stochastic path evolution with LRM for the payoff evaluation