Controlling and understanding chemical-physical phenomena with sensitivity analysis

H. Rabitz
Princeton University, USA
hrabitz@princeton.edu

Random Sampling-High Dimensional Model Representation (RS-HDMR) is a practical approach to HDMR that can provide efficient global sensitivity analysis of complex nonlinear systems with independent or correlated probability density functions of inputs. HDMR expresses the model output f(x) as a finite hierarchical correlated function expansion in terms of the input variables (x1, x2, … , xn): . The HDMR component functions are optimally constructed from a set of generated/measured data for each specific function f(x). The individual RS-HDMR component functions also have a direct statistical correlation interpretation, which permits the model output variance, σ2, in a desired input domain to be decomposed into its input contributions (), due to the independent variable action, the pair correlation action , etc., where , , … are defined as the covariances of fi(xi), fij(xi; xj ), … with f(x), respectively. The global sensitivity indexes, Si, Sij, … are defined as , , ….The magnitudes of Si, Sij, … may be used to quantitatively identify the important inputs and input pairs, triples, etc. After deducing the component functions, the HDMR expansion can be used to explore the input  output relationships of the particular systems. To reduce sampling effort, the RS-HDMR component functions are approximately represented by weighted optimal orthonormal polynomial expansions. The orthonormal polynomial expansion coefficients of RS-HDMR are sequentially determined by least-squares regression. The probability density function of inputs is involved only implicitly through sampling, and the inputs can have an arbitrary independent or correlated probability density function. This feature is especially beneficial for real systems whose input probability density functions are often correlated and unknown. RS-HDMR technique will be illustrated with several examples.