Genyuan Li

Global uncertainty assessments by high dimensional model representations (HDMR)

A general set of quantitative model assessment and analysis tools, termed high-dimensional model representations (HDMR), have been introduced recently for high dimensional input–output systems. HDMR are a particular family of representations where each term in the representation reflects the independent and cooperative contributions of the inputs upon the output. When data are randomly sampled, a RS(random sampling)-HDMR can be constructed, which is an efficient tool to provide a fully global statistical analysis of a model. The individual RS-HDMR component functions have a direct statistical correlation interpretation. This relation permits the model output variance σ2 to be decomposed into its input contributions σ2=∑iσi2+∑i<jσij2+⋯ due to the independent variable action σi2, the pair correlation action σij2, etc. The information gained from this decomposition can be valuable for attaining a physical understanding of the origins of output uncertainty as well as suggesting additional laboratory/field studies or model refinements to best improve the quality of the model. To reduce sampling effort, the RS-HDMR component functions are approximately represented by orthonormal polynomials. Only one randomly sampled set of input–output data is needed to determine all σi, σij, etc. and a few hundred samples may give reliable results. This paper presents its methodology and applications on an atmospheric photochemistry model and a trace metal bioremediation model.

Global Sensitivity Analysis for Systems with Independent and/or Correlated Inputs

The objective of a global sensitivity analysis is to rank the importance of the system inputs considering their uncertainty and the influence they have upon the uncertainty of the system output, typically over a large region of input space. This paper introduces a new unified framework of global sensitivity analysis for systems whose input probability distributions are independent and/or correlated. The new treatment is based on covariance decomposition of the unconditional variance of the output. The treatment can be applied to mathematical models, as well as to measured laboratory and field data. When the input probability distribution is correlated, three sensitivity indices give a full description, respectively, of the total, structural (reflecting the system structure) and correlative (reflecting the correlated input probability distribution) contributions for an input or a subset of inputs. The magnitudes of all three indices need to be considered in order to quantitatively determine the relative importance of the inputs acting either independently or collectively. For independent inputs, these indices reduce to a single index consistent with previous variance-based methods. The estimation of the sensitivity indices is based on a meta-modeling approach, specifically on the random sampling-high dimensional model representation (RS-HDMR). This approach is especially useful for the treatment of laboratory and field data where the input sampling is often uncontrolled.

High dimensional model representations generated from low dimensional data samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x=|x1,x2,...,xn} with n∼102 or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.

A general analysis of approximate lumping in chemical kinetics

A general analysis of approximate lumping is presented. This analysis can be applied to any reaction system with n species described by dy/dt =f(y), where y is an n-dimensional vector in a desired region Ω, and f(y) is an arbitrary n-dimensional function vector. Here we consider lumping by means of a rectangular constant matrix M (i.e. ŷ = My, where M is a row-full rank matrix and ŷ has dimension n not larger than n). The observer theory initiated by Luenberger is formally employed to obtain the kinetic equations and discuss the properties of the approximately lumped system. The approximately lumped kinetic equations have the same form dŷ/dt = Mf/My) as that for exactly lumped ones, but depend on the choice of the generalized inverse M of M. {1,2,3,4}-inverse is a good choice of the generalized inverse of M. The equations to determine the approximate lumping matrices M are presented. These equations can be solved by iteration. An approach for choosing suitable initial iteration values of the equations is illustrated by examples.

The effect of lumping and expanding on kinetic differential equations

Let us consider the differential equation

A general analysis of exact nonlinear lumping in chemical kinetics

\dot{\bf y}(t)={\bf f}({\bf y}(t))

A general analysis of approximate nonlinear lumping in chemical kinetics. II. Constrained lumping

with an

Simulating bioremediation of uranium-contaminated aquifers; uncertainty assessment of model parameters

\bf f

Multicut‐HDMR with an application to an ionospheric model

from

Correlation method for variance reduction of Monte Carlo integration in RS‐HDMR

{\bf R}^N