High-Order Deterministic Sensitivity Analysis and Uncertainty Quantification: Review and New Developments
Abstract
:1. Introduction
2. Mathematical Description of a Generic Nonlinear Physical System with Uncertain Parameters and Boundaries
- (a)
- A mathematical model comprising independent variables (e.g., space, time, etc.); dependent variables (aka “state functions”, e.g., temperature, mass, momentum, etc.) and various parameters (appearing in correlations, coordinates of physical boundaries, etc.), which are all interrelated by equations (linear and/or nonlinear in the state functions) that usually represent conservation laws.
- (b)
- Model parameters, which usually stem from processes that are external to the system under consideration and are seldom, if ever, known precisely. The known characteristics of the model parameters may include their nominal (expected/mean) values and, possibly, higher-order moments or cumulants (i.e., variance/covariances, skewness and kurtosis), which are usually determined from experimental data and/or processes external to the physical system under consideration. Occasionally, only inequality and/or equality constraints that delimit the ranges of the system’s parameters are known.
- (c)
- One or several computational results, customarily called “system responses” (or objective functions or indices of performance), which are computed using the mathematical model.
- (d)
- Experimentally measured values of the responses under consideration, which may be used to infer nominal (expected) values and uncertainties (variances, covariances, skewness, kurtosis, etc.) of the respective measured responses.
- is a -dimensional column vector of dependent variables; the abbreviation “” denotes “Total (number of) Dependent variables.” The functions , denote the system’s “dependent variables” (also called “state functions”); , where is a normed linear space over the scalar field of real numbers.
- denotes a -dimensional column vector The components are operators (including differential, difference, integral, distributions, and/or infinite matrices) acting (usually) nonlinearly on the dependent variables , the independent variables and the model parameters ; is the mapping , where , , , . An arbitrary element has the form .
- is a -dimensional column vector which represents inhomogeneous source terms, which usually depend nonlinearly on the uncertain parameters ; , where is also a normed linear space.
- The equalities in this work are considered to hold in the weak (“distributional”) sense. The right-sides of Equation (1) and of other various equations to be derived in this work may contain “generalized functions/functionals”, particularly Dirac-distributions and derivatives thereof.
3. Sixth-Order Formulas for Sensitivity Analysis of Model Responses to Model Parameters
Illustrating the Need for High-Order Sensitivities: Neutron Scattering in an Infinite Hydrogenous Medium
4. Sixth-Order Moments of the Response Distribution in the Parameter Phase-Space
4.1. Expectation Value of a Response
4.2. Response Parameter Covariances
4.3. Covariance of Two Response
- The 1st-order approximation of the covarianceNotably, the expression of provided in Equation (62) is consistent both in the highest-order (in this case: first-order) of sensitivities and also in the highest-order of parameter standard deviation (in this case, second-order) since all of the terms involving the product are consistently included (i.e., none are missing) in the expression provided. Therefore, the standard deviation will also be correct to first-order in the standard deviations of the parameters, i.e.,
- The 2nd-order approximation of the covariance:
- (i)
- Using Equation (58) yields the following expression:
- (ii)
- Using Equation (59) yields the following expression:
- Notably, the expression of provided in Equation (64) is consistent in the 2nd-order of sensitivities but is inconsistent in the 4th-order of standard deviations of parameters, i.e., . On the other hand, the expression provided in Equation (65) is consistent in the 4th-order (i.e., ) of parameter standard deviations. Therefore, if the 3rd-order sensitivities are available, then the expression provided in Equation (65) should be used, since it is correct up to, and including, the fourth-order terms containing the products . Consequently, the consistent second-order standard deviation will not have any 2nd-order errors in the parameter standard deviations. In contradistinction, the 2nd-order inconsistent standard deviation will have second-order errors in the parameter standard deviations.
- The 3rd-order approximation of the covariance:
- (i)
- Using Equation (58) yields the following expression:
- (ii)
- Using Equation (59) yields the following expression:
The expression of provided in Equation (66) is consistent in the third-order sensitivities but is inconsistent in the highest-order of parameter standard deviations, i.e., , while the expression provided in Equation (67) is consistent in the highest-order of parameter standard deviations. Therefore, if the 4th-order sensitivities are available, then the expression provided in Equation (67) should be used, since it is correct up to, and including, the fifth-order terms containing the products . Also noteworthy is the fact that neither the consistent nor the inconsistent standard deviations computed using the 3rd-order variance approximations provided in Equation (66) or Equation(67), respectively, are correct to third-order (i.e., ) parameter standard deviations. In order to obtain standard deviations which are correct up to and including the 3rd-order terms containing , it is necessary to use the 4th-order variance approximations to be provided below. - The 4th-order approximation of the covariance:
- (i)
- Using Equation (58) yields the following expression:
- (ii)
- Using Equation (59) yields the following expression:
4.4. Triple Correlations among Responses and Parameters
- The triple-correlations (or third-order moment of the distribution of responses), denoted as , among three responses , and , are defined as follows:
- If only first-order response sensitivities are available, the following 1st-order approximate expression, denoted as , can be obtained:
- If 1st- and 2nd-order response sensitivities are available, the following 2nd-order approximate expression, denoted as , can be obtained:As indicated in Equation (74), the expression of consistently contains all of the 1st- and 2nd-order response sensitivities, and all of the 5th-order terms in standard deviations (of the form ) but does not contain all of the 6th-order terms in standard deviations (of the form ) The 6th-order terms in standard deviations, containing the remaining terms of the form are provided below.
- The availability of the third-order response sensitivities enables the computation of additional terms that comprise 6th-order terms containing the products . The following 3rd-order (in sensitivities) approximate expression, denoted as , can be obtained:
- If fourth-order response sensitivities are also available, the following 4th-order (in sensitivities) approximate expression can be obtained:As can be observed from Equation (76), the quantity contains all of the terms involving 6th-order products of standard deviations.
- The triple-correlations among one parameter, , and two responses, and , are defined as follows:Using the Taylor series up to and including the 6th-order in standard deviations, the expression of is as follows:
- The triple-correlations among two parameters , and one response are defined as follows:Using the Taylor series up to and including the 6th-order in standard deviations, the expression of is as follows:
4.5. Quadruple Correlations among Responses and Parameters
- The quadruple-correlations (or fourth-order moment of the distribution of responses), denoted as , among four responses , , and , are defined as follows:
- If only first-order response sensitivities are available, the following 1st-order approximate expression, denoted as , can be obtained:The expression of is consistent in that it comprises all of the 1st-order sensitivities and all of the terms involving 4th-order products of standard deviations of the form .
- If all 1st- and 2nd-order response sensitivities are available, the following approximation of the quadruple-correlations among four responses , and can be obtained:
- If third-order response sensitivities are also available, then the remaining terms containing 6th-order products of standard deviations of the form can also be computed, to obtain the following approximation consistent in 6th-order products of standard deviations, denoted as :
- The quadruple-correlations denoted as , among one parameter, , and three responses , , and , are defined as follows:
- If first- and second-order response sensitivities are available, the following 2nd-order approximate expression, denoted as , can be obtained: