Jacobian sensitivity analysis

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

Figure 17.12 shows some aspects of flame behavior that are revealed through sensitivity analysis (sensitivity analysis is discussed Section 15.5.4). For example, the maximum temperature is relatively insensitive to reaction rates, except very near the extinction point. At the extinction point, all sensitivities become unbounded because at the turning point the Jacobian of the system is singular. Near extinction, the hydrogen-atom concentration is... 

With the complexity of modern pharmacokinetic-pharmacodynamic models, analytical derivation of sensitivity indexes is rarely possible because rarely can these models be expressed as an equation. More often these models are written as a matrix of derivatives and the solution to finding the sensitivity index for these models requires a software package that can do symbolic differentiation of the Jacobian matrix. Hence, the current methodology for sensitivity analysis of complex models is empirical and done by systematically varying the model parameters one at a time and observing how the model outputs change. While easy to do, this approach cannot handle the case where there are interactions between model parameters. For example, two... 

Among many techniques used to obtain skeletons of larger mechanisms, in what follows we discuss the Direct Relation Graph (DRG) technique, coupled with Depth First Search (DFS) technique, the Sensitivity Analysis of the Jacobian matrix for the chemical system, the Intrinsic Low-Dimensional Manifold (ILDM) technique, the Reaction Diffusion Manifolds (REDIM technique) and the flamelet technique. Among these, the flamelet technique is preferred for writing the simplified chemical system for premixed flames and diffusion flames presented in the following sections. 

Operations with the Jacobian matrix are known to consume high computational time in most simulations involving implicit solvers. Based on the sensitivity analysis of the Jacobian matrix, around two-third of reactions and one-third of chemical species can be eliminated from the complete mechanism without significant loss in the quality of results. 

Simplihcations of chemical kinetics generally become an alternative. Small mechanisms of a low number of species are often reduced using the assumptions of steady-state and partial equilibrium. Farge mechanisms are reduced using a combination of techniques such as direct relation graph (DRG), to obtain a skeleton mechanism and techniques based on the sensitivity analysis of the eigenvalues and eigenvectors of the Jacobian matrix of the chemical system to obtain a reduced mechanism. 

As seen from Fig. 5.3, the substrate concentration is most sensitive to the parameters around t = 7 hours. It is therefore advantageous to select more observation points in this region when designing identification experiments (see Section 3.10.2). The sensitivity functions, especially with respect to Ks and Kd, seem to be proportional to each other, and the near—linear dependence of the columns in the Jacobian matrix may lead to ill-conditioned parameter estimation problem. Principal component analysis of the matrix STS is a powerful help in uncovering such parameter dependences. The approach will be discussed in Section 5.8.1. 

Considering that f, g, x, and u are vectors, the differentiation leads to formation of matrices. The matrix A is well known in stability analysis as the jacobian matrix it quantifies the effects of all state variables on their rates of change. A matrix similar to B turns up in metabolic control analysis, as N3v/3p [48, 108], where it denotes the immediate effects of parameter perturbations on the rates of change of all variables. If the function y is scalar and denotes a rate, then C becomes a row vector c harboring unsealed elasticity coefficients and D becomes a row vector d containing so-called n-elasticities - sensitivities of the rates with respect to the parameters [109]. The linearized system is ... 

For the reduction of chemical mechanisms, reaction-rate analysis has probably the largest record of success. A novel way for the inspection of rates is based on the study of algebraic-rate sensitivities and the Jacobian matrix. These methods can be used for the automatic identification of redundant species and reactions, to produce a reduced mechanism consisting of a subset of the original mechanism. The use of algebraic manipulation in techniques such as the QSSA and lumping, make the production of a reduced mechanism essential and make subsequent calculations as simple as possible. 

Let us determine the matrices J and F belonging to the kinetic system of ODEs above. These two types of matrices will be used several dozen times in the following chapters. For example, the Jacobian is used within the solution of stiff differential equations (Sect. 6.7), the calculation of local sensitivities (Sect. 5.2) and in timescale analysis (Sect. 6.2), whilst matrix F is used for the calculation of local sensitivities (Sect. 5.2). Carrying out the appropriate derivations, the following matrices are obtained ... 

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