Parameter estimation differential equation models

We then covered the important topic of estimating parameters for differential-equation models. We employed computational methods for solving differential equations and sensitivities, and solving nonlinear optimization problems in order to tackle this challepging problem. 

This approach is useful when dealing with relatively simple partial differential equation models. Seinfeld and Lapidus (1974) have provided a couple of numerical examples for the estimation of a single parameter by the steepest descent algorithm for systems described by one or two simultaneous PDEs with simple boundary conditions. 

Work has been done to infer differential equation models of cellular networks from time-series data. As we explained in the previous section, the general form of the differential equation model is deceit = f(Cj, c2,. cN), where J] describes how each element of the network affects the concentration rate of the network element. If the functions f are known, that is, the individual reaction and interaction mechanisms in the network are available, a wealth of techniques can be used to fit the model to experimental data and estimate the unknown parameters [Mendes 2002]. In many cases, however, the functions f are unknown, nonlinear functions. A common approach for reverse engineering ordinary differential equations is to linearize the functions f around the equilibrium [Stark, Callard, and Hubank 2003] and obtain... 

On the other hand, these disparate scales often allow us to approximate the complete mechanistic description with simpler rate expressions that retain the essential features of the full problem on the time scale or in the concentration range of interest. Although these approximations were often used in earlier days to allow easier model solution, that is not their primary purpose today. Most models, even stiff differential equation models with fairly disparate time scales, can be solved efficiently with modern ODE solvers. On the other hand, the physical insight provided by these approximations remains valuable. Moreover the reduction of complex mechanisms removes from consideration many parameters that would be difficult to estimate from available data. The next two sections describe, two of the most widely... 

Parameter Estimation with Differential Equation Models... 

Given these expressions for the gradient and Hessian, we can construct a fairly efficient parameter estimation method for differential equation models using standard software for solving nonlinear optimization and solving differential equations with sensitivities. 

The result at this step will be an equality containing at least one concentration gradient and other functions containing concentrations. Typically it will be the so-called third type boundary condition common to differential equations. Consult the appropriate chapters for estimating numerical values of the mass transport parameters in the equality. This is the last step for differential equation model application. 

Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. 

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for T and K so that the postulated model described the reactor s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affec t the v ues of T, and which in turn affects the dynamic response of the reac tor. 

Computer tools can contribute significantly to the optimization of processes. Computer data acquisition allows data to be more readily collected, and easy-to-implement control systems can also be achieved. Mathematical modeling can save personnel time, laboratory time and materials, and the tools for solving differential equations, parameter estimation, and optimization problems can be easy to use and result in great productivity gains. Optimizing the control system resulted in faster startup and consequent productivity gains in the extruder laboratory. 

The methods concerned with differential equation parameter estimation are, of course, the ones of most concern in this book. Generally reactor models are non-linear in their parameters and therefore we are concerned mostly with nonlinear systems. 

The scope of this book deals primarily with the parameter estimation problem. Our focus will be on the estimation of adjustable parameters in nonlinear models described by algebraic or ordinary differential equations. The models describe processes and thus explain the behavior of the observed data. It is assumed that the structure of the model is known. The best parameters are estimated in order to be used in the model for predictive purposes at other conditions where the model is called to describe process behavior. 

Let us first concentrate on dynamic systems described by a set of ordinary differential equations (ODEs). In certain occasions the governing ordinary differential equations can be solved analytically and as far as parameter estimation is concerned, the problem is described by a set of algebraic equations. If however, the ODEs cannot be solved analytically, the mathematical model is more complex. In general, the model equations can be written in the form... 

In this chapter we are concentrating on the Gauss-Newton method for the estimation of unknown parameters in models described by a set of ordinary differential equations (ODEs). 

In summary, at each iteration given the current estimate of the parameters, k , we obtain x(t) and G(t) by integrating the state and sensitivity differential equations. Using these values we compute the model output, y(tj,k ), and the sensitivity coefficients, G(t,), for each data point i=l,...,N which are subsequently used to set up matrix A and vector b. Solution of the linear equation yields Ak M) and hence k ]) is obtained. 

The mathematical model for a hydrocarbon reservoir consists of a number of partial differential equations (PDEs) as well as algebraic equations. The number of equations depends on the scope/capabilities of the model. The set of PDEs is often reduced to a set of ODES by grid discretization. The estimation of the reservoir parameters of each grid cell is the essence ofhistory matching. 

Leung 1993). PBPK models for a particular substance require estimates of the chemical substance-specific physicochemical parameters, and species-specific physiological and biological parameters. The numerical estimates of these model parameters are incorporated within a set of differential and algebraic equations that describe the pharmacokinetic processes. Solving these differential and algebraic equations provides the predictions of tissue dose. Computers then provide process simulations based on these solutions. 

Model-fitting procedures are usually based on analytical solutions of the model however, model parameters may be estimated by fitting the differential equations describing the model. Since the numerical solution of the differential equations introduces another source of error, fitting of differential equations is usually limited to cases where nonlinearities are present. 

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