Numerical methods for optimum experimental design in DAE systems

Bauer, Irene; Bock, Hans Georg; Körkel, Stefan; Schlöder, Johannes P.

doi:10.1016/s0377-0427(00)00300-9

Cited by 167 publications

(151 citation statements)

References 4 publications

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“…Pioneering work on model-based optimal experimental design (OED) was done by Box and Lucas (1959). OED improves the precision of parameter estimates by maximizing the information content of the designed experiments (Bauer et al, 2000). The principle is to minimize the variance-covariance matrix using a design criterion such as A-, D-, or E-optimality.…”

Section: Optimal Experimental Design and Measurement Techniquesmentioning

confidence: 99%

Model-based experimental analysis of enzyme kinetics in aqueous–organic biphasic systems

Zavrel

Schmidt

Michalik

et al. 2007

Journal of Biotechnology

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Section: Optimal Experimental Design and Measurement Techniquesmentioning

confidence: 99%

Model-based experimental analysis of enzyme kinetics in aqueous–organic biphasic systems

Zavrel

Schmidt

Michalik

et al. 2007

Journal of Biotechnology

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“…While these measurements y are discrete in nature, the assumption that y is a continuous variable can be made if the sampling frequency is sufficiently high. Optimal experimental design involves maximizing a criterion function that indicates the quantity of information gained by a given experiment, often in the context of model identification [8][9][10][11][12]14,[17][18][19][20][21][22][23]29,31]. Several commonly used criterion functions for experimental design exist.…”

Section: Optimal Experimental Designmentioning

confidence: 99%

“…Recently, as biological modeling and systems biology have emerged as an important area in biomedical research, optimal experimental design applied to biological experimental systems has become more popular [17][18][19][20][21][22][23][24][25][26][27][28]; additionally, optimal experimental design has been recognized as a valuable tool in optimal control for several decades [29]. For example, Jones et al [13] maximized production of an exogenous commodity chemical in metabolically engineered E. coli using an empirical modeling method similar to those used in [15,16] to maximize the efficacy of drug delivery.…”

Section: Introductionmentioning

confidence: 99%

Optimal Experimental Design for Parameter Estimation of an IL-6 Signaling Model

Sinkoe

Hahn

2017

Processes

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IL-6 signaling plays an important role in inflammatory processes in the body. While a number of models for IL-6 signaling are available, the parameters associated with these models vary from case to case as they are non-trivial to determine. In this study, optimal experimental design is utilized to reduce the parameter uncertainty of an IL-6 signaling model consisting of ordinary differential equations, thereby increasing the accuracy of the estimated parameter values and, potentially, the model itself. The D-optimality criterion, operating on the Fisher information matrix and, separately, on a sensitivity matrix computed from the Morris method, was used as the objective function for the optimal experimental design problem. Optimal input functions for model parameter estimation were identified by solving the optimal experimental design problem, and the resulting input functions were shown to significantly decrease parameter uncertainty in simulated experiments. Interestingly, the determined optimal input functions took on the shape of PRBS signals even though there were no restrictions on their nature. Future work should corroborate these findings by applying the determined optimal experimental design on a real experiment.

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“…Such problems can be solved reliably and efficiently using SQP methods [2], and lead to more complex problems if there are external controls on the differential equation [3,4]. The issue we will address in what follows is the numerical evaluation of the functional Ψ(w) := Tr(M(w)) and its derivatives.…”

Section: Problem Formulationmentioning

confidence: 99%

Differentiable evaluation of objective functions in sampling design with variance‐covariance matrices

Mommer

Sommer

Schlöder

et al. 2011

Proc Appl Math and Mech

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In this short note we consider the differentiable evaluation of the objective function of the sampling design optimization problem based on the inverse of the Fisher information matrix, and where the integer design variables have been converted into real variables using a relaxation technique. To ensure differentiability and cover the full range of the variables, and thus improve the convergence behavior of derivative-based optimization algorithms, we propose applying a Cholesky decomposition on the Fisher information matrix, but using a special higher precision floating point arithmetic to ensure stability. While each evaluation of the functional becomes slower, the algorithm is much simpler, amenable to be used with automatic differentiation directly, and can be shown to be very stable. For many practical situations, this is a valuable trade-off. Problem formulationSuppose you want to estimate the parameters p of a time dependent differential equation model of some phenomenon,by carrying out an experiment. Depending on the situation, measurements might be expensive, and thus there is value in choosing from a predetermined set of candidate measurement times an optimal subset from which to obtain reliable estimates. An approach that works in practice is: assume that you know the correct parameters (this will be the initial guess p 0 for the estimation). Then, perform simulated measurements at the candidate time-points t 0 ≤ t 1 < t 2 < . . . < t N ≤ t f , and set up the expected least-squares estimation problem,where h is the measurement function (for space and simplicity reasons assumed scalar in this note), the variance of the measurement error is assumed to be 1 for simplicity. The w i are weights we give different hypothetical measurements, and are the degrees of freedom subject to optimization in the classical experiment design problem [1]. Ideally, the w i should be in {0, 1}, we consider them in [0, 1] to obtain the so-called relaxed formulation.The variance-covariance matrix associated to the least squares problem is, for a vector of weights w ∈ R N , given by M (w) = F (w) −1 = (J(w) T J(w)) −1 , where the i-th row of J(w) is √ w i ∂ p h(y(t i ; p)). A typical sampling design problem then looks as follows, where we use the trace of the variance-covariance matrix as a quality measure (the so-called A-criterion),where w max denotes the maximum number of measurements to be taken. Such problems can be solved reliably and efficiently using SQP methods [2], and lead to more complex problems if there are external controls on the differential equation [3,4]. The issue we will address in what follows is the numerical evaluation of the functional Ψ(w) := Tr(M(w)) and its derivatives. Numerical evaluation of the sampling design objective functionOne immediately realizes that the presence of the squared matrix induces stability issues. The usual way of computing M (w) is to compute the thin QR decomposition of J(w) and set M (w) = R(w) −1 R(w) −T , a formula which usues the fact that Q(w) T Q(w) = I. This is the f...

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Numerical methods for optimum experimental design in DAE systems

Cited by 167 publications

References 4 publications

Model-based experimental analysis of enzyme kinetics in aqueous–organic biphasic systems

Model-based experimental analysis of enzyme kinetics in aqueous–organic biphasic systems

Optimal Experimental Design for Parameter Estimation of an IL-6 Signaling Model

Differentiable evaluation of objective functions in sampling design with variance‐covariance matrices

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