26-th ECMI Modelling Week Final Report Dresden, Germany

Transcription

1 26-th ECMI Modelling Week Final Report Dresden, Germany

2 Group 2 Parameter Estimation Filipe Casal Department of Mathematics, Instituto Superior Técnico, TU Lisbon, Portugal. Daniel Krähmann Department of Mathematics, TU Dresden, Dresden, Germany Simo Martikainen Department of Mathematics, Tampere University of Technology, Tampere, Finland Beatriz Navarro Garcia Department of Mathematics, Universidad Carlos III de Madrid Madrid, Spain Stefan Schießl Department of Mathematics, Friedrich Alexander Universität Erlangen-Nürnberg, Germany Vladimir Shemyakin Department of Technomathematics, Lappeenranta University of Technology Lappeenranta, Finland Instructor: Kshitij Kulshreshtha Department of Mathematics, University of Paderborn, Paderborn, Germany 2

3 Abstract In this work the concept of inverse problems and Tikhonov regularization are introduced. Two practical parameter estimation problems are formulated as inverse problems and an iteratively regularized Gauss-Newton s method is used to solve the regularized problem. The introduced algorithm was tested with simulated data. The simulations highlight the difficulties in solving inverse problems.

4 2 Parameter Estimation 2.1 Introduction Every single mathematical model has one or more parameters, let them be the capacitances of capacitors in a circuit or the diffusion coefficient of a random walk process. A so called direct problem is to use these given parameters and a given scenario to compute or simulate the measurements. This problem is usually a well-posed one having unique solution which is a continuous mapping of the given scenario assuming the system parameters constant. However, these system parameters are usually unknown and they must be estimated somehow to make further analysis of the considered system. The problems where system parameters are estimated using possibly noisy measurements are called as inverse problems. If one can construct a mathematical relationship between the measurements and the system parameters, one could at least in principle try to solve a system of nonlinear equations by computing an inverse mapping and applying it to solve the unknown parameters. The method has numerous drawbacks. For example, if we limit ourself to consider the case when the amount of measurements is greater than the amount of unknown parameters, the inverse function does not exists. On the other hand, if the inverse mapping exists, it might be highly sensitive to measurement errors. This motivates the study to solve the system parameters in a numerically feasible way which is not sensitive to measurement noise. The structure of this paper is as follows. In section 2.2 a trigger-circuit is described briefly and a typical inverse problem is introduced. In the same section the concept of Tikhonov regularization is defined and an iteratively regularized Gauss-Newton method to solve the regularization problem is described. Furthermore, the section describes the problems faced while solving the inverse problem. In section 2.3 another inverse problem is introduced where the problem is to estimate unknown reaction rates in a system of chemical reaction given noisy measurements of several species. In the end of this work the section 2.4 summarizes the results and observations. 2.2 Trigger-circuit Model description A trigger circuit is an electric circuit consisting of resistors, transistors and power sources. The special case that is being analyzed in this project is a Schmitt Circuit. An example is shown in Figure 2.1 (R1, R2 and R3 are resistors, Opt is an operational amplifier. Normally, u b, the operational voltage, is applied to the upper conduction and the mass is the lower conduction. The clamp on the left would be for a control voltage, on the right is the output voltage). The equations that are used here are taken from [2], representing the

5 26th ECMI modelling week 3 Figure 2.1: Example for a Schmitt Circuit. Kirchhoff s Law for a schmitt circuit with two transistors: F 1 = x 1 R 4 I e (f(s x 1 )) + A i I c (f(s x 2 ) + f(x 3 x 4 )) = 0 F 2 = A n I e f(s x 1 ) I c f(s x 2 ) + x 2 x 3 u b x 2 R 3 t = 0 F 3 = (1 A n )I e f(x 3 x 1 ) x 2 x 3 R 3 + (1 A i )I c f(x 3 x 4 ) = 0 F 4 = A n I e f(x 3 x 1 ) I c f(x 3 x 4 ) u b x 4 R 2 = 0 with f(x) = e 30τ 1. F = [F 1, F 2, F 3, F 4 ] is the residual of the equation, x 1, x 2, x 3 and x 4 are the measurements that are being obtained during the experiment. The equation contains some constants, listed now: R2, R3 and R4 are resistors, I e and I c are emitter and collector powers of a MOS- Transistor. A i and A n are amplifications of the circuit, s is a cut-off voltage for the output voltage where the logical meaning switches from false to true. Variable t represents the impedance of a rheostat. u b is the operational voltage. The physical representation of this trigger circuit is not exactly known, but is irrelevant for the mathematical concerns. Stated as a control problem, the variables are split up in measurements, controls and parameters: x = [x 1, x 2, x 3, x 4 ] : Measurements of experiments. u = [s, t] : Control values for conducting experiments. p = [A i, A n ] : System parameters, normally unknown.

6 4 Parameter Estimation Solving method The system is written as a multidimensional function F = [F 1, F 2, F 3, F 4 ], therefore a solution z = [x, u, p] fullfills F (z ) = 0. To find the solution the zeros of the implicit function need to be found. If u and p are given, that is the direct problem. The interesting case is if one or more measurements x (i) (i = 1,...) and u are given and p has to be found. The measurements normally have some kind of noise, therefore no exact or unique solution can be found and solving of the direct problem does not work. The problem has to be solved as an inverse problem. If G represents the numerical algorithm that is being used to solve the direct problem, an explicit function can be stated with x = G(u, p). In this case, the function G(u, p) solves the direct problem with given u and p using a Newton solver with Armijo step control. p will be the unknown in the inverse problem, therefore the jacobian of G(u, p) with respect to p is needed. With the help of the implicit function theorem, the jacobian can be easily calculated. p G(u, p) = ( x F (x, u, p)) 1 ) p F (x, u, p). Remark: The implicit function theorem only works for a solution of the direct problem. To solve the inverse problem for one or more measurements, a simple newton does not work anymore. The noise of the measurements x and the possibly strong deviances for small changes in p have to be adressed. Approaches to these types of problems are described in [4]. The methods are using Tikhonov Regularization to tackle the difficulties described earlier. For this model, the iteratively regularized Gauss-Newton method is chosen. Details of the method and convergence results can be found in [?] and in [?]. The method adjusts the newton update in the following way: p n+1 = p n (A na n + α n I) 1 (A n(g(p n ) y δ ) + α n (p n p 0 ), where A is the adjoint of A, p 0 is the initial guess for p and α n is a sequence that fulfills A n = p G(u, p n ), G(p n ) = G(u, p n ). α n > 0, 1 α n α n+1 r, lim α n = 0 for some r > 0. n

7 26th ECMI modelling week 5 To find a good initial guess and an appropriate α n sequence is fundamental and differs for each problem. A conservative approach for α n, also implemented in this model, is α n = α 0 1 n. With α n = 5, this is a good compromise between a stable regularization (α n large) and a good approximation (α n small) for this problem. The initial guess is chosen as the mean of all the possible values for p. Tests First, a set of parameters p = [A i, A n ] is chosen at random (the range is [0.85, 1]). With those parameters and a set of different control values u = [s, t] the direct problem is solved and x values obtained for every pair. Those x values then undergo a uniform noise of the magnitude [ 0.005, 0.005] and are then used to solve the inverse problem. For the inverse problem, only the corresponding x and u values are used and the value for p is recreated. The noise seems small, but for the trigger circuit model it is already crucial. The system has large deviances for small changes in x. The most important evaluation criterion is the residual F (x, u, p) of the original system. The residual is calculated for every set of x, u, p and summed up (now called f res). For the noisy data, even the exact parameters might have residuals greater than 1 and therefore the algorithm can not find a solution with fres 0. The stopping criterion for the algorithm is the relative change in fres and also gres = p n+1 p n. Because of the sensitivity of the system and the limited time for the calculation, the accuracy is set to 10 5 and The absolute value of f res is not considered as stopping criterion. Test 1 22 measurements are created. The exact parameters that were used to create the measurements are p = [ , ], no noise is applied, f res = e 11. The found solution is p = [ , ] with a residual of The calculation time is already 2.8 hours. The graphs (Figures 2.2, 2.3, 2.4) show that by increasing the accuracy the results also would improve, which would be the next step. Test 2 10 measurements are created. The exact parameters that were used to create the measurements are p = [ , ], the noise described above is applied, f res = The found solution is p = [ , ] with f res = The algorithm seemingly found a solution that fits the noisy measurements better than the parameters that were used to create them. This is due to the relatively small number of measurements that were used and also due to the large deviances in the

8 6 Parameter Estimation Figure 2.2: Test 1: Convergence of the residuals. Figure 2.3: Test 1: Convergence of the parameters. system. The straightforward way to improve the result is to consider more measurements.

9 26th ECMI modelling week 7 Figure 2.4: Test 1: Absolute error. 2.3 Brusselator Figure 2.5: Test 2: Convergence of the residuals. Mathematical Model Consider the steady state of the following system of chemical reactions [2, 1] A k 0 X 2X + Y k 1 3X B + X k 2 Y + C X k 3 D

10 8 Parameter Estimation Figure 2.6: Test 2: Convergence of the parameters. Figure 2.7: Test 2: Absolute error. in one dimensional space of length L = 8 and under diffusion with diffusion rate D = Moreover it is assumed that the concentration of A at the

11 26th ECMI modelling week 9 boundaries satisfies [A](z = 0) = a 0 = 1, [A] z (z = L) = a L = 0 and the concentration of B is assumed to be a constant with respect to z. That is [B](z) = b = 3. The reaction rates k 0 = 1 and k 3 = 0.8 were assumed to be known. The task in the inverse problem is to estimate the reaction rates k 1:2 given noisy measurements of concentrations [X] and [Y]. The system can be described as a system of partial differential equations as follows [A] = D 2 [A] L 2 z 2 k 0 [A] [X] = D 2 [X] L 2 z 2 + k 0 [A] + k 1 [X] 2 [Y ] k 2 [B][X] k 3 [X] [Y ] = D 2 [Y ] L 2 z 2 k 1 [X] 2 [Y ] + k 2 [B][X] In steady state the time derivatives vanish yielding to a system of ordinary differential equations D 2 [A] L 2 z 2 = k 0 [A] (2.1) D 2 [X] L 2 z 2 = k 0 [A] k 1 [X] 2 [Y ] + k 2 [B][X] + k 3 [X] (2.2) D 2 [Y ] L 2 z 2 = k 1 [X] 2 [Y ] k 2 [B][X] (2.3) The solution to (2.1) given the boundary values is [A](z) = 1 ( ) ( )) 2 a k0 Lz 2 k0 Lz 0 exp (1 + exp D D Substituting this into (2.2) and (2.3) results an ordinary differential equation and its solution can be written as a function of inputs u and parameters p. Formally [ ] [X](z) x(z) = = F (z, u, p) (2.4) [Y ](z) where the following notation has been introduced to highlight the mathematical similarities of the problems introduced before in this work x = [[X](z), [Y ](z)] : Measurements of the concentrations of species X and Y. u = [a 0, a L, [b]] : Control values for conducting experiments. p = [k 1, k 2 ] : Unknown parameters The task is formulated as an inverse problem and thus the solution for the Tikhonov regularized least squares optimization problem can be computed with iteratively regularized Gauss-Newton method.

12 10 Parameter Estimation Computational Methods The Tikhonov regularized least squares solution of the parameter estimation problem was computed with iteratively regularized Gauss-Newton method where α n sequence was defined as α n = 10 n!, n N (2.5) The required Jacobians and the concentrations of species X and Y were computed numerically with Sundials Matlab Toolbox CVODES [3] using sensitivity analysis approach. To compute the Jacobian of the solution x with respect to the parameters p it is assumed that the system is defined as 2 [x] = f(x, u, p) (2.6) z2 Then the following sensitivity system is introduced to compute the Jacobian of y with respect to the unknown parameters p where s is defined as 2 [s] z 2 = f f (x, u, p) s + x p s = dx dp (2.7) (2.8) Solving the differential equations given in (2.6) and (2.7) simultaneously gives the concentrations y together with sensitivities s i. These sensitivities construct the Jacobian of the solution y with respect to parameters p: 2 [x] z 2 = [ s 1 s 2... s dim(x) dim(p) ] The sensitivities s i were computed numerically with CVODES toolbox. Empirical Results (2.9) The proposed computational methods to estimate the reaction rates k 1:2 of the Brusselator was tested with computational simulations. The reaction rates were defined as k = [ ]. The measurements were generated from the equations (2.1) (2.3) with uniformly distributed measurement noise and Gaussian measurement noise. The support of uniform distribution was chosen to be [ 0.1, 0.1] and the standard deviation of Normal distribution was chosen to be 0.1. It was observed that the proposed method converged to the approximately correct values of k 1:2 when the standard deviation of normally distributed measurement error was less or equal than 0.1 with an initial guess k 1:2 = [ ].

13 26th ECMI modelling week 11 2 k 1 1 Empirical Results k 1 estimated k 1 true value k 2 1 k 2 estimated k 2 true value α n iteration number Figure 2.8: Convergence results. The measurement noise was generated from Gaussian distribution with zero mean and standard deviation 0.1. As illustrated in figure 2.3, a low number of iterations is required for the proposed method to converge to the correct values of k 1:2. After ten iterations the estimation error is negligible considering the amount of measurement error. It was also observed that in contrast to the majority of the inverse problems the convergence was not highly sensitive to the choice of α n sequence. 2.4 Conclusion Some aspects of finding the parameters of a given mathematical model were discussed in this work. It was pointed out in section 2.1 that every single mathematical model has some parameters which are typically unknown a priori. The goal of constructing a mathematical model is to perform further analysis of the given system and with no knowledge about the system parameters, it is not possible to say anything useful about the given system. General aspects of inverse problems were discussed in this work and the iteratively regularized Gauss-Newton method for solving Tikhonov regularization problem was introduced briefly. The performance of iteratively regularized Gauss-Newton was demonstrated in solving two practical param-

14 12 Parameter Estimation eter estimation problems with simulated measurements. For the presented problems the proposed method converged with a low number of iterations. However, the computational costs for one iteration is highly dependent on the specific problem. The examples highlighted the key problems of solving inverse problems. For example, it was found that some problems are more sensitive with respect to the system parameters than others. To overcome this problem Tikhonov regularization was applied to make the estimation process less sensitive to the nonlinear behavior of the given system. However, the convergence of iteratively regularized Gauss-Newton method depends on the choice of α n sequence. In this work the choice of α n sequence was done with trial and error. More sophisticated choices are left as future work while the proposed choices result in a good convergence rate under the given scenarios.

15 Bibliography [1] Pönisch, G. Computing hysteresis points of nonlinear equations depending on two parameters. Computing 39(1), 1 17, December ISSN X. doi: /bf URL [2] Pönisch, G; Schnabel, U and Schwetlick, H. Computing multiple turning points by using simple extended systems and computational differentiation. Optimization Methods and Software 10(4), , doi: / URL [3] Sundials. Cvodes tool, September URL description.html#descr_cvodes [4] W, EH; M, H and A, N. Regularization of inverse Problems. Dordrecht: Kluwer,