Cluster Newton Method for Sampling Multiple Solutions of an Underdetermined Inverse Problem: Parameter Identification for Pharmacokinetics

Transcription

1 Cluster Newton Method for Sampling Multiple Solutions of an Underdetermined Inverse Problem: Parameter Identification for Pharmacokinetics 8 8 Yasunori Aoki (University of Waterloo) Uppsala University Ken Hayami National Institute of Informatics Hans De Sterck University of Waterloo Akihiko Konagaya Tokyo Institute of Technology

2 Research funded and supported by Ministry of Education, Sports, Science and Technology, Japan National Institute of Informatics, Tokyo, Japan Natural Science and Engineering Research Council of Canada

3 Model Parameter Estimation Finitely Parameterized Mathematical Model f(x) =y Experimental Data du dt du dt du dt du dt = x u u = x u u x u = x u x u = x x u x u 8 8

4 Model Parameter Estimation underdetermined problem f(x) =y f : R m R n m>n Parameter Space R m underdetermined: many solutions for each right-hand side Simulated Experimental Data Space R n X f y consider the case where many solutions are of interest our goal: compute many solutions efficiently

5 Why Multiple Solutions Can Be Useful

6 Example : Level curve tracing of a rough surface Find a set of points near, s.t. where

7 Levenberg Marquardt method (one-by-one) For all of the initial points, the algorithm terminated with an error Algorithm appears to be converging to a point that is not a root. Initial set Final set

8 Cluster Newton method for underdetermined problems (collective) Stage (Regularized Newton s method applied to a cluster of points) Linear approximation with least squares fitting (Least square solution of an overdetermined system of linear equations) Moore-Penrose inverse using the linear approximation (Min-norm solution of an underdetermined system of linear equations)

9 Stage st iteration step

10 Stage st iteration step compute one collective Jacobian (instead of many Jacobians one-by-one)

11 Stage st iteration step find intersection of collective linearization with

12 Stage st iteration step for each point, find nearest point on intersection line (and add a random perturbation to make new points non-collinear)

13 Stage nd iteration step

14 Stage nd iteration step-

15 Initial set

16 Stage st iteration

17 Stage nd iteration

18 Stage rd iteration

19 Stage th iteration

20 Stage th iteration

21 Cluster Newton method for underdetermined problems Stage (Broyden s method, i.e. multi-dimensional secant method) Use collective linear approximation from Stage for initial Jacobian Use the points found by the Stage as initial points

22 Stage th iteration

23 Stage st iteration

24 Stage nd iteration

25 Stage th iteration

26 Example : Pharmacokinetics Model (Arikuma et al.) unobservable parameters Clinically observed data Slatter et al. 997 Estimate model parameters from non-invasive clinical observation, i.e., find such that where f : R R ( parameters, but only observations)

27 Convergence Levenberg Marquardt method Cluster Newton method Stage Stage * : at least function evaluations / iteration / solution (9,9 total function evaluations (= ODE solves)) * * : only function evaluations / iteration / solution (, total function evaluations) * (we seek solutions)

28 Sensitivity to the function (ODE solve) evaluation error Median relative residual Levenberg Marquardt method accuracy of the function evaluations Number of iterations Median relative residual Cluster Newton method Stage Stage Number of iterations accuracy of the function evaluations (in practice, target accuracy is determined by experimental error)

29 Model Parameter Estimation underdetermined problem f(x) y Parameter Space R m f : R m R n m<n R n X Experimental Error f y (underdetermined: many solutions for right-hand side, but computation can be stopped once solutions are in blue region; the size of the blue region is determined by the experimental error)

30 Computation Speed Comparison : Levenberg-Marquadt method vs. Cluster Newton Method Median relative residual Cluster Newton method (left) Levenberg Marquardt method (right) Computation time (hours)

31 Overdetermined Parameter Identification Problem

32 overdetermined: finding multiple solutions collectively is often still very useful, to quantify the effect of experimental error, and to investigate identifiability of parameters Model Parameter Estimation overdetermined problem f : R m f(x) R n y m<n R n Parameter Space R m Experimental Error y overdetermined: unique solution for each right-hand side, except for parameters that are not identifiable

33 Example : Influenza Kinetics model (Baccam et al.) Finitely Parameterized Mathematical Model du dt du dt du dt du dt = x u u = x u u x u = x u x u = x x u x u f(x) y f : R 7 R Experimental Data u (t = ) = x u (t = ) = u (t = ) = u (t = ) = x overdetermined, but some parameters are not identifiable

34 8 8 modified Cluster-Newton method can be used to find multiple solutions covering the blue region efficiently

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55 Cluster Newton method Genetic Algorithm + Monte Carlo method After 7,88 function (= ODE) evaluations After 9,8 +,97 function evaluations

56 Conclusion

57 Cluster Newton method is a computationally efficient algorithm to find multiple reasonable parameters of finitely parameterized mathematical models. It is efficient and robust due to collective fitting of the Jacobian. [] Yasunori Aoki, Ken Hayami, Hans De Sterck and Akihiko Konagaya, Cluster Newton Method for Sampling Multiple Solutions of an Underdetermined Inverse Problem: Parameter Identification for Pharmacokinetics, Technical Report NII--E, National Institute of Informatics,Tokyo, Japan (). [] Yasunori Aoki, Ken Hayami, Hans De Sterck, and Akihiko Konagaya, 'Cluster Newton Method for Sampling Multiple Solutions of Underdetermined Inverse Problems: Application to a Parameter Identification Problem in Pharmacokinetics', SIAM Journal on Scientific Computing, accepted,.