The parameter s 0 denotes the initial sugar concentration. These images provide information about the localization of the lymphatic vessels encoded in Q and the concentration of immobilized CCL As the measured intensity values are corrupted by background fluorescence and as the data are not normalized, we model the readout following [ 25 ] as. All parameters are assumed to be non-negative due to their biological meaning. For the numerical simulation of the biological process, we employed a finite element discretization of the PDE model.
The mesh consists of elements and the concentrations in these elements are the state variables of the discretization. The simulation-based method for parameter estimation was implemented in MATLAB extending the routine published in [ 16 ]. The simulation of the continuous analogue was terminated, if the gradient of the right-hand side became small, i.
Furthermore, simulations were interrupted whenever the objective function value became complex, which can happen due to the log-transformation of the output. A Geometry of a lymphoid vessel obtained from biological imaging data [ 27 ]. B Simulated data of the CCL21 gradient generated by simulating model To evaluate the convergence properties of the proposed algorithm for the models, we considered published simulated data for the ground truth similar to [ 28 ].
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The objective function for most parameter estimation problems is non-convex and can be multimodal. For this reason, we employed multi-start local optimization using the continuous analogue for which we have established local convergence in this paper. We did not implement any bounds for values of parameter or states. The implementation of the objective function and finite element schemes was adapted from [ 25 ]. For the local optimization with the continuous analogue, we chose the negative gradient as descent direction.
As a reference, we performed also multi-start local optimization using a discrete iterative optimization method. We used the state-of-the-art optimizer fmincon. This interior point algorithm employs either a Newton step, where the Hessian is approximated by the Broyden—Fletcher—Goldfarb—Shanno BFGS algorithm, or a conjugate gradient step using a trust region [ 31—33 ].
The optimizer was provided with the objective function, the nonlinear constraint, as well as the corresponding derivatives. A total of iterations and function evaluations was allowed. The assessment of the results revealed a good convergence of the continuous analogue.
Results of parameter estimation for CCL21 model. Converged runs are indicated in blue. B CPU time needed per optimizer run for the optimization using the continuous analogue and the discrete iterative procedure lighter grey colour indicates runs which stopped because the maximal number of iterations was reached.
The box covers the range between the 25th and the 75th percentile of the distribution. The median CPU time is indicated by a line. Accordingly, the success rate was substantially lower than for the proposed continuous analogue. Of the runs which did not converge to the global optimum 25 runs were stopped because the maximal number of iterations was reached. We found a median CPU time of 15 minutes for the continuous solver and minutes for the discrete iterative procedure.
In light of the fact that the discrete iterative method uses second-order information, it is interesting to observe that a continuous analogue using the negative gradient is more efficient. One possible explanation is that the efficiency of the continuous analogue is a result of the application of sophisticated numerical solvers. The adaptive, implicit solver ode15s , which is provided with the analytical Jacobian of the ODE—PDE model, might facilitate large step-sizes and fast convergence.
Indeed, the Jacobian also provides second-order information. This result indicated that for different starting points very different retraction factors might be ideal. Yet, for large retraction factors many of the completed runs also converged, while for small retraction factors no runs converged as the maximum number of iterations becomes too large. Notably, for the small values, the median CPU time was nearly six to seven times higher than the smallest one.
These results indicated that the retraction factor should be chosen large enough but not too large. In an intermediate regime, which could here also be found by random sampling, we found the best convergence properties.
Parameter estimation is an important problem in a wide range of applications. Robustness and performance of the available iterative methods is, however, often limited. In this study, we introduced continuous analogues of descent methods for optimization with PDE constraints. For these continuous analogues, we proved local convergence of their solutions to the optima. The necessary assumptions are fulfilled for a wide range of application problems, rendering the results interesting for several research fields. We demonstrated the applicability of continuous analogues for a model of gradient formation in biological tissues and compared them with an iterative discrete procedure.
The results highlight the potential of the continuous analogues, e. As fmincon is a generic interior point method, there might apparently be approaches which are efficient for the considered PDE-constrained problems see also the no free lunch theorem [ 34 ]. The evaluation of the influence of the retraction factor revealed the importance of an appropriate choice of the retraction factor as well as the issue of premature stopping.
As this bound might, however, be conservative and can only be assessed pointwise, the use of adaptive methods might be interesting.
To address the issue of premature stopping, bounds for parameters and state variables have to be implemented, e. In the application problem, we only considered elliptic PDE constraints as for the proposed continuous analogues parabolic constraints can be encapsulated in the objective function. This changes the objective function landscape and indirectly influences the convergence.
Conceptually, it should also be possible to formulate continuous analogues which do not require a solution operator for the parabolic PDE but also have the solution of the parabolic PDE as a state variable. This mathematically more elegant approach is left for future research.
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In conclusion, this study presented continuous analogues for a new problem class. Similar to other problem classes for which continuous analogues have been established [ 15 , 16 ], we expect an improvement of convergence and computation time. The continuous analogues for optimization also complement recent work on simulation-based methods for uncertainty analysis [ 36 ].
The efficient implementation of these methods in easily accessible software packages should be a focus of future research as it would render the methods available to a broad community. Future research in this context will be concerned with globalization strategies, such as those proposed in [ 20 , 21 ], cf. No potential conflict of interest was reported by the authors. National Center for Biotechnology Information , U. Inverse Problems in Science and Engineering.
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Inverse Probl Sci Eng. Published online Jul Kaltenbacher a. Author information Article notes Copyright and License information Disclaimer. Hasenauer ed. Received Jun 29; Accepted May Introduction Partial differential equations PDEs are used in various application areas to describe physical, chemical, biological, economic, or social processes. Mathematical model We consider parameter estimation for models with elliptic and parabolic PDE constraints. Assumption 2. Elliptic and parabolic PDE constraints In the general case, observations are available for the initial state and the transient phase.
Continuous analogue to iterative optimization for PDE-constrained inverse problems
Elliptic PDE constraint In many applications, only experimental data for the steady state of a process are available. Open in a separate window. Figure 1. Assumption 5. Theorem 5.
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Figure 2. Remark 5. Remark 6. Proposition 6.