(635d) Robust Simulation of Transient PDE Models Under Uncertainty

Transient partial differential equations (PDEs) are ubiquitous in model-based systems engineering for modeling spatiotemporally varying phenomena. Such models are encountered in a wide range of applications from traditional chemical engineering applications of reacting flow systems and thermofluid systems to new applications of drug delivery in metastatic tumors [1,2]. Many applications are deemed “safety critical” and therefore strict guarantees of performance and safety must be provided from modeling and simulation efforts prior to the deployment of a design (e.g., a new exothermic reactor, a new airplane, or a new drug therapy). To provide such rigorous guarantees, uncertainty in the environment and uncertainty in the model must be accounted for at the design stage.

Within the context of transient PDE models, robust simulation is the ability to calculate rigorous bounds on the system’s states for all realizations of parametric uncertainty [3]. This approach has been applied to ordinary differential equation (ODE) initial value problems (IVPs) and referred to as bounding the reachable set. The common methods for bounding the reachable set are differential inequalities (DI) [4,5], finite difference approximations and Taylor series expansion with remainder [6,7]. However, there have limited applications to transient PDE systems using these methods [8].

We propose using discrete-time DI [9] and interval-based finite-differencing methods [10] for efficiently computing tight enclosures of the spatiotemporally varying parametric solutions of transient PDE systems under uncertainty. Specifically, we use centered finite-differencing for the spatial derivatives and reformulate the parametric PDE as a large coupled system of IVPs using the method of lines. Then, we use the interval methods for finite differences to bound the spatial derivatives. Next, we use discrete-time DI to calculate the state bounds at each integration time step of the explicit integration method of choice. This approach is demonstrated to have desirable properties of computational efficiency, high accuracy of bounds, and overall effectiveness for applications modeled as transient PDE systems.

In this paper, we demonstrate the robust simulation approach on two motivating examples: a transient PFR and a spherical breast tumor model both exhibiting reaction-convection-diffusion phenomena. We calculate the efficient enclosures of the PFR system modeled as a transient one-dimensional PDE. We also applied the algorithms to the spherical breast tumor model with uncertainty in the physiological parameters to compute the rigorous global bounds of the reachable sets [11,12]. We implemented these methods using the Julia programming language [13] within the EAGO package developed for applications in global optimization [14]. The next steps are to use these methods within a deterministic global optimization framework for rigorous worst-case design under uncertainty.

References

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[8] P. Azunre, “Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations,” Optimal Control Applications and Methods, vol. 38, no. 4, pp. 618-633, 2017.

[9] X. Yang and J. K. Scott, “Efficient reachability bounds for discrete-time nonlinear systems by extending the continuous-time theory of differential inequalities,” in 2018 Annual American Control Conference (ACC), pp. 6242-6247, IEEE, 2018.

[10] C. Wang and Z.-P. Qiu, “Interval finite difference method for steady-state temperature field prediction with interval parameters,” Acta Mechanica Sinica, vol. 30, no. 2, pp. 161-166, 2014.

[11] C. Wang, J. D. Martin, H. Cabral, and M. D. Stuber, “Rigorous parameter estimation for model validation in oncological systems,” in AIChE Annual Meeting 2018, Pittsburgh, 2018.

[12] J. D. Martin, M. Panagi, C. Wang, T. T. Khan, M. R. Martin, C. Voutouri, K. Toh, P. Papageorgis, F. Mpekris, C. Polydorou, G. Ishii, S. Takahashi, N. Gotohda, T. Suzuki, M. Wilhelm, V. A. Melo, S. Quader, J. Norimatsu, R. M. Lanning, M. Kojima, M. D. Stuber, T. Stylianopoulos, H. Cabral, and K. Kataoka, “Dexamethasone increases nanocarrier delivery and efficacy in metastatic breast cancer by normalizing the tumor microenvironment,” Under review, 2018.

[13] J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, “Julia: A fresh approach to numerical computing,” SIAM review, vol. 59, no. 1, pp. 65-98, 2017.

[14] M. Wilhelm and M. D. Stuber, “EAGO: Easy advanced global optimization Julia package,” 2018. https://www.github.com/PSORLab/EAGO.jl