Numerical simulation of skin transport using Parareal. (English) Zbl 1360.65337

Summary: In silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination’s performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal’s convergence. The influence of load imbalances in time due to differences in number of iterations required by the spatial solver as well as spatio-temporal weak scaling is discussed.


65Y05 Parallel numerical computation
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
92-08 Computational methods for problems pertaining to biology
92C05 Biophysics
92C50 Medical applications (general)
Full Text: DOI arXiv Link


[1] Anissimov, YG; Roberts, MS, Diffusion modeling of percutaneous absorption kinetics: 3. variable diffusion and partition coefficients, consequences for stratum corneum depth profiles and desorption kinetics, J. Pharm. Sci., 93, 470-487, (2004)
[2] Anissimov, YG; Roberts, MS, Diffusion modelling of percutaneous absorption kinetics: 4. effects of slow equilibration process within stratum corneum on absorbtion and desorption kinetics, J. Pharm. Sci., 98, 772-781, (2009)
[3] Arteaga, A., Ruprecht, D., Krause, R.: A stencil-based implementation of Parareal in the C\(++\) domain specific embedded language STELLA. Appl. Math. Comput. (2015). doi:10.1016/j.amc.2014.12.055 · Zbl 1411.65173
[4] Aubanel, E, Scheduling of tasks in the parareal algorithm, Parallel Comput., 37, 172-182, (2011) · Zbl 1216.65124
[5] Bal, G; Kornhuber, R (ed.); etal., On the convergence and the stability of the parareal algorithm to solve partial differential equations, No. 40, 426-432, (2005), Berlin · Zbl 1066.65091
[6] Bylaska, EJ; Weare, JQ; Weare, JH, Extending molecular simulation time scales: parallel in time integrations for high-level quantum chemistry and complex force representations, J. Chem. Phys., 139, 074114, (2013)
[7] Celledoni, E; Kvamsdal, T, Parallelization in time for thermo-viscoplastic problems in extrusion of aluminium, Int. J. Numer. Methods Eng., 79, 576-598, (2009) · Zbl 1171.74460
[8] Demmel, JW; Eisenstat, SC; Gilbert, JR; Li, XS; Liu, JWH, A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20, 720-755, (1999) · Zbl 0931.65022
[9] Dick, B., Vogel, A., Khabi, D., Rupp, M., Küster, U., Wittum, G.: Utilization of empirically determined energy-optimal CPU-frequencies in a numerical simulation code. Comput. Vis. Sci. (2015). doi:10.1007/s00791-015-0249-8
[10] Dongarra, J., et al.: Applied Mathematics Research for Exascale Computing. Technical Report LLNL-TR-651000, Lawrence Livermore National Laboratory (2014). http://science.energy.gov/ /media/ascr/pdf/research/am/docs/EMWGreport.pdf
[11] Elwasif, W.R., Foley, S.S., Bernholdt, D.E., Berry, L.A., Samaddar, D., Newman, D.E., Snchez, R.S.: A dependency-driven formulation of parareal: parallel-in-time solution of PDEs as a many-task application. In: Proceedings of the 2011 ACM International Workshop on Many Task Computing on Grids and Supercomputers, p. 1524 (2011). doi:10.1145/2132876.2132883
[12] Emmett, M; Minion, ML, Toward an efficient parallel in time method for partial differential equations, Commun. Appl. Math. Comput. Sci., 7, 105132, (2012) · Zbl 1248.65106
[13] Falgout, RD; Friedhoff, S; Kolev, TV; MacLachlan, SP; Schroder, JB, Parallel time integration with multigrid, SIAM J. Sci. Comput., 36, c635c661, (2014)
[14] Farhat, C; Chandesris, M, Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications, Int. J. Numer. Methods Eng., 58, 13971434, (2003) · Zbl 1032.74701
[15] Gander, M.J., Vandewalle, S.: On the superlinear and linear convergence of the Parareal algorithm. In: Widlund, O.B., Keyes, D.E. (eds.) Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 55, pp. 291-298. Springer, Berlin (2007). doi:10.1007/978-3-540-34469-8_34
[16] Hansen, S., Lehr, C.M., Schaefer, U.F.: Modeling the human skin barrier—towards a better understanding of dermal absorption. Adv. Drug Deliv. Rev. (2013). doi:10.1016/j.addr.2012.12.002
[17] Kreienbuehl, A., Benedusi, P., Ruprecht, D., Krause, R.: Time parallel gravitational collapse simulation (2015) (in preparation) · Zbl 1349.76730
[18] Li, X., Demmel, J., Gilbert, J., iL. Grigori, Shao, M., Yamazaki, I.: SuperLU Users’ Guide. Technical Report LBNL-44289, Lawrence Berkeley National Laboratory (1999). http://crd.lbl.gov/ xiaoye/SuperLU/. Last update: August 2011
[19] Lions, JL; Maday, Y; Turinici, G, A “parareal” in time discretization of PDE’s, C. R. l’Acad. Sci. Ser. I Math., 332, 661668, (2001) · Zbl 0984.65085
[20] Minion, M.L., Speck, R., Bolten, M., Emmett, M., Ruprecht, D.: Interweaving PFASST and parallel multigrid. SIAM J. Sci. Comput. (2015). arxiv:1407.6486 · Zbl 1325.65193
[21] Minion, ML, A hybrid parareal spectral deferred corrections method, Commun. Appl. Math. Comput. Sci., 5, 265301, (2010) · Zbl 1208.65101
[22] Mula, O.: Some contributions towards the parallel simulation of time dependent neutron transport and the integration of observed data in real time. Ph.D. Thesis, Université Pierre et Marie Curie - Paris VI (2014). https://tel.archives-ouvertes.fr/tel-01081601 · Zbl 0931.65022
[23] Naegel, A., Heisig, M., Wittum, G.: A comparison of two- and three-dimensional models for the simulation of the permeability of human stratum corneum. Eur. J. Pharm. Biopharm. 72(2), 332-338 (2009). doi:10.1016/j.ejpb.2008.11.009. http://www.sciencedirect.com/science/article/B6T6C-4V1KMMP-1/2/b906a3a90140385ba35b48bed48fdef7 · Zbl 1171.74460
[24] Querleux, B. (ed.): Computational Biophysics of the Skin. Pan Stanford Publishing, Singapore (2014)
[25] Randles, A; Kaxiras, E, Parallel in time approximation of the lattice Boltzmann method for laminar flows, J. Comput. Phys., 270, 577586, (2014) · Zbl 1349.76730
[26] Reiter, S; Vogel, A; Heppner, I; Rupp, M; Wittum, G, A massively parallel geometric multigrid solver on hierarchically distributed grids, Comput. Vis. Sci., 16, 151-164, (2013) · Zbl 1380.65463
[27] Rim, J.E., Pinsky, P.M., van Osdol, W.W.: Using the method of homogenization to calculate the effective diffusivity of the stratum corneum with permeable corneocytes. J. Biomech. 41(4), 788-796 (2008). doi:10.1016/j.jbiomech.2007.11.011. http://www.sciencedirect.com/science/article/B6T82-4RWHXFR-2/2/bfe8e93f74d145a105071a106d6d227c
[28] Rim, JE; Pinsky, PM; Osdol, WW, Multiscale modeling framework of transdermal drug delivery, Ann. Biomed. Eng., 37, 1217-1229, (2009)
[29] Ruprecht, D., Speck, R., Emmett, M., Bolten, M., Krause, R.: Poster: Extreme-scale space-time parallelism. In: Proceedings of the 2013 Conference on High Performance Computing Networking, Storage and Analysis Companion, SC’13 Companion (2013). http://sc13.supercomputing.org/sites/default/files/PostersArchive/tech_posters/post148s2-file3.pdf
[30] Ruprecht, D., Speck, R., Krause, R.: Parareal for diffusion problems with space- and time-dependent coefficients. In: Domain Decomposition Methods in Science and Engineering XXII, Lecture Notes in Computational Science and Engineering, vol. 104, pp. 3-10. Springer, Switzerland (2015). doi:10.1007/978-3-319-18827-0_1 · Zbl 1343.65113
[31] Ruprecht, D, Convergence of parareal with spatial coarsening, PAMM, 14, 1031-1034, (2014)
[32] Samaddar, D; Newman, DE; Snchez, RS, Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm, J. Comput. Phys., 229, 65586573, (2010) · Zbl 1425.76090
[33] Speck, R., Ruprecht, D., Krause, R., Emmett, M., Minion, M.L., Winkel, M., Gibbon, P.: A massively space-time parallel N-body solver. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, SC’12, p. 92:1-92:11. IEEE Computer Society Press, Los Alamitos, CA, USA (2012). doi:10.1109/SC.2012.6 · Zbl 1443.76185
[34] Vogel, A., Calotoiu, A., Strube, A., Reiter, S., Nägel, A., Wolf, F., Wittum, G.: 10,000 performance models per minute-scalability of the UG4 simulation framework. In: Träff, J.L., Hunold, S., Versaci, F. (eds.) Euro-Par 2015: parallel processing, pp. 519-531. Springer, Berlin (2015)
[35] Vogel, A; Reiter, S; Rupp, M; Nägel, A; Wittum, G, UG4: A novel flexible software system for simulating PDE based models on high performance computers, Comput. Vis. Sci., 16, 165-179, (2013) · Zbl 1375.35003
[36] Wang, TF; Kasting, GB; Nitsche, JM, A multiphase microscopic diffusion model for stratum corneum permeability. I. formulation, solution, and illustrative results for representative compounds, J. Pharm. Sci., 95, 620-648, (2006)
[37] Wang, TF; Kasting, GB; Nitsche, JM, A multiphase microscopic diffusion model for stratum corneum permeability. II. estimation of physicochemical parameters, and application to a large permeability database, J. Pharm. Sci., 96, 3024-3051, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.