Inscrição na biblioteca: Guest
Portal Digital Begell Biblioteca digital da Begell eBooks Diários Referências e Anais Coleções de pesquisa
International Journal for Uncertainty Quantification
Fator do impacto: 4.911 FI de cinco anos: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Imprimir: 2152-5080
ISSN On-line: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2020033344
pages 351-373

YIELD OPTIMIZATION BASED ON ADAPTIVE NEWTON-MONTE CARLO AND POLYNOMIAL SURROGATES

Mona Fuhrländer
Institut für Teilchenbeschleunigung und Elektromagnetische Felder (TEMF), Technische Universität Darmstadt, Schlossgartenstr. 8, 64289 Darmstadt, Germany; Centre for Computational Engineering, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany
Niklas Georg
Institut für Dynamik und Schwingungen, Technische Universität Braunschweig, Braunschweig, Germany; Centre for Computational Engineering, Technische Universität Darmstadt, Darmstadt, Germany; Institut für Teilchenbeschleunigung und Elektromagnetische Felder (TEMF), Technische Universität Darmstadt, Darmstadt, Germany
Ulrich Römer
Institut für Dynamik und Schwingungen, Technische Universität Braunschweig, Schleinitzstraße 20, 38106 Braunschweig, Germany
Sebastian Schöps
Institut für Teilchenbeschleunigung und Elektromagnetische Felder (TEMF), Technische Universität Darmstadt, Schlossgartenstr. 8, 64289 Darmstadt, Germany; Centre for Computational Engineering, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany

RESUMO

In this paper we present an algorithm for yield estimation and optimization consisting of Hessian-based optimization methods, an adaptive Monte Carlo (MC) strategy, polynomial surrogates, and several error indicators. Yield estimation is used to quantify the impact of uncertainty in a manufacturing process. Since computational efficiency is one main issue in uncertainty quantification, we propose a hybrid method, where a large part of a MC sample is evaluated with a surrogate model, and only a small subset of the sample is reevaluated with a high-fidelity finite element model. In order to determine this critical fraction of the sample, an adjoint error indicator is used for both the surrogate error and the finite element error. For yield optimization we propose an adaptive Newton-MC method. We reduce computational effort and control the MC error by adaptively increasing the sample size. The proposed method minimizes the impact of uncertainty by optimizing the yield. It allows one to control the finite element error, surrogate error, and MC error. At the same time it is much more efficient than standard MC approaches combined with standard Newton algorithms.

Referências

  1. Graeb, H.E., Analog Design Centering and Sizing, Dordrecht: Springer, 2007.

  2. Hammersley, J., Monte Carlo Methods, New York: Springer Science & Business Media, 2013.

  3. Giles, M.B., Multilevel Monte Carlo Methods, Acta Numer, 24:259-328, 2015.

  4. Choi, C.K. and Yoo, H.H., Uncertainty Analysis of Nonlinear Systems Employing the First-Order Reliability Method, J. Mech. Sci. Technol., 26(1):39-44, 2012.

  5. Breitung, K., Asymptotic Approximations for Multinormal Integrals, J. Eng. Mech, 110(3):357-366,1984.

  6. Gallimard, L., Adaptive Reduced Basis Strategy for Rare-Event Simulations, Int. J. Numer. Methods Eng., 1:1-20, 2019.

  7. Au, S.K. and Beck, J.L., Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation, Probab. Eng. Mech., 16:263-277, 2001.

  8. Bect, J., Li, L., and Vazquez, E., Bayesian Subset Simulation, SIAM/ASA J. Uncertainty Quantif., 5(1):762-786, 2017.

  9. Bogoclu, C. and Roos, D., A Benchmark of Contemporary Metamodeling Algorithms, in VIIEur. Cong. Comput. Methods in Appl. Sci. Eng., Crete, Greece, 2016.

  10. Rao, C.R. and Toutenburg, H., Linear Models: Least Squares and Alternatives, 2nd ed., New York: Springer, 1999.

  11. Rasmussen, C.E. and Williams, C.K., Gaussian Processes for Machine Learning, Cambridge: The MIT Press, 2006.

  12. Babuska, I., Nobile, F., and Tempone, R., A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAMJ Numer. Anal., 45(3):1005-1034,2007.

  13. Leifsson, L., Du, X., and Koziel, S., Efficient Yield Estimation of Multiband Patch Antennas by Polynomial Chaos-Based Kriging, Int. J. Numer. Modell. Electron. Networks Dev. Fields, pp. 1-10, 2020.

  14. Nobile, F., Tempone, R., and Webster, C.G., A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data, SIAMJ. Numer. Anal., 46(5):2309-2345, 2008.

  15. Li, J. and Xiu, D., Evaluation of Failure Probability via Surrogate Models, J. Comput. Phys., 229(23):8966-8980, 2010.

  16. Butler, T. and Wildey, T., Utilizing Adjoint-Based Error Estimates for Surrogate Models to Accurately Predict Probabilities of Events, Int. J. Uncertainty Quantif., 8(2):143-159, 2018.

  17. Ulbrich, M. and Ulbrich, S., Nichtlineare Optimierung, Basel, Switzerland: Birkhauser, 2012.

  18. Babuska, I., Nobile, F., and Tempone, R., A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAMJ. Numer. Anal, 45(3):1005-1034,2007.

  19. Chkifa, A., Cohen, A., and Schwab, C., Breaking the Curse of Dimensionality in Sparse Polynomial Approximation of Parametric PDEs, J. Math. Pures Appl., 103(2):400-428,2015.

  20. Papaioannou, I., Daub, M., Drieschner, M., Duddeck, F., Ehre, M., Eichner, L., Eigel, M., Gotz, M., Graf, W., and Grasedyck, L., Assessment and Design of an Engineering Structure with Polymorphic Uncertainty Quantification, GAMM, 42(2):e201900009, 2019.

  21. Adams, B., Bauman, L., Bohnhoff, W., Dalbey, K., Ebeida, M., Eddy, J., Eldred, M., Hough, P., Hu, K., and Jakeman, J., Dakota: A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, etc.: Version 6.0 User's Manual, Sandia Technical Report SAND2014-4633,2015.

  22. Caflisch, R.E., Monte Carlo and Quasi-Monte Carlo Methods, Acta Numer, 7:1-49, 1998.

  23. Xiu, D. and Karniadakis, G.E., The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM J. Sci. Comput, 24(2):619-644,2002.

  24. Xiu, D. and Hesthaven, J.S., High-Order Collocation Methods for Differential Equations with Random Inputs, SIAM J. Sci. Comput, 27(3):1118-1139, 2005.

  25. Bungartz, H.J. and Griebel, M., Sparse Grids, Acta Numur., 13:147-269, 2004.

  26. Georg, N., Loukrezis, D., Romer, U., and Schops, S., Enhanced Adaptive Surrogate Models with Applications in Uncertainty Quantification for Nanoplasmonics, Int. J. Uncertainty Quantif, 10(2):165-193, 2020.

  27. Narayan, A. and Jakeman, J.D., Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation, SIAMJ. Sci. Comput, 36(6):A495-A521,2014.

  28. Butler, T., Constantine, P., and Wildey, T., A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods, SIAMJ. MatrxAnal. Appl, 33(1):195-209, 2012.

  29. Butler, T., Dawson, C., and Wildey, T., Propagation of Uncertainties Using Improved Surrogate Models, SIAM/ASA J. Uncer-tainty Quantif, 1(1):164-191, 2013.

  30. Jakeman, J.D. and Wildey, T., Enhancing Adaptive Sparse Grid Approximations and Improving Refinement Strategies Using Adjoint-Based a Posteriori Error Estimates, J. Comput. Phys., 280:54-71, 2015.

  31. Becker, R. and Rannacher, R., An Optimal Control Approach to a Posteriori Error Estimation in Finite Element Methods, Acta Numer., 10:1-102,2001.

  32. Eriksson, K., Estep, D., Hansbo, P., and Johnson, C., Introduction to Adaptive Methods for Differential Equations, Acta Numer, 4:105-158, 1995.

  33. Zienkiewicz, O.C. andZhu, J.Z., The Superconvergent Patch Recovery and a Posteriori Error Estimates. Part 1: The Recovery Technique, Int. J. Numer. Methods Eng., 33(7):1331-1364, 1992.

  34. Bellman, R., Curse ofDimensionality, Adaptive Control Processes: A Guided Tour, Princeton, NJ: Princeton University Press, 1961.

  35. Geiersbach, C. and Wollner, W., A Stochastic Gradient Method with Mesh Refinement for PDE Constrained Optimization under Uncertainty, Math. Opt. Control, arXiv:1905.08650, 2019.

  36. Loukrezis, D., Romer, U., and De Gersem, H., Assessing the Performance of Leja and Clenshaw-Curtis Collocation for Computational Electromagnetics with Random Input Data, Int. J. Uncertainty Quantif., 9(1):33-57, 2019.

  37. Loukrezis, D., Adaptive Approximations for High-Dimensional Uncertainty Quantification in Stochastic Parametric Electro-magnetic Field Simulations, PhD, Technische Universitat, 2019.

  38. Jin, J.M., The Finite Element Method in Electromagnetics, New York: Wiley & Sons, 2015.

  39. Jackson, J.D., Classical Electrodynamics, 3rd ed., New York: Wiley & Sons, 1998.

  40. Monk, P., Finite Element Methods for Maxwell's Equations, Oxford: Oxford University Press, 2003.

  41. Nedelec, J.C., Mixed Finite Elements in R3, Numer. Math., 35(3):315-341, 1980.

  42. Alnss, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., and Wells, G.N., The FEniCS Project Version 1.5, Arch. Numer. Software, 3(100):9-23, 2015.

  43. Dembo, R.S., Eisenstat, S.C., and Steihaug, T., Inexact Newton Methods, SIAMJ. Numer. Anal, 19(2):400-408, 1982.


Articles with similar content:

A NOVEL INDEPENDENT PROCESSING LLL ALGORITHM FOR MIMO DETECTION
Telecommunications and Radio Engineering, Vol.75, 2016, issue 7
L. Jianping, L. Huazhang
COMPUTING GREEN'S FUNCTIONS FOR FLOW IN HETEROGENEOUS COMPOSITE MEDIA
International Journal for Uncertainty Quantification, Vol.3, 2013, issue 1
David A. Barajas-Solano, Daniel M. Tartakovsky
FAST AND FLEXIBLE UNCERTAINTY QUANTIFICATION THROUGH A DATA-DRIVEN SURROGATE MODEL
International Journal for Uncertainty Quantification, Vol.8, 2018, issue 2
Gerta Köster, Hans-Joachim Bungartz, Felix Dietrich, Tobias Neckel, Florian Künzner
A MULTILEVEL APPROACH FOR SEQUENTIAL INFERENCE ON PARTIALLY OBSERVED DETERMINISTIC SYSTEMS
International Journal for Uncertainty Quantification, Vol.9, 2019, issue 4
Ajay Jasra, Yi Xu, Kody J.H. Law
GRADIENT-BASED STOCHASTIC OPTIMIZATION METHODS IN BAYESIAN EXPERIMENTAL DESIGN
International Journal for Uncertainty Quantification, Vol.4, 2014, issue 6
Youssef Marzouk, Xun Huan