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doi: 10.3934/dcdsb.2022095
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Quantify uncertainty by estimating the probability density function of the output of interest using MLMC based Bayes method

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author: Ju Ming

Received  September 2021 Revised  March 2022 Early access May 2022

In uncertainty quantification, the quantity of interest is usually the statistics of the space and/or time integration of system solution. In order to reduce the computational cost, a Bayes estimator based on multilevel Monte Carlo (MLMC) is introduced in this paper. The cumulative distribution function of the output of interest, that is, the expectation of the indicator function, is estimated by MLMC method instead of the classic Monte Carlo simulation. Then, combined with the corresponding probability density function, the quantity of interest is obtained by using some specific quadrature rules. In addition, the smoothing of indicator function and Latin hypercube sampling are used to accelerate the reduction of variance. An elliptic stochastic partial differential equation is used to provide a research context for this model. Numerical experiments are performed to verify the advantage of computational reduction and accuracy improvement of our MLMC-Bayes method.

Citation: Meixin Xiong, Liuhong Chen, Ju Ming. Quantify uncertainty by estimating the probability density function of the output of interest using MLMC based Bayes method. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022095
References:
[1]

A. A. AliE. Ullmann and M. Hinze, Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients, SIAM-ASA J. Uncertain. Quantif., 5 (2017), 466-492.  doi: 10.1137/16M109870X.

[2]

I. BabuškaF. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034.  doi: 10.1137/050645142.

[3]

A. BarthC. Schwab and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math., 119 (2011), 123-161.  doi: 10.1007/s00211-011-0377-0.

[4]

C. Bierig and A. Chernov, Approximation of probability density functions by the multilevel Monte Carlo maximum entropy method, J. Comput. Phys., 314 (2016), 661-681.  doi: 10.1016/j.jcp.2016.03.027.

[5] S. BrooksA. GelmanG. Jones and X. Meng, Handbook of Markov Chain Monte Carlo, CRC Press, Boca Raton, FL, 2011.  doi: 10.1201/b10905.
[6]

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing, Springer-Verlag, New York, 2007.

[7]

J. CharrierR. Scheichl and A. Teckentrup., Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM J. Numer. Anal., 51 (2013), 322-352.  doi: 10.1137/110853054.

[8]

Q. Chen and J. Ming, The multilevel Monte Carlo method for simulations of turbulent flows, Mon. Weather. Rev., 146 (2018), 2933-2947. 

[9]

C. C. DrovandiM. T. Moores and R. J. Boys, Accelerating pseudo-marginal MCMC using Gaussian processes, Comput. Statist. Data Anal., 118 (2018), 1-17.  doi: 10.1016/j.csda.2017.09.002.

[10]

D. ElfversonF. Hellman and A. Målqvist, A multilevel Monte Carlo method for computing failure probabilities, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 312-330.  doi: 10.1137/140984294.

[11]

F. FagerlundF. HellmanA. Målqvist and A. Niemi, Multilevel Monte Carlo methods for computing failure probability of porous media flow systems, Adv. Water Resour., 94 (2016), 498-509. 

[12]

H. R. Fairbanks, S. Osborn and P. S. Vassilevski, Estimating posterior quantity of interest expectations in a multilevel scalable framework, Numer. Linear Algebra Appl., 28 (2021), Paper No. e2352, 20 pp. doi: 10.1002/nla.2352.

[13]

G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2553-7.

[14]

Y. Gal and Z. Ghahramani, Dropout as a Bayesian approximation: Representing model uncertainty in deep learning, in International Conference on Machine Learning, 48 (2016), 1050–1059. arXiv: 1506.02142.

[15]

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3094-6.

[16]

M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.

[17]

M. B. GilesT. Nagapetyan and K. Ritter, Multilevel Monte Carlo approximation of distribution functions and densities, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 267-295.  doi: 10.1137/140960086.

[18]

S. Heinrich, Multilevel Monte Carlo method, in Large-Scale Scientific Computing (eds. S. Margenov and J. Waśniewski and P. Yalamov), Springer Berlin Heidelberg, (2001), 58–67. doi: 10.1007/3-540-45346-6_5.

[19]

T. HironakaM. B. GilesT. Goda and H. Thom, Multilevel Monte Carlo estimation of the expected value of sample information, SIAM/ASA J. Uncertain. Quantif., 8 (2020), 1236-1259.  doi: 10.1137/19M1284981.

[20]

D. C. Knill and A. Pouget, The Bayesian brain: The role of uncertainty in neural coding and computation, Trends Neurosci., 27 (2004), 712-719. 

[21]

R. Krzysztofowicz, The case for probabilistic forecasting in hydrology, J. Hydrol., 249 (2001), 2-9. 

[22]

W. K. LiuT. Belytschko and A. Mani, Random field finite elements, Internat. J. Numer. Methods Engrg., 23 (1986), 1831-1845.  doi: 10.1002/nme.1620231004.

[23]

D. LuG. ZhangC. Webster and C. Barbier, An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations, Water Resour. Res., 52 (2016), 9642-9660. 

[24]

O. L. Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, Springer, New York, 2010. doi: 10.1007/978-90-481-3520-2.

[25]

M. D. MckayR. J. Beckman and W. J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 42 (2000), 55-61. 

[26]

J. Neyman, On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection, in Breakthroughs in Statistics (eds. S. Kotz and N. L. Johnson), Springer, New York, (1992), 123–150.

[27]

H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970081.

[28]

F. NobileR. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345.  doi: 10.1137/060663660.

[29]

T. N. PalmerG. J. ShuttsR. HagedornF. J. Doblas-ReyesT. Jung and M. Leutbecher, Representing model uncertainty in weather and climate prediction, Annu. Rev. Earth Planet. Sci., 33 (2005), 163-193. 

[30] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804.
[31]

P. J. Roache, Quantification of uncertainty in computational fluid dynamics, Annu. Rev. Fluid Mech., 29 (1997), 123-160.  doi: 10.1146/annurev.fluid.29.1.123.

[32]

C. Scheidt, L. Li and J. Caers, Quantifying Uncertainty in Subsurface Systems, John Wiley & Sons: New York, 2018. doi: 10.1002/9781119325888.

[33]

R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM Computational Science & Engineering Series: Philadelphia, USA, 2014.

[34]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.

[35]

S. Taverniers and D. M. Tartakovsky, Estimation of distributions via multilevel Monte Carlo with stratified sampling, J. Comput. Phys., 419 (2020), 109572, 21 pp. doi: 10.1016/j.jcp.2020.109572.

[36]

S. T. Tokdar and R. E. Kass, Importance sampling: A review, Wiley Interdiscip. Rev. Comput. Stat., 2 (2010), 54-60. 

[37]

P. Wesseling, Introduction to Multigrid Methods, John Wiley & Sons, Chichester, 1992.

[38]

D. Wilson and R. E. Baker, Multi-level methods and approximating distribution functions, AIP Adv., 6 (2016), 075020. 

[39]

D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139.  doi: 10.1137/040615201.

[40]

D. Xiu and G. E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.

[41]

Z. YangX. GuiJ. Ming and G. Hu, Bayesian approach to inverse time-harmonic acoustic obstacle scattering with phaseless data generated by point source waves, Comput. Methods Appl. Mech. Engrg., 386 (2021), 114073.  doi: 10.1016/j.cma.2021.114073.

[42]

Z. Yang, X. Gui, J. Ming and G. Hu, Bayesian approach to inverse time-harmonic acoustic scattering with phaseless far-field data, Inverse Problems, 36 (2020), 065012, 30 pp. doi: 10.1088/1361-6420/ab82ee.

[43] D. Zhang, Stochastic Methods for Flow in Porous Media: Coping with Uncertainties, Academic Press, San Diego, CA, 2002.  doi: 10.2136/vzj2005.0133br.
[44]

J. Zhang and A. A. Taflanidis, Accelerating MCMC via kriging-based adaptive independent proposals and delayed rejection, Comput. Methods Appl. Mech. Eng., 355 (2019), 1124-1147.  doi: 10.1016/j.cma.2019.07.016.

show all references

References:
[1]

A. A. AliE. Ullmann and M. Hinze, Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients, SIAM-ASA J. Uncertain. Quantif., 5 (2017), 466-492.  doi: 10.1137/16M109870X.

[2]

I. BabuškaF. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034.  doi: 10.1137/050645142.

[3]

A. BarthC. Schwab and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math., 119 (2011), 123-161.  doi: 10.1007/s00211-011-0377-0.

[4]

C. Bierig and A. Chernov, Approximation of probability density functions by the multilevel Monte Carlo maximum entropy method, J. Comput. Phys., 314 (2016), 661-681.  doi: 10.1016/j.jcp.2016.03.027.

[5] S. BrooksA. GelmanG. Jones and X. Meng, Handbook of Markov Chain Monte Carlo, CRC Press, Boca Raton, FL, 2011.  doi: 10.1201/b10905.
[6]

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing, Springer-Verlag, New York, 2007.

[7]

J. CharrierR. Scheichl and A. Teckentrup., Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM J. Numer. Anal., 51 (2013), 322-352.  doi: 10.1137/110853054.

[8]

Q. Chen and J. Ming, The multilevel Monte Carlo method for simulations of turbulent flows, Mon. Weather. Rev., 146 (2018), 2933-2947. 

[9]

C. C. DrovandiM. T. Moores and R. J. Boys, Accelerating pseudo-marginal MCMC using Gaussian processes, Comput. Statist. Data Anal., 118 (2018), 1-17.  doi: 10.1016/j.csda.2017.09.002.

[10]

D. ElfversonF. Hellman and A. Målqvist, A multilevel Monte Carlo method for computing failure probabilities, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 312-330.  doi: 10.1137/140984294.

[11]

F. FagerlundF. HellmanA. Målqvist and A. Niemi, Multilevel Monte Carlo methods for computing failure probability of porous media flow systems, Adv. Water Resour., 94 (2016), 498-509. 

[12]

H. R. Fairbanks, S. Osborn and P. S. Vassilevski, Estimating posterior quantity of interest expectations in a multilevel scalable framework, Numer. Linear Algebra Appl., 28 (2021), Paper No. e2352, 20 pp. doi: 10.1002/nla.2352.

[13]

G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2553-7.

[14]

Y. Gal and Z. Ghahramani, Dropout as a Bayesian approximation: Representing model uncertainty in deep learning, in International Conference on Machine Learning, 48 (2016), 1050–1059. arXiv: 1506.02142.

[15]

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3094-6.

[16]

M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.

[17]

M. B. GilesT. Nagapetyan and K. Ritter, Multilevel Monte Carlo approximation of distribution functions and densities, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 267-295.  doi: 10.1137/140960086.

[18]

S. Heinrich, Multilevel Monte Carlo method, in Large-Scale Scientific Computing (eds. S. Margenov and J. Waśniewski and P. Yalamov), Springer Berlin Heidelberg, (2001), 58–67. doi: 10.1007/3-540-45346-6_5.

[19]

T. HironakaM. B. GilesT. Goda and H. Thom, Multilevel Monte Carlo estimation of the expected value of sample information, SIAM/ASA J. Uncertain. Quantif., 8 (2020), 1236-1259.  doi: 10.1137/19M1284981.

[20]

D. C. Knill and A. Pouget, The Bayesian brain: The role of uncertainty in neural coding and computation, Trends Neurosci., 27 (2004), 712-719. 

[21]

R. Krzysztofowicz, The case for probabilistic forecasting in hydrology, J. Hydrol., 249 (2001), 2-9. 

[22]

W. K. LiuT. Belytschko and A. Mani, Random field finite elements, Internat. J. Numer. Methods Engrg., 23 (1986), 1831-1845.  doi: 10.1002/nme.1620231004.

[23]

D. LuG. ZhangC. Webster and C. Barbier, An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations, Water Resour. Res., 52 (2016), 9642-9660. 

[24]

O. L. Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, Springer, New York, 2010. doi: 10.1007/978-90-481-3520-2.

[25]

M. D. MckayR. J. Beckman and W. J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 42 (2000), 55-61. 

[26]

J. Neyman, On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection, in Breakthroughs in Statistics (eds. S. Kotz and N. L. Johnson), Springer, New York, (1992), 123–150.

[27]

H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970081.

[28]

F. NobileR. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345.  doi: 10.1137/060663660.

[29]

T. N. PalmerG. J. ShuttsR. HagedornF. J. Doblas-ReyesT. Jung and M. Leutbecher, Representing model uncertainty in weather and climate prediction, Annu. Rev. Earth Planet. Sci., 33 (2005), 163-193. 

[30] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804.
[31]

P. J. Roache, Quantification of uncertainty in computational fluid dynamics, Annu. Rev. Fluid Mech., 29 (1997), 123-160.  doi: 10.1146/annurev.fluid.29.1.123.

[32]

C. Scheidt, L. Li and J. Caers, Quantifying Uncertainty in Subsurface Systems, John Wiley & Sons: New York, 2018. doi: 10.1002/9781119325888.

[33]

R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM Computational Science & Engineering Series: Philadelphia, USA, 2014.

[34]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.

[35]

S. Taverniers and D. M. Tartakovsky, Estimation of distributions via multilevel Monte Carlo with stratified sampling, J. Comput. Phys., 419 (2020), 109572, 21 pp. doi: 10.1016/j.jcp.2020.109572.

[36]

S. T. Tokdar and R. E. Kass, Importance sampling: A review, Wiley Interdiscip. Rev. Comput. Stat., 2 (2010), 54-60. 

[37]

P. Wesseling, Introduction to Multigrid Methods, John Wiley & Sons, Chichester, 1992.

[38]

D. Wilson and R. E. Baker, Multi-level methods and approximating distribution functions, AIP Adv., 6 (2016), 075020. 

[39]

D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139.  doi: 10.1137/040615201.

[40]

D. Xiu and G. E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.

[41]

Z. YangX. GuiJ. Ming and G. Hu, Bayesian approach to inverse time-harmonic acoustic obstacle scattering with phaseless data generated by point source waves, Comput. Methods Appl. Mech. Engrg., 386 (2021), 114073.  doi: 10.1016/j.cma.2021.114073.

[42]

Z. Yang, X. Gui, J. Ming and G. Hu, Bayesian approach to inverse time-harmonic acoustic scattering with phaseless far-field data, Inverse Problems, 36 (2020), 065012, 30 pp. doi: 10.1088/1361-6420/ab82ee.

[43] D. Zhang, Stochastic Methods for Flow in Porous Media: Coping with Uncertainties, Academic Press, San Diego, CA, 2002.  doi: 10.2136/vzj2005.0133br.
[44]

J. Zhang and A. A. Taflanidis, Accelerating MCMC via kriging-based adaptive independent proposals and delayed rejection, Comput. Methods Appl. Mech. Eng., 355 (2019), 1124-1147.  doi: 10.1016/j.cma.2019.07.016.

Figure 1.  (a) The computational cost, (b) variance, and (c) sample size versus the level of the MLMC-Bayes estimator
Figure 2.  (a) The CDFs of $ G $ estimated using the MLMC-Bayes method, and (b) their $ L^{\infty} $-norm between adjacent levels
Figure 3.  The PDFs of $ G $ estimated using the MLMC-Bayes method: (a) $ G \in [0,3] $; (b) $ G \in [0,1] $
Figure 4.  (a) $ L^{\infty} $-norm and (b) KLD between adjacent levels of the estimated PDF
Figure 5.  The estimated (a) CDFs and (b) PDFs of $ G $ by using MC-smooth and MC-nonsmooth methods
Table 1.  The mesh size, calculation cost, variance, sample size, and bandwidth in each level of the MLMC-Bayes estimator
$ {\boldsymbol{L}} $ 0 1 2 3 4
$ h_l $ 1/2 1/4 1/8 1/16 1/32
$ \mathcal{C}_l $ 0.0007 0.0024 0.0092 0.0372 0.1578
$ \mathcal{V}_l $ 0.2200 0.3974 0.1447 0.1046 0.0737
$ N_l $ 4636 3384 1079 338 101
$ \delta_l $ $ 9.4604 \times 10^{-5} $ $ 3.0518 \times 10^{-6} $ $ 3.0518 \times 10^{-6} $ $ 3.0518 \times 10^{-6} $ $ 3.0518 \times 10^{-6} $
$ {\boldsymbol{L}} $ 0 1 2 3 4
$ h_l $ 1/2 1/4 1/8 1/16 1/32
$ \mathcal{C}_l $ 0.0007 0.0024 0.0092 0.0372 0.1578
$ \mathcal{V}_l $ 0.2200 0.3974 0.1447 0.1046 0.0737
$ N_l $ 4636 3384 1079 338 101
$ \delta_l $ $ 9.4604 \times 10^{-5} $ $ 3.0518 \times 10^{-6} $ $ 3.0518 \times 10^{-6} $ $ 3.0518 \times 10^{-6} $ $ 3.0518 \times 10^{-6} $
Table 2.  Simulation results obtained by different methods
MLMC-Bayes MC-Nonsmooth MC-Smooth MC MC-Equal
QoI 0.2341 0.2313 0.2308 0.2348 0.0177
$ N_L $ 101 62493 124918 62493 316
$ \mathcal{V}_L $ - 0.2500 0.2498 - -
$ \mathcal{C}_t $ 49.8050 $ 9.8614 \times 10^{3} $ $ 1.9712\times 10^{4} $ $ 9.8614\times 10^{3} $ 49.8050
MLMC-Bayes MC-Nonsmooth MC-Smooth MC MC-Equal
QoI 0.2341 0.2313 0.2308 0.2348 0.0177
$ N_L $ 101 62493 124918 62493 316
$ \mathcal{V}_L $ - 0.2500 0.2498 - -
$ \mathcal{C}_t $ 49.8050 $ 9.8614 \times 10^{3} $ $ 1.9712\times 10^{4} $ $ 9.8614\times 10^{3} $ 49.8050
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