November  2016, 10(4): 943-975. doi: 10.3934/ipi.2016028

FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems

1. 

Department of Aerospace Engineering and Engineering Mechanics, Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, TX 78712

2. 

Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712, United States

Received  August 2015 Revised  May 2016 Published  October 2016

We present a systematic construction of FEM-based dimension-independent (discretization-invariant) Markov chain Monte Carlo (MCMC) approaches to explore PDE-constrained Bayesian inverse problems in infinite dimensional parameter spaces. In particular, we consider two frameworks to achieve this goal: Metropolize-then-discretize and discretize-then-Metropolize. The former refers to the method of discretizing function-space MCMC methods. The latter, on the other hand, first discretizes the Bayesian inverse problem and then proposes MCMC methods for the resulting discretized posterior probability density. In general, these two frameworks do not commute, that is, the resulting finite dimensional MCMC algorithms are not identical. The discretization step of the former may not be trivial since it involves both numerical analysis and probability theory, while the latter, perhaps ``easier'', may not be discretization-invariant using traditional approaches. This paper constructively develops finite element (FEM) discretization schemes for both frameworks and shows that both commutativity and discretization-invariance are attained. In particular, it shows how to construct discretize-then-Metropolize approaches for both Metropolis-adjusted Langevin algorithm and the hybrid Monte Carlo method that commute with their Metropolize-then-discretize counterparts. The key that enables this achievement is a proper FEM discretization of the prior, the likelihood, and the Bayes' formula, together with a correct definition of quantities such as the gradient and the covariance matrix in discretized finite dimensional parameter spaces. The implication is that practitioners can take advantage of the developments in this paper to straightforwardly construct discretization-invariant discretize-then-Metropolize MCMC for large-scale inverse problems. Numerical results for one- and two-dimensional elliptic inverse problems with up to $17899$ parameters are presented to support the proposed developments.
Citation: Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems & Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028
References:
[1]

A. Beskos, N. Pillai, G. Roberts, J-M Sanz-Serna and A. Stuart, Optimal tuning of the hybrid Monte Carlo algorithm,, Bernoulli, 19 (2013), 1501. doi: 10.3150/12-BEJ414. Google Scholar

[2]

A. Beskos, F. J. Pinski, J. M. Sanz-Serna and A. M. Stuart, Hybrid Monte Carlo on Hilbert spaces,, Stochastic Processes and their Applications, 121 (2011), 2201. doi: 10.1016/j.spa.2011.06.003. Google Scholar

[3]

A. Beskos and A. M. Stuart, MCMC methods for sampling function space,, in Invited Lectures: Sixth International Congress on Industrial and Applied Mathematics, (2007), 337. doi: 10.4171/056-1/16. Google Scholar

[4]

A. Borzí and V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations,, SIAM, (2012). Google Scholar

[5]

T. Bui-Thanh and O. Ghattas, Analysis of the Hessian for inverse scattering problems. Part I: Inverse shape scattering of acoustic waves,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/5/055001. Google Scholar

[6]

_________, Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves,, Inverse Problems, 28 (2012). Google Scholar

[7]

_________, Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves., Inverse Problems and Imaging, (2013). Google Scholar

[8]

_________, Randomized maximum likelihood sampling for large-scale Bayesian inverse problems,, In preparation, (2013). Google Scholar

[9]

_________, An analysis of infinite dimensional Bayesian inverse shape acoustic scattering and its numerical approximation,, SIAM Journal of Uncertainty Quantification, 2 (2014), 203. doi: 10.1137/120894877. Google Scholar

[10]

T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion,, SIAM Journal on Scientific Computing, 35 (2013). doi: 10.1137/12089586X. Google Scholar

[11]

T. Bui-Thanh and M. A. Girolami, Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo,, Inverse Problems, 30 (2014). doi: 10.1088/0266-5611/30/11/114014. Google Scholar

[12]

F. P. Casey, J. J. Waterfall, R. N. Gutenkunst, C. R. Myers and J. P. Sethna, Variational method for estimating the rate of convergence of Marko-chain Monte Carlo algorithms,, Phy. Rev. E., 78 (2008). Google Scholar

[13]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978). Google Scholar

[14]

S. L. Cotter, G. O. Roberts, A. M. Stuart and D. White, MCMC methods for functions: Modifying old algorithms to make them faster,, Statistical Science, 28 (2013), 424. doi: 10.1214/13-STS421. Google Scholar

[15]

M. Dashti, K. J. H. Law, A. M. Stuart and J. Voss, MAP estimators and their consistency in Bayesian nonparametric inverse problems,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/9/095017. Google Scholar

[16]

S. Duane, A. D. Kennedy, B. Pendleton and D. Roweth, Hybrid Monte Carlo,, Phys. Lett. B, 195 (1987), 216. doi: 10.1016/0370-2693(87)91197-X. Google Scholar

[17]

J. N. Franklin, Well-posed stochastic extensions of ill-posed linear problems,, Journal of Mathematical Analysis and Applications, 31 (1970), 682. doi: 10.1016/0022-247X(70)90017-X. Google Scholar

[18]

A. Gelman, G. O. Roberts and W. R. Gilks, Efficient Metropolis jumping rules,, in Bayesian Statistics, 5 (1996), 599. Google Scholar

[19]

M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods,, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 123. doi: 10.1111/j.1467-9868.2010.00765.x. Google Scholar

[20]

M. D. Gunzburger, Perspectives in Flow Control and Optimization,, SIAM, (2003). Google Scholar

[21]

H. Haario, M. Laine, A. Miravete and E. Saksman, DRAM: Efficient adaptive MCMC,, Statistics and Computing, 16 (2006), 339. doi: 10.1007/s11222-006-9438-0. Google Scholar

[22]

M. Hairer, A. M. Stuart and J. Voss, Analysis of SPDEs arising in path sampling. part II: The nonlinear case,, Annals of Applied Probability, 17 (2007), 1657. doi: 10.1214/07-AAP441. Google Scholar

[23]

W. Keith Hastings, Monte Carlo sampling methods using Markov chains and their applications,, Biometrika, 57 (1970), 97. doi: 10.1093/biomet/57.1.97. Google Scholar

[24]

E. Herbst, Gradient and Hessian-based MCMC for DSGE models,, (2010). Unpublished manuscript., (2010). Google Scholar

[25]

M. Ilić, F. Liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation,, Frac. Calc. and A Anal., 8 (2005), 323. Google Scholar

[26]

S. Lasanen, Discretizations of Generalized Random Variables with Applications to Inverse Problems,, PhD thesis, (2002). Google Scholar

[27]

M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalized random variables,, Inverse Problems, 5 (1989), 599. doi: 10.1088/0266-5611/5/4/011. Google Scholar

[28]

F. Lindgren, H. Rue and J. Lindström, An explicit link between gaussian fields and gaussian markov random fields: The stochastic partial differential equation approach,, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 423. doi: 10.1111/j.1467-9868.2011.00777.x. Google Scholar

[29]

J. Martin, L. C. Wilcox, C. Burstedde and O. Ghattas, A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion,, SIAM Journal on Scientific Computing, 34 (2012). doi: 10.1137/110845598. Google Scholar

[30]

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines,, The Journal of Chemical Physics, 21 (1953), 1087. doi: 10.1063/1.1699114. Google Scholar

[31]

R. M. Neal, Handbook of Markov Chain Monte Carlo,, Chapman & Hall / CRC Press, (2010). Google Scholar

[32]

J. Nocedal and S. J. Wright, Numerical Optimization,, Springer Verlag, (2006). Google Scholar

[33]

M. Ottobre, N. S. Pillai, F. J. Pinski and A. M. Stuart, A function space HMC algorithm with second order langevin diffusion limit,, Bernoulli, 22 (2016), 60. doi: 10.3150/14-BEJ621. Google Scholar

[34]

P. Piiroinen, Statistical Measurements, Experiments, and Applications,, PhD thesis, (2005). Google Scholar

[35]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambidge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[36]

Y. Qi and T. P. Minka, Hessian-based Markov chain Monte-Carlo algorithms,, in First Cape Cod Workshop on Monte Carlo Methods, (2002). Google Scholar

[37]

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer Texts in Statistics),, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-4145-2. Google Scholar

[38]

G. O. Roberts, A. Gelman and W. R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms,, The Annals of Applied Probability, 7 (1997), 110. doi: 10.1214/aoap/1034625254. Google Scholar

[39]

G. O. Roberts and J. S. Rosenthal, Optimal scaling of discrete approximations to Langevin diffusions,, J. R. Statist. Soc. B, 60 (1997), 255. doi: 10.1111/1467-9868.00123. Google Scholar

[40]

J. Rosenthal, Optimal proposal distributions and adaptive MCMC,, in Handbook of Markov chain Monte Carlo, (2011), 93. Google Scholar

[41]

P. J. Rossky, J. D. Doll and H. L. Friedman, Brownian dynamics as smart Monte Carlo simulation,, J. Chem. Phys., 69 (1978). doi: 10.1063/1.436415. Google Scholar

[42]

D. P. Simpson, Krylov subpsace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous Diffusion,, PhD thesis, (2008). Google Scholar

[43]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numerica, 19 (2010), 451. doi: 10.1017/S0962492910000061. Google Scholar

[44]

A. M. Stuart, J. Voss and P. Wiberg, Conditional path sampling of SDEs and the Langevin MCMC method,, Communications in Mathematical Sciences, 2 (2004), 685. doi: 10.4310/CMS.2004.v2.n4.a7. Google Scholar

[45]

L. Tierney, A note on Metropolis-Hastings kernels for general state spaces,, Annals of Applied Probability, 8 (1998), 1. doi: 10.1214/aoap/1027961031. Google Scholar

[46]

C. Vacar, J.-F. Giovannelli and Y. Berthoumieu, Langevin and Hessian with Fisher approximation stochastic sampling for parameter estimation of structured covariance,, IEEE International Conference on Acoustics, (2011), 3964. doi: 10.1109/ICASSP.2011.5947220. Google Scholar

[47]

B. Vexler, Adaptive finite element methods for parameter identification problems,, Model Based Parameter Estimation, (2012), 31. doi: 10.1007/978-3-642-30367-8_2. Google Scholar

[48]

C. R. Vogel, Computational Methods for Inverse Problems,, Frontiers in Applied Mathematics, (2002). doi: 10.1137/1.9780898717570. Google Scholar

[49]

Y. Zhang and C. Sutton, Quasi-Newton methods for Markov chain Monte Carlo,, in Advances in Neural Information Processing Systems, (2011). Google Scholar

show all references

References:
[1]

A. Beskos, N. Pillai, G. Roberts, J-M Sanz-Serna and A. Stuart, Optimal tuning of the hybrid Monte Carlo algorithm,, Bernoulli, 19 (2013), 1501. doi: 10.3150/12-BEJ414. Google Scholar

[2]

A. Beskos, F. J. Pinski, J. M. Sanz-Serna and A. M. Stuart, Hybrid Monte Carlo on Hilbert spaces,, Stochastic Processes and their Applications, 121 (2011), 2201. doi: 10.1016/j.spa.2011.06.003. Google Scholar

[3]

A. Beskos and A. M. Stuart, MCMC methods for sampling function space,, in Invited Lectures: Sixth International Congress on Industrial and Applied Mathematics, (2007), 337. doi: 10.4171/056-1/16. Google Scholar

[4]

A. Borzí and V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations,, SIAM, (2012). Google Scholar

[5]

T. Bui-Thanh and O. Ghattas, Analysis of the Hessian for inverse scattering problems. Part I: Inverse shape scattering of acoustic waves,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/5/055001. Google Scholar

[6]

_________, Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves,, Inverse Problems, 28 (2012). Google Scholar

[7]

_________, Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves., Inverse Problems and Imaging, (2013). Google Scholar

[8]

_________, Randomized maximum likelihood sampling for large-scale Bayesian inverse problems,, In preparation, (2013). Google Scholar

[9]

_________, An analysis of infinite dimensional Bayesian inverse shape acoustic scattering and its numerical approximation,, SIAM Journal of Uncertainty Quantification, 2 (2014), 203. doi: 10.1137/120894877. Google Scholar

[10]

T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion,, SIAM Journal on Scientific Computing, 35 (2013). doi: 10.1137/12089586X. Google Scholar

[11]

T. Bui-Thanh and M. A. Girolami, Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo,, Inverse Problems, 30 (2014). doi: 10.1088/0266-5611/30/11/114014. Google Scholar

[12]

F. P. Casey, J. J. Waterfall, R. N. Gutenkunst, C. R. Myers and J. P. Sethna, Variational method for estimating the rate of convergence of Marko-chain Monte Carlo algorithms,, Phy. Rev. E., 78 (2008). Google Scholar

[13]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978). Google Scholar

[14]

S. L. Cotter, G. O. Roberts, A. M. Stuart and D. White, MCMC methods for functions: Modifying old algorithms to make them faster,, Statistical Science, 28 (2013), 424. doi: 10.1214/13-STS421. Google Scholar

[15]

M. Dashti, K. J. H. Law, A. M. Stuart and J. Voss, MAP estimators and their consistency in Bayesian nonparametric inverse problems,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/9/095017. Google Scholar

[16]

S. Duane, A. D. Kennedy, B. Pendleton and D. Roweth, Hybrid Monte Carlo,, Phys. Lett. B, 195 (1987), 216. doi: 10.1016/0370-2693(87)91197-X. Google Scholar

[17]

J. N. Franklin, Well-posed stochastic extensions of ill-posed linear problems,, Journal of Mathematical Analysis and Applications, 31 (1970), 682. doi: 10.1016/0022-247X(70)90017-X. Google Scholar

[18]

A. Gelman, G. O. Roberts and W. R. Gilks, Efficient Metropolis jumping rules,, in Bayesian Statistics, 5 (1996), 599. Google Scholar

[19]

M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods,, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 123. doi: 10.1111/j.1467-9868.2010.00765.x. Google Scholar

[20]

M. D. Gunzburger, Perspectives in Flow Control and Optimization,, SIAM, (2003). Google Scholar

[21]

H. Haario, M. Laine, A. Miravete and E. Saksman, DRAM: Efficient adaptive MCMC,, Statistics and Computing, 16 (2006), 339. doi: 10.1007/s11222-006-9438-0. Google Scholar

[22]

M. Hairer, A. M. Stuart and J. Voss, Analysis of SPDEs arising in path sampling. part II: The nonlinear case,, Annals of Applied Probability, 17 (2007), 1657. doi: 10.1214/07-AAP441. Google Scholar

[23]

W. Keith Hastings, Monte Carlo sampling methods using Markov chains and their applications,, Biometrika, 57 (1970), 97. doi: 10.1093/biomet/57.1.97. Google Scholar

[24]

E. Herbst, Gradient and Hessian-based MCMC for DSGE models,, (2010). Unpublished manuscript., (2010). Google Scholar

[25]

M. Ilić, F. Liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation,, Frac. Calc. and A Anal., 8 (2005), 323. Google Scholar

[26]

S. Lasanen, Discretizations of Generalized Random Variables with Applications to Inverse Problems,, PhD thesis, (2002). Google Scholar

[27]

M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalized random variables,, Inverse Problems, 5 (1989), 599. doi: 10.1088/0266-5611/5/4/011. Google Scholar

[28]

F. Lindgren, H. Rue and J. Lindström, An explicit link between gaussian fields and gaussian markov random fields: The stochastic partial differential equation approach,, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 423. doi: 10.1111/j.1467-9868.2011.00777.x. Google Scholar

[29]

J. Martin, L. C. Wilcox, C. Burstedde and O. Ghattas, A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion,, SIAM Journal on Scientific Computing, 34 (2012). doi: 10.1137/110845598. Google Scholar

[30]

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines,, The Journal of Chemical Physics, 21 (1953), 1087. doi: 10.1063/1.1699114. Google Scholar

[31]

R. M. Neal, Handbook of Markov Chain Monte Carlo,, Chapman & Hall / CRC Press, (2010). Google Scholar

[32]

J. Nocedal and S. J. Wright, Numerical Optimization,, Springer Verlag, (2006). Google Scholar

[33]

M. Ottobre, N. S. Pillai, F. J. Pinski and A. M. Stuart, A function space HMC algorithm with second order langevin diffusion limit,, Bernoulli, 22 (2016), 60. doi: 10.3150/14-BEJ621. Google Scholar

[34]

P. Piiroinen, Statistical Measurements, Experiments, and Applications,, PhD thesis, (2005). Google Scholar

[35]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambidge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[36]

Y. Qi and T. P. Minka, Hessian-based Markov chain Monte-Carlo algorithms,, in First Cape Cod Workshop on Monte Carlo Methods, (2002). Google Scholar

[37]

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer Texts in Statistics),, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-4145-2. Google Scholar

[38]

G. O. Roberts, A. Gelman and W. R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms,, The Annals of Applied Probability, 7 (1997), 110. doi: 10.1214/aoap/1034625254. Google Scholar

[39]

G. O. Roberts and J. S. Rosenthal, Optimal scaling of discrete approximations to Langevin diffusions,, J. R. Statist. Soc. B, 60 (1997), 255. doi: 10.1111/1467-9868.00123. Google Scholar

[40]

J. Rosenthal, Optimal proposal distributions and adaptive MCMC,, in Handbook of Markov chain Monte Carlo, (2011), 93. Google Scholar

[41]

P. J. Rossky, J. D. Doll and H. L. Friedman, Brownian dynamics as smart Monte Carlo simulation,, J. Chem. Phys., 69 (1978). doi: 10.1063/1.436415. Google Scholar

[42]

D. P. Simpson, Krylov subpsace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous Diffusion,, PhD thesis, (2008). Google Scholar

[43]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numerica, 19 (2010), 451. doi: 10.1017/S0962492910000061. Google Scholar

[44]

A. M. Stuart, J. Voss and P. Wiberg, Conditional path sampling of SDEs and the Langevin MCMC method,, Communications in Mathematical Sciences, 2 (2004), 685. doi: 10.4310/CMS.2004.v2.n4.a7. Google Scholar

[45]

L. Tierney, A note on Metropolis-Hastings kernels for general state spaces,, Annals of Applied Probability, 8 (1998), 1. doi: 10.1214/aoap/1027961031. Google Scholar

[46]

C. Vacar, J.-F. Giovannelli and Y. Berthoumieu, Langevin and Hessian with Fisher approximation stochastic sampling for parameter estimation of structured covariance,, IEEE International Conference on Acoustics, (2011), 3964. doi: 10.1109/ICASSP.2011.5947220. Google Scholar

[47]

B. Vexler, Adaptive finite element methods for parameter identification problems,, Model Based Parameter Estimation, (2012), 31. doi: 10.1007/978-3-642-30367-8_2. Google Scholar

[48]

C. R. Vogel, Computational Methods for Inverse Problems,, Frontiers in Applied Mathematics, (2002). doi: 10.1137/1.9780898717570. Google Scholar

[49]

Y. Zhang and C. Sutton, Quasi-Newton methods for Markov chain Monte Carlo,, in Advances in Neural Information Processing Systems, (2011). Google Scholar

[1]

Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81

[2]

Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803

[3]

Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291

[4]

Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683

[5]

Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335

[6]

Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125

[7]

Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313

[8]

Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025

[9]

Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems & Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567

[10]

Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems & Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27

[11]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[12]

Vincent Renault, Michèle Thieullen, Emmanuel Trélat. Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks & Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019

[13]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024

[14]

Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183

[15]

Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial & Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503

[16]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489

[17]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[18]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[19]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[20]

Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]