• Previous Article
    Flexible online multivariate regression with variational Bayes and the matrix-variate Dirichlet process
  • FoDS Home
  • This Issue
  • Next Article
    Levels and trends in the sex ratio at birth and missing female births for 29 states and union territories in India 1990–2016: A Bayesian modeling study
June  2019, 1(2): 157-176. doi: 10.3934/fods.2019007

Estimation and uncertainty quantification for the output from quantum simulators

1. 

Computational Science and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

2. 

Department of Statistics and Applied Probability, National University of Singapore, Singapore

3. 

School of Mathematics, University of Manchester, Manchester, UK, M13 9PL

* Corresponding author

Published  May 2019

The problem of estimating certain distributions over {0, 1}d is considered here. The distribution represents a quantum system of d qubits, where there are non-trivial dependencies between the qubits. A maximum entropy approach is adopted to reconstruct the distribution from exact moments or observed empirical moments. The Robbins Monro algorithm is used to solve the intractable maximum entropy problem, by constructing an unbiased estimator of the un-normalized target with a sequential Monte Carlo sampler at each iteration. In the case of empirical moments, this coincides with a maximum likelihood estimator. A Bayesian formulation is also considered in order to quantify uncertainty a posteriori. Several approaches are proposed in order to tackle this challenging problem, based on recently developed methodologies. In particular, unbiased estimators of the gradient of the log posterior are constructed and used within a provably convergent Langevin-based Markov chain Monte Carlo method. The methods are illustrated on classically simulated output from quantum simulators.

Citation: Ryan Bennink, Ajay Jasra, Kody J. H. Law, Pavel Lougovski. Estimation and uncertainty quantification for the output from quantum simulators. Foundations of Data Science, 2019, 1 (2) : 157-176. doi: 10.3934/fods.2019007
References:
[1]

C. Andrieu and G. O. Roberts, The pseudo-marginal approach for efficient Monte Carlo computations, The Annals of Statistics, 37 (2009), 697-725. doi: 10.1214/07-AOS574. Google Scholar

[2]

M. A. Beaumont, Estimation of population growth or decline in genetically monitored populations, Genetics, 164 (2003), 1139-1160. Google Scholar

[3]

A. BeskosD. Crisan and A. Jasra, On the stability of sequential monte carlo methods in high dimensions, The Annals of Applied Probability, 24 (2014), 1396-1445. doi: 10.1214/13-AAP951. Google Scholar

[4]

J. Bierkens, P. Fearnhead and G. Roberts, The zig-zag process and super-efficient sampling for bayesian analysis of big data, preprint, arXiv: 1607.03188. doi: 10.1214/18-AOS1715. Google Scholar

[5]

R. Blume-Kohout, Optimal, reliable estimation of quantum states, New Journal of Physics, 12 (2010), 043034. doi: 10.1088/1367-2630/12/4/043034. Google Scholar

[6]

A. Bouchard-Côté, S. J. Vollmer and A. Doucet, The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method, Journal of the American Statistical Association, 1–13. doi: 10.1080/01621459.2017.1294075. Google Scholar

[7] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441.
[8]

E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717. doi: 10.1007/s10208-009-9045-5. Google Scholar

[9]

A. M. Childs, R. B. Patterson and D. J. MacKay, Exact sampling from nonattractive distributions using summary states, Physical Review E, 63 (2001), 036113. doi: 10.1103/PhysRevE.63.036113. Google Scholar

[10]

N. Chopin, A sequential particle filter method for static models, Biometrika, 89 (2002), 539-552. doi: 10.1093/biomet/89.3.539. Google Scholar

[11]

T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 2012. Google Scholar

[12]

M. H. Davis, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models, Journal of the Royal Statistical Society. Series B (Methodological), 353–388. doi: 10.1111/j.2517-6161.1984.tb01308.x. Google Scholar

[13]

P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications, Probability and its Applications (New York), Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4684-9393-1. Google Scholar

[14]

P. Del MoralA. Doucet and A. Jasra, Sequential Monte Carlo samplers, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68 (2006), 411-436. doi: 10.1111/j.1467-9868.2006.00553.x. Google Scholar

[15]

P. Del MoralA. Doucet and A. Jasra, Sequential Monte Carlo samplers, J. R. Stat. Soc. Ser. B Stat. Methodol., 68 (2006), 411-436. doi: 10.1111/j.1467-9868.2006.00553.x. Google Scholar

[16]

J. Franks, A. Jasra, K. Law and M. Vihola, Unbiased inference for discretely observed hidden markov model diffusions, preprint, arXiv: 1807.10259.Google Scholar

[17]

M. Giles, T. Nagapetyan, L. Szpruch, S. Vollmer and K. Zygalakis, Multilevel Monte Carlo for scalable Bayesian computations, preprint, arXiv: 1609.06144. doi: 10.1017/S096249291500001X. Google Scholar

[18]

W. R. Gilks, Markov Chain Monte Carlo, Wiley Online Library, 2005. doi: 10.1002/0470011815.b2a14021. 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), 123-214. doi: 10.1111/j.1467-9868.2010.00765.x. Google Scholar

[20]

C. Granade, J. Combes and D. Cory, Practical bayesian tomography, New Journal of Physics, 18 (2016), 033024. doi: 10.1088/1367-2630/18/3/033024. Google Scholar

[21]

P. J. GreenK. ŁatuszyńskiM. Pereyra and C. P. Robert, Bayesian computation: a summary of the current state, and samples backwards and forwards, Statistics and Computing, 25 (2015), 835-862. doi: 10.1007/s11222-015-9574-5. Google Scholar

[22]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker and J. Eisert, Quantum state tomography via compressed sensing, Physical Review Letters, 105 (2010), 150401. doi: 10.1103/PhysRevLett.105.150401. Google Scholar

[23]

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97-109. doi: 10.1093/biomet/57.1.97. Google Scholar

[24]

Z. Hradil, Quantum-state estimation, Physical Review A, 55 (1997), R1561. doi: 10.1103/PhysRevA.55.R1561. Google Scholar

[25]

F. Huszár and N. M. Houlsby, Adaptive bayesian quantum tomography, Physical Review A, 85 (2012), 052120.Google Scholar

[26]

C. Jarzynski, Nonequilibrium equality for free energy differences, Physical Review Letters, 78 (1997), 2690. doi: 10.1103/PhysRevLett.78.2690. Google Scholar

[27]

E. T. Jaynes, Information theory and statistical mechanics, Physical Review, 106 (1957), 620. doi: 10.1103/PhysRev.106.620. Google Scholar

[28]

E. T. Jaynes, Information theory and statistical mechanics. Ⅱ, Physical Review, 108 (1957), 171. doi: 10.1103/PhysRev.108.171. Google Scholar

[29]

K. Jones, Principles of quantum inference, Annals of Physics, 207 (1991), 140-170. doi: 10.1016/0003-4916(91)90182-8. Google Scholar

[30]

H. Kushner and G. G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, vol. 35, Springer Science & Business Media, 2003. Google Scholar

[31]

A.-M. LyneM. GirolamiY. AtchadéH. Strathmann and D. Simpson, On Russian roulette estimates for bayesian inference with doubly-intractable likelihoods, Statistical Science, 30 (2015), 443-467. doi: 10.1214/15-STS523. Google Scholar

[32]

D. McLeish, A general method for debiasing a Monte Carlo estimator, Monte Carlo Methods and Applications, 17 (2011), 301–315. doi: 10.1515/mcma.2011.013. Google Scholar

[33]

J. MøllerA. N. PettittR. Reeves and K. K. Berthelsen, An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants, Biometrika, 93 (2006), 451-458. doi: 10.1093/biomet/93.2.451. Google Scholar

[34]

K. P. Murphy, Machine learning: A probabilistic perspective.Google Scholar

[35]

I. Murray, Z. Ghahramani and D. J. MacKay, Mcmc for doubly-intractable distributions, in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence, AUAI Press, 2006, 359–366.Google Scholar

[36]

R. M. Neal, Annealed importance sampling, Statistics and Computing, 11 (2001), 125-139. doi: 10.1023/A:1008923215028. Google Scholar

[37]

M. Paris and J. Rehacek, Quantum State Estimation, 1st edition, Springer Publishing Company, Incorporated, 2010. doi: 10.1007/b98673. Google Scholar

[38]

S. Patterson and Y. W. Teh, Stochastic gradient riemannian Langevin dynamics on the probability simplex, in Advances in Neural Information Processing Systems, 2013, 3102–3110.Google Scholar

[39]

E. A. Peters, et al., Rejection-free Monte Carlo sampling for general potentials, Physical Review E, 85 (2012), 026703.Google Scholar

[40]

J. Preskill, Quantum computing in the NISQ era and beyond, preprint, arXiv: 1801.00862. doi: 10.1098/rspa.1998.0171. Google Scholar

[41]

J. G. Propp and D. B. Wilson, Exact sampling with coupled markov chains and applications to statistical mechanics, Random Structures & Algorithms, 9 (1996), 223-252. doi: 10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O. Google Scholar

[42]

C.-H. Rhee and P. W. Glynn, A new approach to unbiased estimation for sde's, in Proceedings of the Winter Simulation Conference, Winter Simulation Conference, 2012, 17. doi: 10.1109/WSC.2012.6465150. Google Scholar

[43]

C.-H. Rhee and P. W. Glynn, Unbiased estimation with square root convergence for sde models, Operations Research, 63 (2015), 1026-1043. doi: 10.1287/opre.2015.1404. Google Scholar

[44]

H. Robbins and S. Monro, A stochastic approximation method, The Annals of Mathematical Statistics, 400–407. doi: 10.1214/aoms/1177729586. Google Scholar

[45]

G. O. Roberts and J. S. Rosenthal, Optimal scaling for various Metropolis-Hastings algorithms, Statistical Science, 16 (2001), 351-367. doi: 10.1214/ss/1015346320. Google Scholar

[46]

Y. W. TehA. H. Thiery and S. J. Vollmer, Consistency and fluctuations for stochastic gradient Langevin dynamics, The Journal of Machine Learning Research, 17 (2016), 193-225. Google Scholar

[47]

M. Vihola, Unbiased estimators and multilevel Monte Carlo, Operations Research, 66 (2017), 448-462. doi: 10.1287/opre.2017.1670. Google Scholar

[48]

S. J. VollmerK. C. Zygalakis and Y. W. Teh, Exploration of the (non-) asymptotic bias and variance of stochastic gradient langevin dynamics, The Journal of Machine Learning Research, 17 (2016), 5504-5548. Google Scholar

[49]

C. Wei and I. Murray, Markov chain truncation for doubly-intractable inference, in Artificial Intelligence and Statistics, 2017,776–784.Google Scholar

[50]

M. Welling and Y. W. Teh, Bayesian learning via stochastic gradient Langevin dynamics, in Proceedings of the 28th International Conference on Machine Learning (ICML-11), 2011,681–688. doi: 10.4310/CIS.2012.v12.n3.a3. Google Scholar

show all references

References:
[1]

C. Andrieu and G. O. Roberts, The pseudo-marginal approach for efficient Monte Carlo computations, The Annals of Statistics, 37 (2009), 697-725. doi: 10.1214/07-AOS574. Google Scholar

[2]

M. A. Beaumont, Estimation of population growth or decline in genetically monitored populations, Genetics, 164 (2003), 1139-1160. Google Scholar

[3]

A. BeskosD. Crisan and A. Jasra, On the stability of sequential monte carlo methods in high dimensions, The Annals of Applied Probability, 24 (2014), 1396-1445. doi: 10.1214/13-AAP951. Google Scholar

[4]

J. Bierkens, P. Fearnhead and G. Roberts, The zig-zag process and super-efficient sampling for bayesian analysis of big data, preprint, arXiv: 1607.03188. doi: 10.1214/18-AOS1715. Google Scholar

[5]

R. Blume-Kohout, Optimal, reliable estimation of quantum states, New Journal of Physics, 12 (2010), 043034. doi: 10.1088/1367-2630/12/4/043034. Google Scholar

[6]

A. Bouchard-Côté, S. J. Vollmer and A. Doucet, The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method, Journal of the American Statistical Association, 1–13. doi: 10.1080/01621459.2017.1294075. Google Scholar

[7] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441.
[8]

E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717. doi: 10.1007/s10208-009-9045-5. Google Scholar

[9]

A. M. Childs, R. B. Patterson and D. J. MacKay, Exact sampling from nonattractive distributions using summary states, Physical Review E, 63 (2001), 036113. doi: 10.1103/PhysRevE.63.036113. Google Scholar

[10]

N. Chopin, A sequential particle filter method for static models, Biometrika, 89 (2002), 539-552. doi: 10.1093/biomet/89.3.539. Google Scholar

[11]

T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 2012. Google Scholar

[12]

M. H. Davis, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models, Journal of the Royal Statistical Society. Series B (Methodological), 353–388. doi: 10.1111/j.2517-6161.1984.tb01308.x. Google Scholar

[13]

P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications, Probability and its Applications (New York), Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4684-9393-1. Google Scholar

[14]

P. Del MoralA. Doucet and A. Jasra, Sequential Monte Carlo samplers, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68 (2006), 411-436. doi: 10.1111/j.1467-9868.2006.00553.x. Google Scholar

[15]

P. Del MoralA. Doucet and A. Jasra, Sequential Monte Carlo samplers, J. R. Stat. Soc. Ser. B Stat. Methodol., 68 (2006), 411-436. doi: 10.1111/j.1467-9868.2006.00553.x. Google Scholar

[16]

J. Franks, A. Jasra, K. Law and M. Vihola, Unbiased inference for discretely observed hidden markov model diffusions, preprint, arXiv: 1807.10259.Google Scholar

[17]

M. Giles, T. Nagapetyan, L. Szpruch, S. Vollmer and K. Zygalakis, Multilevel Monte Carlo for scalable Bayesian computations, preprint, arXiv: 1609.06144. doi: 10.1017/S096249291500001X. Google Scholar

[18]

W. R. Gilks, Markov Chain Monte Carlo, Wiley Online Library, 2005. doi: 10.1002/0470011815.b2a14021. 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), 123-214. doi: 10.1111/j.1467-9868.2010.00765.x. Google Scholar

[20]

C. Granade, J. Combes and D. Cory, Practical bayesian tomography, New Journal of Physics, 18 (2016), 033024. doi: 10.1088/1367-2630/18/3/033024. Google Scholar

[21]

P. J. GreenK. ŁatuszyńskiM. Pereyra and C. P. Robert, Bayesian computation: a summary of the current state, and samples backwards and forwards, Statistics and Computing, 25 (2015), 835-862. doi: 10.1007/s11222-015-9574-5. Google Scholar

[22]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker and J. Eisert, Quantum state tomography via compressed sensing, Physical Review Letters, 105 (2010), 150401. doi: 10.1103/PhysRevLett.105.150401. Google Scholar

[23]

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97-109. doi: 10.1093/biomet/57.1.97. Google Scholar

[24]

Z. Hradil, Quantum-state estimation, Physical Review A, 55 (1997), R1561. doi: 10.1103/PhysRevA.55.R1561. Google Scholar

[25]

F. Huszár and N. M. Houlsby, Adaptive bayesian quantum tomography, Physical Review A, 85 (2012), 052120.Google Scholar

[26]

C. Jarzynski, Nonequilibrium equality for free energy differences, Physical Review Letters, 78 (1997), 2690. doi: 10.1103/PhysRevLett.78.2690. Google Scholar

[27]

E. T. Jaynes, Information theory and statistical mechanics, Physical Review, 106 (1957), 620. doi: 10.1103/PhysRev.106.620. Google Scholar

[28]

E. T. Jaynes, Information theory and statistical mechanics. Ⅱ, Physical Review, 108 (1957), 171. doi: 10.1103/PhysRev.108.171. Google Scholar

[29]

K. Jones, Principles of quantum inference, Annals of Physics, 207 (1991), 140-170. doi: 10.1016/0003-4916(91)90182-8. Google Scholar

[30]

H. Kushner and G. G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, vol. 35, Springer Science & Business Media, 2003. Google Scholar

[31]

A.-M. LyneM. GirolamiY. AtchadéH. Strathmann and D. Simpson, On Russian roulette estimates for bayesian inference with doubly-intractable likelihoods, Statistical Science, 30 (2015), 443-467. doi: 10.1214/15-STS523. Google Scholar

[32]

D. McLeish, A general method for debiasing a Monte Carlo estimator, Monte Carlo Methods and Applications, 17 (2011), 301–315. doi: 10.1515/mcma.2011.013. Google Scholar

[33]

J. MøllerA. N. PettittR. Reeves and K. K. Berthelsen, An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants, Biometrika, 93 (2006), 451-458. doi: 10.1093/biomet/93.2.451. Google Scholar

[34]

K. P. Murphy, Machine learning: A probabilistic perspective.Google Scholar

[35]

I. Murray, Z. Ghahramani and D. J. MacKay, Mcmc for doubly-intractable distributions, in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence, AUAI Press, 2006, 359–366.Google Scholar

[36]

R. M. Neal, Annealed importance sampling, Statistics and Computing, 11 (2001), 125-139. doi: 10.1023/A:1008923215028. Google Scholar

[37]

M. Paris and J. Rehacek, Quantum State Estimation, 1st edition, Springer Publishing Company, Incorporated, 2010. doi: 10.1007/b98673. Google Scholar

[38]

S. Patterson and Y. W. Teh, Stochastic gradient riemannian Langevin dynamics on the probability simplex, in Advances in Neural Information Processing Systems, 2013, 3102–3110.Google Scholar

[39]

E. A. Peters, et al., Rejection-free Monte Carlo sampling for general potentials, Physical Review E, 85 (2012), 026703.Google Scholar

[40]

J. Preskill, Quantum computing in the NISQ era and beyond, preprint, arXiv: 1801.00862. doi: 10.1098/rspa.1998.0171. Google Scholar

[41]

J. G. Propp and D. B. Wilson, Exact sampling with coupled markov chains and applications to statistical mechanics, Random Structures & Algorithms, 9 (1996), 223-252. doi: 10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O. Google Scholar

[42]

C.-H. Rhee and P. W. Glynn, A new approach to unbiased estimation for sde's, in Proceedings of the Winter Simulation Conference, Winter Simulation Conference, 2012, 17. doi: 10.1109/WSC.2012.6465150. Google Scholar

[43]

C.-H. Rhee and P. W. Glynn, Unbiased estimation with square root convergence for sde models, Operations Research, 63 (2015), 1026-1043. doi: 10.1287/opre.2015.1404. Google Scholar

[44]

H. Robbins and S. Monro, A stochastic approximation method, The Annals of Mathematical Statistics, 400–407. doi: 10.1214/aoms/1177729586. Google Scholar

[45]

G. O. Roberts and J. S. Rosenthal, Optimal scaling for various Metropolis-Hastings algorithms, Statistical Science, 16 (2001), 351-367. doi: 10.1214/ss/1015346320. Google Scholar

[46]

Y. W. TehA. H. Thiery and S. J. Vollmer, Consistency and fluctuations for stochastic gradient Langevin dynamics, The Journal of Machine Learning Research, 17 (2016), 193-225. Google Scholar

[47]

M. Vihola, Unbiased estimators and multilevel Monte Carlo, Operations Research, 66 (2017), 448-462. doi: 10.1287/opre.2017.1670. Google Scholar

[48]

S. J. VollmerK. C. Zygalakis and Y. W. Teh, Exploration of the (non-) asymptotic bias and variance of stochastic gradient langevin dynamics, The Journal of Machine Learning Research, 17 (2016), 5504-5548. Google Scholar

[49]

C. Wei and I. Murray, Markov chain truncation for doubly-intractable inference, in Artificial Intelligence and Statistics, 2017,776–784.Google Scholar

[50]

M. Welling and Y. W. Teh, Bayesian learning via stochastic gradient Langevin dynamics, in Proceedings of the 28th International Conference on Machine Learning (ICML-11), 2011,681–688. doi: 10.4310/CIS.2012.v12.n3.a3. Google Scholar

Figure 1.  Truth (left) and reconstruction after $ K = 10^4 $ steps (middle) for known truth. The error as a function of iteration is given in the right plot
Figure 2.  The reconstruction with $ 1000 $ (left) and $ 50 $ (right) observations are given in the top row, along with the corresponding convergence plots in the second row. The bottom row shows the actual second moments $ \widehat m $ (left), with $ M = 50 $ observations, which are used to train the model, and the moments under the reconstruction (right)
Figure 3.  Convergence of SGD with the debiased estimator described in Section 6.3 (left), the original unbiased estimator $ \widehat F $ (middle), and the simple consistent but biased estimator (right)
Figure 4.  Illustration of pairwise marginal UQ for the posterior on the diagonal of $ \Lambda $ with $ M = 1000 $ observations and $ d = 4 $ qubits. The true value of the parameters is indicated in red
Figure 5.  Illustration of pairwise marginal UQ for the posterior on the diagonal of $ \Lambda $ with $ M = 10^6 $ observations and $ d = 4 $ qubits. The true value of the parameters is indicated in red
[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]

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

[3]

Evangelos Evangelou. Approximate Bayesian inference for geostatistical generalised linear models. Foundations of Data Science, 2019, 1 (1) : 39-60. doi: 10.3934/fods.2019002

[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]

Jie Huang, Xiaoping Yang, Yunmei Chen. A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation. Inverse Problems & Imaging, 2011, 5 (3) : 645-657. doi: 10.3934/ipi.2011.5.645

[7]

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

[8]

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

[9]

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

[10]

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

[11]

Julia Piantadosi, Phil Howlett, Jonathan Borwein, John Henstridge. Maximum entropy methods for generating simulated rainfall. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 233-256. doi: 10.3934/naco.2012.2.233

[12]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499

[13]

K.H. Wong, C. Myburgh, L. Omari. A gradient flow approach for computing jump linear quadratic optimal feedback gains. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 803-808. doi: 10.3934/dcds.2000.6.803

[14]

Yaxian Xu, Ajay Jasra. Particle filters for inference of high-dimensional multivariate stochastic volatility models with cross-leverage effects. Foundations of Data Science, 2019, 1 (1) : 61-85. doi: 10.3934/fods.2019003

[15]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285

[16]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565

[17]

Péter Koltai. A stochastic approach for computing the domain of attraction without trajectory simulation. Conference Publications, 2011, 2011 (Special) : 854-863. doi: 10.3934/proc.2011.2011.854

[18]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[19]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[20]

Julia Amador, Mariajesus Lopez-Herrero. Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3137-3151. doi: 10.3934/dcdsb.2017211

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]