Jonathan Weare
Warren Weaver Hall, Room 1103
251 Mercer Street
New York, NY 10012-1185
[my last name]@nyu.edu
Warren Weaver Hall, Room 1103
251 Mercer Street
New York, NY 10012-1185
[my last name]@nyu.edu
Our group is primarily focused on
the design, mathematical analysis, and application of stochastic
algorithms and models. Our work draws on tools from probability
theory, the theory of partial differential equations, and
numerical analysis. Research in the group is informed by our
long term collaborations with applications experts in areas such
as biophysics, computational chemistry, and weather and climate
science.
I am currently an associate
professor of mathematics in the Courant Institute of
Mathematical Sciences at New York University. Previously I was
an associate professor in the statistics department and in the
James Franck Institute at the University of Chicago and, before
that, an assistant professor in the mathematics department
there. Before moving to Chicago I was a Courant Instructor of
mathematics at NYU and a PhD student in mathematics at the
University of California at Berkeley.
Research
areas
Monte Carlo
sampling
multiscale analysis and simulation
randomized numerical linear algebra
rare event analysis and simulation
statistical and machine learning for scientific computing
multiscale analysis and simulation
randomized numerical linear algebra
rare event analysis and simulation
statistical and machine learning for scientific computing
People
High school students
Anna Zhang,
Stuyvesant High School 2020-2021
Undergraduate
students
James Butler,
mathematics, University of Chicago (co-mentored with D. Abbot)
2020-2021
Runxin Ni, mathematics, New York University (co-mentored with D. Abbot) Summer 2020
Charlie Marshall, mathematics, University of Chicago (co-mentored with D. Abbot) 2019-2020
Douglas Dow, mathematics, University of Chicago 2019-2020
Bradley Stadie, mathematics, University of Chicago 2013-2014
Runxin Ni, mathematics, New York University (co-mentored with D. Abbot) Summer 2020
Charlie Marshall, mathematics, University of Chicago (co-mentored with D. Abbot) 2019-2020
Douglas Dow, mathematics, University of Chicago 2019-2020
Bradley Stadie, mathematics, University of Chicago 2013-2014
Masters students
Bixing Qiao,
mathematics, New York University 2019-2020
Eileen Li, statistics, University of Chicago (co-mentored with A. Dinner) 2016-2017
Eileen Li, statistics, University of Chicago (co-mentored with A. Dinner) 2016-2017
Doctoral students
Xiaoou Cheng, New
York University 2020-
Huan Zhang, mathematics, New York University 2019-
Zhengqu Wan, mathematics, New York University 2018-
Chatipat Lorpaiboon, chemistry, University of Chicago (co-mentored with A. Dinner) 2018-
John Strahan, chemistry, University of Chicago (co-mentored with A. Dinner) 2018-
Anya Katsevich, mathematics, New York University 2017-2022
Justin Finkel, applied math, University of Chicago (co-mentored with D. Abbot) 2017-2022
Adam Antoszewski, chemistry, University of Chicago (co-mentored with A. Dinner) 2017-2022
Sam Greene, chemistry, Columbia (co-mentored with T. Berkelbach) 2017-2022
Robert Webber, mathematics, New York University 2015-2021
Bodhi Vani, chemistry, University of Chicago (co-mentored with A. Dinner) 2015-2021
Erik Thiede, chemistry, University of Chicago (co-mentored with A. Dinner) 2013-2019
David Plotkin, geoscience, University of Chicago (co-mentored with D. Abbot) 2012-2018
Jeremy Tempkin, chemistry, University of Chicago (co-mentored with A. Dinner) 2012-2017
Huan Zhang, mathematics, New York University 2019-
Zhengqu Wan, mathematics, New York University 2018-
Chatipat Lorpaiboon, chemistry, University of Chicago (co-mentored with A. Dinner) 2018-
John Strahan, chemistry, University of Chicago (co-mentored with A. Dinner) 2018-
Anya Katsevich, mathematics, New York University 2017-2022
Justin Finkel, applied math, University of Chicago (co-mentored with D. Abbot) 2017-2022
Adam Antoszewski, chemistry, University of Chicago (co-mentored with A. Dinner) 2017-2022
Sam Greene, chemistry, Columbia (co-mentored with T. Berkelbach) 2017-2022
Robert Webber, mathematics, New York University 2015-2021
Bodhi Vani, chemistry, University of Chicago (co-mentored with A. Dinner) 2015-2021
Erik Thiede, chemistry, University of Chicago (co-mentored with A. Dinner) 2013-2019
David Plotkin, geoscience, University of Chicago (co-mentored with D. Abbot) 2012-2018
Jeremy Tempkin, chemistry, University of Chicago (co-mentored with A. Dinner) 2012-2017
Instructors and postdoctoral scholars
Michael
Lindsey, mathematics, New York University 2019-
Brian Van Koten, applied math, University of Chicago 2014-2018
Charles Matthews, applied math, University of Chicago 2014-2018
Seyit Kale, chemistry, University of Chicago 2012-2015
Brian Van Koten, applied math, University of Chicago 2014-2018
Charles Matthews, applied math, University of Chicago 2014-2018
Seyit Kale, chemistry, University of Chicago 2012-2015
Editorial
work
Stochastics and Partial Differential Equations: Analysis and Computations
SIAM/ASA Journal on Uncertainty Quantification
Publications
(sorted roughly by area)
Preprints can be found on arXiv
Monte Carlo, Markov chains, and related
Ensemble Markov
chain Monte Carlo with teleporting walkers
with M. Lindsey and A. Zhang, SIAM Journal on Uncertainty Quantification, accepted
A metric on directed graphs and Markov chains based on hitting probabilities
with Z.M. Boyd, N. Fraiman, J.L. Marzuola, P.J. Mucha, and B. Osting, SIAM Journal on Mathematics of Data Science, 3(2) [2021], 467-493
Stratification as a general variance reduction method for Markov chain Monte Carlo
with A.R. Dinner, E.H. Thiede, and B. Van Koten, SIAM/ASA Journal on Uncertainty Quantification (JUQ), 8(3) [2020], 1139-1188
Langevin Markov chain Monte Carlo with stochastic gradients
with C. Matthews [2018]
Umbrella sampling: a powerful method to sample tails of distributions
with C. Matthews, A. Kravstov, and E. Jennings, Monthly Notices of the Royal Astronomical Society, 480(3) [2018], 4069-4079
Ensemble preconditioning for Markov chain Monte Carlo simulation
with B. Leimkuhler and C. Matthews, Statistics and Computing, 28(2) [2017], 277-290
Eigenvector method for umbrella sampling enables error analysis
with E. Thiede, B. Van Koten, and A. Dinner, Journal of Chemical Physics, 145(8) [2016], 084115
Sharp entrywise perturbation bounds for Markov chains
with E. Thiede and B. Van Koten, SIAM Journal on Matrix Analysis and Applications, 36(3) [2015], 917-941
On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method
with B. Roux, Journal of Chemical Physics, 138(8) [2013], 084107
An affine-invariant sampler for exoplanet fitting and discovery in radial velocity data
with F. Hou, J. Goodman, D. Hogg, and C. Schwab, The Astrophysical Journal, 75 [2012], 198
Ensemble samplers with affine invariance
with J. Goodman, Communications in Applied Mathematics and Computational Science, 5 [2010], 65-80
Efficient Monte Carlo sampling by parallel marginalization
Proceedings of the National Academy of Science, 104(31) [2007], 12657-12662
with M. Lindsey and A. Zhang, SIAM Journal on Uncertainty Quantification, accepted
A metric on directed graphs and Markov chains based on hitting probabilities
with Z.M. Boyd, N. Fraiman, J.L. Marzuola, P.J. Mucha, and B. Osting, SIAM Journal on Mathematics of Data Science, 3(2) [2021], 467-493
Stratification as a general variance reduction method for Markov chain Monte Carlo
with A.R. Dinner, E.H. Thiede, and B. Van Koten, SIAM/ASA Journal on Uncertainty Quantification (JUQ), 8(3) [2020], 1139-1188
Langevin Markov chain Monte Carlo with stochastic gradients
with C. Matthews [2018]
Umbrella sampling: a powerful method to sample tails of distributions
with C. Matthews, A. Kravstov, and E. Jennings, Monthly Notices of the Royal Astronomical Society, 480(3) [2018], 4069-4079
Ensemble preconditioning for Markov chain Monte Carlo simulation
with B. Leimkuhler and C. Matthews, Statistics and Computing, 28(2) [2017], 277-290
Eigenvector method for umbrella sampling enables error analysis
with E. Thiede, B. Van Koten, and A. Dinner, Journal of Chemical Physics, 145(8) [2016], 084115
Sharp entrywise perturbation bounds for Markov chains
with E. Thiede and B. Van Koten, SIAM Journal on Matrix Analysis and Applications, 36(3) [2015], 917-941
On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method
with B. Roux, Journal of Chemical Physics, 138(8) [2013], 084107
An affine-invariant sampler for exoplanet fitting and discovery in radial velocity data
with F. Hou, J. Goodman, D. Hogg, and C. Schwab, The Astrophysical Journal, 75 [2012], 198
Ensemble samplers with affine invariance
with J. Goodman, Communications in Applied Mathematics and Computational Science, 5 [2010], 65-80
Efficient Monte Carlo sampling by parallel marginalization
Proceedings of the National Academy of Science, 104(31) [2007], 12657-12662
Multiscale analysis and simulation
A
Kinetic Monte Carlo approach for simulating cascading
transmission line failure
with J. Roth, D.A. Barajas-Solano, P. Stinis, and Mihai Anitescu, SIAM Multiscale Modeling and Simulation, 19(1) [2021], 208-241
Multiple time-step dual-Hamiltonian hybrid molecular dynamics Monte Carlo canonical propagation algorithm
with Y. Chen, S. Kale, A. Dinner, and B. Roux, Journal of Chemical Theory and Computation, 12(4) [2016], 1449-1458
Finding chemical reaction paths with a multilevel preconditioning protocol
with S. Kale, S. Olaseni, and A. Dinner, Journal of Chemical Theory and Simulation, 10(12) [2014], 5467-5475
Using multiscale preconditioning to accelerate the convergence of iterative molecular calculations
with J. Tempkin, B. Qui, M. Saunders, B. Roux, and A. Dinner, Journal of Chemical Physics, 140(18) [2014], 184114
Nucleotide regulation of the structure and dynamics of G-actin
with M. Saunders, J. Tempkin, A. Dinner, B. Roux, and G. Voth, Biophysical Journal, 106(8) [2014], 1710-1720
The relaxation of a family of broken bond crystal surface models
with J. Marzuola, Physical Review E, 88 [2013], 032403
The theory of ultra coarse graining I, general principles
with J. Dama, A. Sinitskiy, M. McCullagh, B. Roux, A. Dinner, and G. Voth, Journal of Chemical Theory and Computation, 9(5) [2013], 2466-2480
Minimizing memory as an objective for coarse-graining
with N. Guttenberg, J. Dama, M. Saunders, A. Dinner, and G. Voth, Journal of Chemical Physics, 138(9) [2013], 094111
Extending molecular simulation time scales: Parallel-in-time integration for high-level quantum chemistry and complex force representations
with E. Bylaska and J.H. Weare, Journal of Chemical Physics, 139 [2013], 074114
The evolution of a crystal surface: analysis of a 1D step train connecting two facets in the ADL regime
with H. Al Hajj Shehadeh and R.V. Kohn, Physica D, 240 [2011], 1771-1784
Variance reduction for particle filters of systems with time-scale separation
with D. Givon and P. Stinis, IEEE Transactions on Signal Processing, 57(2) [2009], 424-435
with J. Roth, D.A. Barajas-Solano, P. Stinis, and Mihai Anitescu, SIAM Multiscale Modeling and Simulation, 19(1) [2021], 208-241
Multiple time-step dual-Hamiltonian hybrid molecular dynamics Monte Carlo canonical propagation algorithm
with Y. Chen, S. Kale, A. Dinner, and B. Roux, Journal of Chemical Theory and Computation, 12(4) [2016], 1449-1458
Finding chemical reaction paths with a multilevel preconditioning protocol
with S. Kale, S. Olaseni, and A. Dinner, Journal of Chemical Theory and Simulation, 10(12) [2014], 5467-5475
Using multiscale preconditioning to accelerate the convergence of iterative molecular calculations
with J. Tempkin, B. Qui, M. Saunders, B. Roux, and A. Dinner, Journal of Chemical Physics, 140(18) [2014], 184114
Nucleotide regulation of the structure and dynamics of G-actin
with M. Saunders, J. Tempkin, A. Dinner, B. Roux, and G. Voth, Biophysical Journal, 106(8) [2014], 1710-1720
The relaxation of a family of broken bond crystal surface models
with J. Marzuola, Physical Review E, 88 [2013], 032403
The theory of ultra coarse graining I, general principles
with J. Dama, A. Sinitskiy, M. McCullagh, B. Roux, A. Dinner, and G. Voth, Journal of Chemical Theory and Computation, 9(5) [2013], 2466-2480
Minimizing memory as an objective for coarse-graining
with N. Guttenberg, J. Dama, M. Saunders, A. Dinner, and G. Voth, Journal of Chemical Physics, 138(9) [2013], 094111
Extending molecular simulation time scales: Parallel-in-time integration for high-level quantum chemistry and complex force representations
with E. Bylaska and J.H. Weare, Journal of Chemical Physics, 139 [2013], 074114
The evolution of a crystal surface: analysis of a 1D step train connecting two facets in the ADL regime
with H. Al Hajj Shehadeh and R.V. Kohn, Physica D, 240 [2011], 1771-1784
Variance reduction for particle filters of systems with time-scale separation
with D. Givon and P. Stinis, IEEE Transactions on Signal Processing, 57(2) [2009], 424-435
Randomized numerical linear algebra
Approximating matrix eigenvalues by subspace iteration with
repeated random sparsification
with S.M. Greene, R.J. Webber, and T.C. Berkelbach, SIAM Journal on Scientific Computing, accepted
Improved Fast Randomized Iteration approach to Full Configuration Interaction
with S.M. Greene, R.J. Webber, and T.C. Berkelbach, Journal of Chemical Theory and Computation, 16(9) [2020], 5572–5585
Beyond walkers in stochastic quantum chemistry: reducing error using Fast Randomized Iteration
with S.M. Greene, R.J. Webber, and T.C. Berkelbach, Journal of Chemical Theory and Computation, 15(9) [2019], 4834-4850
Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra
with L.H. Lim, SIAM Reviews: Research Spotlight, 59(3) [2017], 547-587
with S.M. Greene, R.J. Webber, and T.C. Berkelbach, SIAM Journal on Scientific Computing, accepted
Improved Fast Randomized Iteration approach to Full Configuration Interaction
with S.M. Greene, R.J. Webber, and T.C. Berkelbach, Journal of Chemical Theory and Computation, 16(9) [2020], 5572–5585
Beyond walkers in stochastic quantum chemistry: reducing error using Fast Randomized Iteration
with S.M. Greene, R.J. Webber, and T.C. Berkelbach, Journal of Chemical Theory and Computation, 15(9) [2019], 4834-4850
Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra
with L.H. Lim, SIAM Reviews: Research Spotlight, 59(3) [2017], 547-587
Rare event analysis and simulation
Augmented transition path theory for sequences of events
with C. Lorpaiboon and A.R. Dinner, Journal of Chemical Physics, accepted
Computing transition path theory quantities with trajectory stratification
with B.P. Vani and A.R. Dinner, Journal of Chemical Physics, accepted
Rare event sampling improves Mercury instability statistics
with D.S. Abott, R.J. Webber, and S. Hadden, Astrophysical Journal, 923(2) [2021], 236
Insulin dissociates by diverse mechanisms of coupled unfolding and unbinding
with A. Antoszewski, C.-J. Feng, B.P. Vani, E.H. Thiede, L. Hong, A. Tokmakoff, and A.R. Dinner, Journal of Physical Chemistry B, 124(27) [2020], 5571-5587
Path properties of atmospheric transitions: illustration with a low-order sudden stratospheric warming model
with J. Finkel, and D.S. Abbot, Journal of Atmospheric Science, 77(7) [2020], 2327-2347
Practical rare event simulation for extreme mesoscale weather
with R.J. Webber, D.A. Plotkin, M.E O'Neill, and D.S. Abbot, Chaos, 29 [2019], 053109
Maximizing simulated tropical cyclone intensity with action minimization
with D.A. Plotkin, R.J. Webber, M.E O'Neill, and D.S. Abbot, Journal of Advances in Modeling Earth Systems (JAMES), 11(4) [2019], 863-891
Trajectory stratification of stochastic dynamics
with A.R. Dinner, J.C. Mattingly, J. Tempkin, and B. Van Koten, SIAM Reviews: Research Spotlight, 60(4) [2018], 909–938
Simulating the stochastic dynamics and cascade failure of power networks
with C. Matthews, B. Stadie, M. Anitescu, and C. Demarco, [2017]
The Brownian fan
with M. Hairer, Communications in Pure and Applied Mathematics, 68(1) [2015], 1-60
Improved diffusion Monte Carlo
with M. Hairer, Communications in Pure and Applied Mathematics, 67(12) [2014], 1995-2021
Data assimilation in the low noise regime with applications to the Kuroshio
with E. Vanden-Eijnden, Monthly Weather Review, 141 [2013], 1822-1841
Steered transition path sampling
with N. Guttenberg and A. Dinner, Journal of Chemical Physics, 136 [2012], 234103
Rare event simulation for small noise diffusions
with E. Vanden-Eijnden, Communications in Pure and Applied Mathematics, 65(12) [2012], 1770-1803
Particle filtering with path sampling and an application to a bimodal ocean current model
Journal of Computational Physics, 228 [2009], 4312-4331
Efficient Conditional Path Sampling of Stochastic Differential Equations
Thesis (Ph.D.) - University of California, Berkeley, ProQuest, [2007], ISBN: 978-0549-73732-2
Statistical and machine learning for
scientific computing
with C. Lorpaiboon and A.R. Dinner, Journal of Chemical Physics, accepted
Computing transition path theory quantities with trajectory stratification
with B.P. Vani and A.R. Dinner, Journal of Chemical Physics, accepted
Rare event sampling improves Mercury instability statistics
with D.S. Abott, R.J. Webber, and S. Hadden, Astrophysical Journal, 923(2) [2021], 236
Insulin dissociates by diverse mechanisms of coupled unfolding and unbinding
with A. Antoszewski, C.-J. Feng, B.P. Vani, E.H. Thiede, L. Hong, A. Tokmakoff, and A.R. Dinner, Journal of Physical Chemistry B, 124(27) [2020], 5571-5587
Path properties of atmospheric transitions: illustration with a low-order sudden stratospheric warming model
with J. Finkel, and D.S. Abbot, Journal of Atmospheric Science, 77(7) [2020], 2327-2347
Practical rare event simulation for extreme mesoscale weather
with R.J. Webber, D.A. Plotkin, M.E O'Neill, and D.S. Abbot, Chaos, 29 [2019], 053109
Maximizing simulated tropical cyclone intensity with action minimization
with D.A. Plotkin, R.J. Webber, M.E O'Neill, and D.S. Abbot, Journal of Advances in Modeling Earth Systems (JAMES), 11(4) [2019], 863-891
Trajectory stratification of stochastic dynamics
with A.R. Dinner, J.C. Mattingly, J. Tempkin, and B. Van Koten, SIAM Reviews: Research Spotlight, 60(4) [2018], 909–938
Simulating the stochastic dynamics and cascade failure of power networks
with C. Matthews, B. Stadie, M. Anitescu, and C. Demarco, [2017]
The Brownian fan
with M. Hairer, Communications in Pure and Applied Mathematics, 68(1) [2015], 1-60
Improved diffusion Monte Carlo
with M. Hairer, Communications in Pure and Applied Mathematics, 67(12) [2014], 1995-2021
Data assimilation in the low noise regime with applications to the Kuroshio
with E. Vanden-Eijnden, Monthly Weather Review, 141 [2013], 1822-1841
Steered transition path sampling
with N. Guttenberg and A. Dinner, Journal of Chemical Physics, 136 [2012], 234103
Rare event simulation for small noise diffusions
with E. Vanden-Eijnden, Communications in Pure and Applied Mathematics, 65(12) [2012], 1770-1803
Particle filtering with path sampling and an application to a bimodal ocean current model
Journal of Computational Physics, 228 [2009], 4312-4331
Efficient Conditional Path Sampling of Stochastic Differential Equations
Thesis (Ph.D.) - University of California, Berkeley, ProQuest, [2007], ISBN: 978-0549-73732-2
Learning
forecasts of rare stratospheric transitions from short
simulations
with J. Finkel, R.J. Webber, D.S. Abbot, and E.P. Gerber, Monthly Weather Review, 149(11) [2021], 3649-3669
Long-timescale predictions from short-trajectory data: A benchmark analysis of the trp-cage miniprotein
with J. Strahan, A. Antoszewski, C. Lorpaiboon, B. Vani, and A.R. Dinner, Journal of Chemical Theory and Computation, 17(5) [2021], 2948-2963
Error bounds for dynamical spectral estimation
with R.J. Webber, E.H. Thiede, D. Dow, and A.R. Dinner, SIAM Journal on Mathematics of Data Science, 3(1) [2021], 225-252
Integrated VAC: A robust strategy for identifying eigenfunctions of dynamical operators
with C. Lorpaiboon, E.H. Thiede, R.J. Webber, and A.R. Dinner, Journal of Physical Chemistry B, 124(42) [2020], 9354-9364
Galerkin approximation of dynamical quantities using trajectory data
with E.H. Thiede, D. Giannakis, and A.R. Dinner, Journal of Chemical Physics, 150 [2019], 24111
Distinguishing meanders of the Kuroshio using machine learning
with D. Plotkin and D. Abbot, Journal of Geophysical Research - Oceans, 119(10) [2014], 6593-6604
with J. Finkel, R.J. Webber, D.S. Abbot, and E.P. Gerber, Monthly Weather Review, 149(11) [2021], 3649-3669
Long-timescale predictions from short-trajectory data: A benchmark analysis of the trp-cage miniprotein
with J. Strahan, A. Antoszewski, C. Lorpaiboon, B. Vani, and A.R. Dinner, Journal of Chemical Theory and Computation, 17(5) [2021], 2948-2963
Error bounds for dynamical spectral estimation
with R.J. Webber, E.H. Thiede, D. Dow, and A.R. Dinner, SIAM Journal on Mathematics of Data Science, 3(1) [2021], 225-252
Integrated VAC: A robust strategy for identifying eigenfunctions of dynamical operators
with C. Lorpaiboon, E.H. Thiede, R.J. Webber, and A.R. Dinner, Journal of Physical Chemistry B, 124(42) [2020], 9354-9364
Galerkin approximation of dynamical quantities using trajectory data
with E.H. Thiede, D. Giannakis, and A.R. Dinner, Journal of Chemical Physics, 150 [2019], 24111
Distinguishing meanders of the Kuroshio using machine learning
with D. Plotkin and D. Abbot, Journal of Geophysical Research - Oceans, 119(10) [2014], 6593-6604
Software
Fast Randomized Iteration for Electronic Structure (FRIES)
S.M. Greene (I am not an author)
C++ implementations of various methods within the Fast Randomized Iteration (FRI) framework for performing Full Configuration Interaction calculations on molecular systems and the Hubbard model.
pyEDGAR
E.H. Thiede (I am not an author)
Python implementation of Dynamic Galerkin Approximation (DGA) which builds predictions of long-timescale phenomena from short trajectory data.
Eigenvector Method for Umbrella Sampling (EMUS)
E.H. Thiede (I am not an author)
Python implementation of a stratification approach to MCMC
Umbrella Sampling for Data Science Applications
C. Matthews (I am not an author)
Lightweight Python implementation of EMUS for data science applications
Ensemble QuasiNewton MCMC (EQN)
C. Matthews (I am not an author)
Python implementation of an emsemble preconditioning approach to MCMC
Fast Randomized Iteration (FRI)
with J. Dama
Demonstration C++ implementation of randomized power iteration.
Enhanced Sampling Toolkit
J. Tempkin (I am not an author)
The Enhanced Sampling Toolkit provides a flexible and extensible toolkit for rapidly prototyping rare event simulation algorithms. The code is written entirely in Python and acts as a wrapper to various well-established molecular dynamics codes.
Ticketed Diffusion Monte Carlo (TDMC)
J. Dama (I am not an author)
Demonstration C++ implementation of an improved diffusion Monte Carlo method.
emcee: The MCMC Hammer
D. Foreman-Mackey, D. Hogg, D. Lang, and J. Goodman (I am not an author)
Python implementation of an affine invariant ensemble MCMC scheme.