TY  - JOUR
A1  - de Wiljes, Jana
A1  - Pathiraja, Sahani Darschika
A1  - Reich, Sebastian
T1  - Ensemble transform algorithms for nonlinear smoothing problems
JF  - SIAM journal on scientific computing
N2  - Several numerical tools designed to overcome the challenges of smoothing in a non-linear and non-Gaussian setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of linear ensemble transform filters which contains classical filters such as the stochastic ensemble Kalman filter, the ensemble square root filter, and the recently introduced nonlinear ensemble transform filter. Further the ensemble transform particle smoother is introduced and particularly highlighted as it is consistent in the particle limit and does not require assumptions with respect to the family of the posterior distribution. The linear update pattern of the considered class of linear ensemble transform smoothers allows one to implement important supplementary techniques such as adaptive spread corrections, hybrid formulations, and localization in order to facilitate their application to complex estimation problems. These additional features are derived and numerically investigated for a sequence of increasingly challenging test problems.
KW  - data assimilation
KW  - smoother
KW  - localization
KW  - optimal transport
KW  - adaptive
KW  - spread correction
Y1  - 2019
U6  - https://doi.org/10.1137/19M1239544
SN  - 1064-8275
SN  - 1095-7197
VL  - 42
IS  - 1
SP  - A87
EP  - A114
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Bourne, D. P.
A1  - Cushing, D.
A1  - Liu, S.
A1  - Münch, Florentin
A1  - Peyerimhoff, Norbert
T1  - Ollivier-Ricci idleness functions of graphs
JF  - SIAM Journal on Discrete Mathematics
N2  - We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most three linear parts, and at most two linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.
KW  - Ollivier-Ricci
KW  - idleness
KW  - optimal transport
Y1  - 2018
U6  - https://doi.org/10.1137/17M1134469
SN  - 0895-4801
SN  - 1095-7146
VL  - 32
IS  - 2
SP  - 1408
EP  - 1424
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Nüsken, Nikolas
A1  - Pavhotis, Grigorios A.
T1  - Constructing Sampling Schemes via Coupling
BT  - Markov Semigroups and Optimal Transport
JF  - SIAM ASA journal on uncertainty quantification / Society for Industrial and Applied Mathematics ; American Statistical Association
N2  - In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincare inequality.
KW  - sampling
KW  - optimal transport
KW  - particle methods
KW  - Markov semigroups
KW  - MCMC
Y1  - 2019
U6  - https://doi.org/10.1137/18M119896X
SN  - 2166-2525
VL  - 7
IS  - 1
SP  - 324
EP  - 382
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Gregory, A.
A1  - Cotter, C. J.
A1  - Reich, Sebastian
T1  - MULTILEVEL ENSEMBLE TRANSFORM PARTICLE FILTERING
JF  - SIAM journal on scientific computing
N2  - This paper extends the multilevel Monte Carlo variance reduction technique to nonlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particle filter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transport methods to re-establish correlation between coarse and fine ensembles after resampling; this controls the variance of the estimator. Numerical examples present a proof of concept of the effectiveness of the proposed method, demonstrating significant computational cost reductions (relative to the single-level ETPF counterpart) in the propagation of ensembles.
KW  - multilevel Monte Carlo
KW  - sequential data assimilation
KW  - optimal transport
Y1  - 2016
U6  - https://doi.org/10.1137/15M1038232
SN  - 1064-8275
SN  - 1095-7197
VL  - 38
SP  - A1317
EP  - A1338
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -