Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
9
3
2019
PREFACE: A SPECIAL ISSUE CELEBRATING A NEW UQ ACTIVITY GROUP IN CHINA
v
Tao
Zhou
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,
China
AN ADAPTIVE MULTIFIDELITY PC-BASED ENSEMBLE KALMAN INVERSION FOR INVERSE PROBLEMS
205-220
Liang
Yan
Department of Mathematics, Southeast University, Nanjing, 210096, China
Tao
Zhou
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,
China
The ensemble Kalman inversion (EKI), as a derivative-free methodology, has been widely used in the parameter estimation of inverse problems. Unfortunately, its cost may become moderately large for systems described by highdimensional nonlinear PDEs, as EKI requires a relatively large ensemble size to guarantee its performance. In this
paper, we propose an adaptive multifidelity polynomial chaos (PC) based EKI technique to address this challenge. Our
new strategy combines a large number of low-order PC surrogate model evaluations and a small number of high-fidelity forward model evaluations, yielding a multifidelity approach. Specifically, we present a new approach that adaptively constructs and refines a local multifidelity PC surrogate during the EKI simulation. Since the forward model evaluations are only required for updating the low-order local multifidelity PC model, whose number can be much smaller than the total ensemble size of the classic EKI, the entire computational costs are thus significantly reduced. The new algorithm was tested through the two-dimensional time fractional inverse diffusion problems and demonstrated great effectiveness in comparison with PC-based EKI and classic EKI.
A GENERAL FRAMEWORK FOR ENHANCING SPARSITY OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS
221-243
Xiu
Yang
Xiaoliang
Wan
Department of Mathematics and Center of Computation and Technology, Louisiana State
University, Baton Rouge, LA, 70803
Lin
Lin
Department of Mathematics, University of California, Berkeley and Computational Research
Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
Huan
Lei
Advanced Computing, Mathematics and Data Division, Pacific Northwest National
Laboratory, Richland, WA, 99352
Compressive sensing has become a powerful addition to uncertainty quantification when only limited data are available. In this paper, we provide a general framework to enhance the sparsity of the representation of uncertainty in
the form of generalized polynomial chaos expansion. We use an alternating direction method to identify new sets of
random variables through iterative rotations so the new representation of the uncertainty is sparser. Consequently,
we increase both the efficiency and accuracy of the compressive-sensing-based uncertainty quantification method. We demonstrate that the previously developed rotation-based methods to enhance the sparsity of Hermite polynomial expansion is a special case of this general framework. Moreover, we use Legendre and Chebyshev polynomial expansions to demonstrate the effectiveness of this method with applications in solving stochastic partial differential equations and high-dimensional (O (100)) problems.
VARIABLE-SEPARATION BASED ITERATIVE ENSEMBLE SMOOTHER FOR BAYESIAN INVERSE PROBLEMS IN ANOMALOUS DIFFUSION REACTION MODELS
245-273
Yuming
Ba
College of Mathematics and Econometrics, Hunan University 1, Changsha 410082, China
Lijian
Jiang
School of Mathematical Sciences, Tongji University, Shanghai 200092, China
Na
Ou
College of Mathematics and Econometrics, Hunan University 1, Changsha 410082, China
The iterative ensemble smoother (IES) has been widely used to estimate parameters and states of dynamic models
where the data are collected at all observation steps simultaneously. A large number of IES ensemble samples may
be required in the estimation. This implies that we need to repeatedly compute the forward model corresponding to
the ensemble samples. This leads to slow efficiency for large-scale and strongly nonlinear models. To accelerate the
posterior inference in the estimation, a low rank approximation using a variable-separation (VS) method is presented to reduce the cost of computing the forward model. It will be efficient to construct a surrogate model based on the low rank approximation, which gives a separated representation of the solution for the stochastic partial differential equations (SPDEs). The separated representation is the product of deterministic basis functions and stochastic basis functions. For the anomalous diffusion reaction equations, the solution of the next moment depends on all of the previous moments, and this causes expensive computation for the Bayesian inverse problem. The presented VS can avoid this process through a few deterministic basis functions. The surrogate model can work well as the iteration moves on because the stochastic basis becomes more accurate when the uncertainty of random parameters decreases. To enhance the applicability in Bayesian inverse problems, we apply the VS-based IES method to complex structure patterns, which can be parameterized by discrete cosine transform (DCT). The post-processing technique based on a regularization method is employed after the iterations to improve the connectivity of the main features. In the paper, we focus on the time fractional diffusion reaction models in porous media and investigate their Bayesian inverse problems using the VS-based IES. A few numerical examples are presented to show the performance of the proposed IES method by taking account of structure inversion in permeability fields, parameters in permeability and reaction fields, and source functions.
AN EFFICIENT NUMERICAL METHOD FOR UNCERTAINTY QUANTIFICATION IN CARDIOLOGY MODELS
275-294
Xindan
Gao
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road,
Minhang, Shanghai, P.R. China, 200240
Wenjun
Ying
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong,
SAR, China
Zhiwen
Zhang
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong,
SAR, China
Mathematical models of cardiology involve conductivity and massive parameters describing the dynamics of ionic
channels. The conductivity is space dependent and cannot be measured directly. The dynamics of ionic channels are
highly nonlinear, and the parameters have unavoidable uncertainties because they are estimated using repeated experimental data. Such uncertainties can impact model dependability and credibility since they spread to model parameters during model calibration. It is necessary to study how the uncertainties influence the solution compared to the deterministic solution and to quantify the difference resulting from uncertainty. In this paper, the generalized polynomial chaos method and stochastic collocation method are used to solve the corresponding stochastic partial differential equations. Numerical results are shown to demonstrate that each parameter has different effects on the model responses. More importantly, a quadratic convergence of the expectation is exhibited in the numerical results. The amplitude of standard deviation of the stochastic solution can be controlled by the parameter uncertainty. More precisely, the standard deviation of the stochastic solution is positively linear to the standard deviation of the random parameter. We utilized monodomain equations, which are representative mathematical models to demonstrate the results with the most widely used ionic models, the Hodgkin-Huxley model and Fitz-Hugh Nagumo model.
USING PARALLEL MARKOV CHAIN MONTE CARLO TO QUANTIFY UNCERTAINTIES IN GEOTHERMAL RESERVOIR CALIBRATION
295-310
Tiangang
Cui
School of Mathematical Sciences, Monash University, VIC 3800, Australia
C.
Fox
Department of Physics, University of Otago, Dunedin 9016, New Zealand
G. K.
Nicholls
Department of Statistics, University of Oxford, Oxford, OX1 3LG, United Kingdom
M. J.
O'Sullivan
Department of Engineering Sciences, The University of Auckland, Auckland 1010, New
Zealand
We introduce a parallel rejection scheme to give a simple but reliable way to parallelize the Metropolis-Hastings algorithm. This method can be particularly useful when the target density is computationally expensive to evaluate and the acceptance rate of the Metropolis-Hastings is low. We apply the resulting method to quantify uncertainties of inverse problems, in which we aim to calibrate a challenging nonlinear geothermal reservoir model using real measurements
from well tests. We demonstrate the parallelized method on various well-test scenarios. In some scenarios, the sample-based statistics obtained by our scheme shows clear advantages in providing robust model calibration and prediction compared with those obtained by nonlinear optimization methods.
A WEIGHT-BOUNDED IMPORTANCE SAMPLING METHOD FOR VARIANCE REDUCTION
311-319
Tenchao
Yu
School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong
University, 800 Dongchuan Rd, Shanghai 200240, China
Linjun
Lu
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University,
Shanghai 200240, China
Jinglai
Li
Institute of Natural Sciences and School of
Mathematical Sciences
Shanghai Jiaotong University, Shanghai 200240, China
Importance sampling (IS) is an important technique to reduce the estimation variance in Monte Carlo simulations. In
many practical problems, however, the use of the IS method may result in unbounded variance, and thus fail to provide
reliable estimates. To address the issue, we propose a method which can prevent the risk of unbounded variance; the
proposed method performs the standard IS for the integral of interest in a region only in which the IS weight is bounded
and we use the result as an approximation to the original integral. It can be verified that the resulting estimator has a finite variance. Moreover, we also provide a normality test based method to identify the region with bounded IS weight (termed as the safe region) from the samples drawn from the standard IS distribution. With numerical examples,
we demonstrate that the proposed method can yield a rather reliable estimate when the standard IS fails, and it also
outperforms the defensive IS, a popular method to prevent unbounded variance.