Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
9
2
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
HESSIAN-BASED SAMPLING FOR HIGH-DIMENSIONAL MODEL REDUCTION
103-121
Peng
Chen
Institute for Computational Engineering & Sciences, The University of Texas at Austin,
Austin, Texas 78712, USA
Omar
Ghattas
Institute for Computational Engineering & Sciences, The University of Texas at Austin,
Austin, Texas 78712, USA; Department of Mechanical Engineering, and Department of Geological Sciences, The University of Texas at Austin, Austin, Texas 78712, USA
In this work we develop a Hessian-based sampling method for the construction of goal-oriented reduced order models
with high-dimensional parameter inputs. Model reduction is known to be very challenging for high-dimensional parametric
problems whose solutions also live in high-dimensional manifolds. However, the manifold of some quantity of
interest (QoI) depending on the parametric solutions may be low-dimensional. We use the Hessian of the QoI with respect
to the parameter to detect this low-dimensionality, and draw training samples by projecting the high-dimensional
parameter to a low-dimensional subspace spanned by the eigenvectors of the Hessian corresponding to its dominating
eigenvalues. Instead of forming the full Hessian, which is computationally intractable for a high-dimensional parameter,
we employ a randomized algorithm to efficiently compute the dominating eigenpairs of the Hessian whose cost does
not depend on the nominal dimension of the parameter but only on the intrinsic dimension of the QoI.We demonstrate
that the Hessian-based sampling leads to much smaller errors of the reduced basis approximation for the QoI compared
to a random sampling for a diffusion equation with random input obeying either uniform or Gaussian distributions.
RANDOM REGULARITY OF A NONLINEAR LANDAU DAMPING SOLUTION FOR THE VLASOV-POISSON EQUATIONS WITH RANDOM INPUTS
123-142
Zhiyan
Ding
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA
Shi
Jin
School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC,
Shanghai Jiao Tong University, Shanghai 200240, China
In this paper, we study the nonlinear Landau damping solution of the Vlasov-Poisson equations with random inputs
from the initial data or equilibrium, for the solution studied by Hwang and VelÃ¡zquez smoothly on the random input,
if the long-time limit distribution function has the same smoothness, under some smallness assumptions. We also
establish the decay of the higher-order derivatives of the solution in the random variable, with the same decay rate as
its deterministic counterpart.
ADJOINT FORWARD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY JUMP DIFFUSION PROCESSES AND ITS APPLICATION TO NONLINEAR FILTERING PROBLEMS
143-159
Feng
Bao
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
Yanzhao
Cao
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849; School of Mathematics, Sun Yat Sun University, China
Hongmei
Chi
Department of Computer and Information Sciences, Florida A&M University, Tallahassee,
Florida 32306, USA
Forward backward stochastic differential equations (FBSDEs) were first introduced as a probabilistic interpretation
for the Kolmogorov backward equation, and the solution of FBSDEs is equivalent to the solution of quasilinear partial
differential equations. In this work, we introduce the adjoint relation between a generalized FBSDE system driven
by jump diffusion processes and its time inverse adjoint FBSDE system under the probabilistic framework without
translating them into their corresponding PDEs. The "exact solution" of a nonlinear filtering problem is derived as an
application.
NUMERICAL APPROXIMATION OF ELLIPTIC PROBLEMS WITH LOG-NORMAL RANDOM COEFFICIENTS
161-186
Xiaoliang
Wan
Department of Mathematics and Center of Computation and Technology, Louisiana State
University, Baton Rouge, LA, 70803
Haijun
Yu
NCMIS & LSEC, Institute of Computational Mathematics and Scientific/Engineering
Computing, Academy of Mathematics and Systems Science, Beijing 100190; School of
Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical
elliptic equations with log-normal random coefficients. In earlier work, a Wick-type elliptic model was proposed by
modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such
a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard
deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian
process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.
REDUCING FRACTURE PREDICTION UNCERTAINTY BASED ON TIME-LAPSE SEISMIC (4D) AND DETERMINISTIC INVERSION ALGORITHM
187-204
Liming
Zhang
China University of Petroleum, Qingdao, Shandong 266580, China
Chenyu
Cui
China University of Petroleum, Qingdao, Shandong 266580, China
Kai
Zhang
China University of Petroleum, Qingdao, Shandong 266580, China
Yi
Wang
Sinopec Research Institute of Petroleum Engineering, Beijing 100000, China
Zhixue
Sun
China University of Petroleum, Qingdao, Shandong 266580, China
Jun
Yao
School of Petroleum Engineering, China University of Petroleum (East China), No. 66 Changjiang West Road, Huangdao Zone, Qingdao City, Shandong Province, 266580 P.R. China
Qin
Luo
Southwest Petroleum University, Chengdu 610000, China
The uncertainty of hydraulic fracture is high due to the complex geological features of which there is limited accurate understanding, and the limitations of the fracture diagnosis method. However, hydraulic fractures are one of the main driving forces for oilfields to improve economic benefit and important reference imformation for further development and adjustment of oilfields. Therefore, reducing fracture morphology uncertainty is a key challenge for the further development of oilfields. To improve this situation, we present a novel method based on the time-lapse (4D) seismic and discrete network deterministic inversion (DNDI) algorithm for mapping the geometry of hydraulic fracture. The time-lapse (4D) seismic method can provide spatial and dynamic change of reservoir; this information is used by DNDI to optimize fracture geometry continually, where the embedded discrete fracture model (EDFM) is implied to simulate reservoir production, and objective function is constructed using Bayesian theory for reaching iterative convergence quickly. An uncertainty analysis of results based on the posterior probability is also presented in this paper. Finally, this method has been validated in different scale study cases.