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
Chinese Academy of Sciences
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.