Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quasi-random sequences and their discrepancies
SIAM Journal on Scientific Computing
An efficient dynamically adaptive mesh for potentially singular solutions
Journal of Computational Physics
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Modeling uncertainty in flow simulations via generalized polynomial chaos
Journal of Computational Physics
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Multilevel Monte Carlo Path Simulation
Operations Research
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
IEEE Transactions on Audio, Speech, and Language Processing
Journal of Computational Physics
Journal of Computational Physics
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This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O((m/N"p)^3) for a quadratic nonlinear SPDE, where m is the number of mode pairs used in the DyBO method and N"p is the number of elements in the polynomial basis in gPC. The effective dimensions of the stochastic solutions have been found to be small in many applications, so we can expect m is much smaller than N"p and computational savings of our DyBO method against gPC are dramatic. The adaptive strategy plays an essential role for the DyBO method to be effective in solving some challenging problems. Another important contribution of this paper is the generalization of the DyBO formulation for a system of time-dependent SPDEs. Several numerical examples are provided to demonstrate the effectiveness of our method, including the Navier-Stokes equations and the Boussinesq approximation with Brownian forcing.