173 citations · 1.2k across the 31 of their papers we have counts for
30 papers · 1 filter
Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations
Hong-lin Liao, Tao Tang, Tao Zhou
This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BD…
A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations
Jun Liu, Xiang-Sheng Wang, Shu-Lin Wu +1
In this paper, we propose a direct parallel-in-time (PinT) algorithm for time-dependent problems with first- or second-order derivative. We use a second-order boundary value method…
Optimal design for kernel interpolation: applications to uncertainty quantification
Akil Narayan, Liang Yan, Tao Zhou
The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation us…
A new discrete energy technique for multi-step backward difference formulas
Honglin Liao, Tao Tang, Tao Zhou
The backward differentiation formula (BDF) is a useful family of implicit methods for the numerical integration of stiff differential equations. It is well noticed that the stabili…
Stability implies robust convergence of a class of preconditioned parallel-in-time iterative algorithms
Shu-Lin Wu, Tao Zhou, Zhi Zhou
Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block -circ…
An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation
Hong-lin Liao, Tao Tang, Tao Zhou
In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in…
Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators
Hong-lin Liao, Tao Tang, Tao Zhou
The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this…
Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection
Hong-lin Liao, Xuehua Song, Tao Tang +1
In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without…
ParaDiag: parallel-in-time algorithms based on the diagonalization technique
Martin J. Gander, Jun Liu, Shu-Lin Wu +2
In 2008, Maday and Ronquist introduced an interesting new approach for the direct parallel-in-time (PinT) solution of time-dependent PDEs. The idea is to diagonalize the time stepp…
On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation
Hong-lin Liao, Tao Tang, Tao Zhou
In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is ener…
A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations
Hong-lin Liao, Tao Tang, Tao Zhou
In this work, we present a second-order nonuniform time-stepping scheme for the time-fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete maximum…
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Tao Tang, Li-Lian Wang, Huifang Yuan +1
Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solution…
An adaptive multi-fidelity PC-based ensemble Kalman inversion for inverse problems
Liang Yan, Tao Zhou
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 mode…
On energy dissipation theory and numerical stability for time-fractional phase field equations
Tao Tang, Haijun Yu, Tao Zhou
For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shal…
Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems
Liang Yan, Tao Zhou
The polynomial chaos (PC) expansion has been widely used as a surrogate model in the Bayesian inference to speed up the Markov chain Monte Carlo (MCMC) calculations. However, the u…
Data-driven polynomial chaos expansions: a weighted least-square approximation
Ling Guo, Yongle Liu, Tao Zhou
In this work, we combine the idea of data-driven polynomial chaos expansions with the weighted least-square approach to solve uncertainty quantification (UQ) problems. The idea of…
A gradient enhanced -minimization for sparse approximation of polynomial chaos expansions
Ling Guo, Akil Narayan, Tao Zhou
We investigate a gradient-enhanced -minimization for constructing sparse polynomial chaos expansions. In addition to function evaluations, measurements of the function grad…
Hermite spectral collocation methods for fractional PDEs in unbounded domains
Tao Tang, Huifang Yuan, Tao Zhou
This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions…
Weighted approximate Fekete points: Sampling for least-squares polynomial approximation
Ling Guo, Akil Narayan, Liang Yan +1
We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations o…
Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs
Yu Fu, Weidong Zhao, Tao Zhou
This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our…
A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions
John D. Jakeman, Akil Narayan, Tao Zhou
In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomi…
Stochastic collocation methods via minimization using randomized quadratures
Ling Guo, Akil Narayan, Tao Zhou +1
In this work, we discuss the problem of approximating a multivariate function via minimization method, using a random chosen sub-grid of the corresponding tensor grid of G…
High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control
Kong Tao, Weidong Zhao, Tao Zhou
This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial…
Stochastic collocation on unstructured multivariate meshes
Akil Narayan, Tao Zhou
Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification (UQ) community. Techniques for least-squares regularization, com…