In this work,we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of orderα(0,1)in time.The fully discrete scheme is obtained using the standard G...In this work,we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of orderα(0,1)in time.The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time.At each time step,instead of solving the linear algebraic system exactly,we employ a multigrid iteration with a Gauss–Seidel smoother to approximate the solution efficiently.Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.展开更多
We study multi-parameter regularization(multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regu...We study multi-parameter regularization(multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters from the viewpoint of augmented Tikhonov regularization, and derive a new parameter choice strategy called the balanced discrepancy principle. A priori and a posteriori error estimates are provided to theoretically justify the principles, and numerical algorithms for efficiently implementing the principles are also provided. Numerical results on deblurring are presented to illustrate the feasibility of the balanced discrepancy principle.展开更多
In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GM...In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)methods.The former provides a hierarchy of approximations of different resolution,whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels.The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost,and to efficiently generate samples at different levels.In particular,it is cheap to generate samples on coarse grids but with low resolution,and it is expensive to generate samples on fine grids with high accuracy.By suitably choosing the number of samples at different levels,one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces,while retaining the accuracy of the final Monte Carlo estimate.Further,we describe a multilevel Markov chain Monte Carlo method,which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids,while combining the samples at different levels to arrive at an accurate estimate.The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in[26],and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.展开更多
文摘In this work,we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of orderα(0,1)in time.The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time.At each time step,instead of solving the linear algebraic system exactly,we employ a multigrid iteration with a Gauss–Seidel smoother to approximate the solution efficiently.Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.
基金supported by the Army Research Office under DAAD19-02-1-0394,US-ARO grant 49308MA and US-AFSOR grant FA9550-06-1-0241
文摘We study multi-parameter regularization(multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters from the viewpoint of augmented Tikhonov regularization, and derive a new parameter choice strategy called the balanced discrepancy principle. A priori and a posteriori error estimates are provided to theoretically justify the principles, and numerical algorithms for efficiently implementing the principles are also provided. Numerical results on deblurring are presented to illustrate the feasibility of the balanced discrepancy principle.
基金Y.Efendiev’s work is partially supported by the U.S.Department of Energy Office of Science,Office of Advanced Scientific Computing Research,Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO ProjectThe research of B.Jin is partly supported by NSF Grant DMS-1319052.
文摘In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)methods.The former provides a hierarchy of approximations of different resolution,whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels.The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost,and to efficiently generate samples at different levels.In particular,it is cheap to generate samples on coarse grids but with low resolution,and it is expensive to generate samples on fine grids with high accuracy.By suitably choosing the number of samples at different levels,one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces,while retaining the accuracy of the final Monte Carlo estimate.Further,we describe a multilevel Markov chain Monte Carlo method,which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids,while combining the samples at different levels to arrive at an accurate estimate.The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in[26],and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.
