128 research outputs found
An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion
In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion
model with a Caputo fractional derivative of order in time,
which is often used to describe anomalous diffusion processes in heterogeneous
media. The nonlocality of the fractional derivative requires storing all the
solutions from time zero. The proposed scheme is based on continuous piecewise
linear finite elements, L1 time stepping, and proper orthogonal decomposition
(POD). By constructing an effective reduced-order scheme using problem-adapted
basis functions, it can significantly reduce the computational complexity and
storage requirement. We shall provide a complete error analysis of the scheme
under realistic regularity assumptions by means of a novel energy argument.
Extensive numerical experiments are presented to verify the convergence
analysis and the efficiency of the proposed scheme.Comment: 25 pp, 5 figure
A Finite Element Method With Singularity Reconstruction for Fractional Boundary Value Problems
We consider a two-point boundary value problem involving a Riemann-Liouville
fractional derivative of order \al\in (1,2) in the leading term on the unit
interval . Generally the standard Galerkin finite element method can
only give a low-order convergence even if the source term is very smooth due to
the presence of the singularity term x^{\al-1} in the solution
representation. In order to enhance the convergence, we develop a simple
singularity reconstruction strategy by splitting the solution into a singular
part and a regular part, where the former captures explicitly the singularity.
We derive a new variational formulation for the regular part, and establish
that the Galerkin approximation of the regular part can achieve a better
convergence order in the , H^{\al/2}(0,1) and -norms
than the standard Galerkin approach, with a convergence rate for the recovered
singularity strength identical with the error estimate. The
reconstruction approach is very flexible in handling explicit singularity, and
it is further extended to the case of a Neumann type boundary condition on the
left end point, which involves a strong singularity x^{\al-2}. Extensive
numerical results confirm the theoretical study and efficiency of the proposed
approach.Comment: 23 pp. ESAIM: Math. Model. Numer. Anal., to appea
Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
In this work we propose and analyze a numerical method for electrical
impedance tomography of recovering a piecewise constant conductivity from
boundary voltage measurements. It is based on standard Tikhonov regularization
with a Modica-Mortola penalty functional and adaptive mesh refinement using
suitable a posteriori error estimators of residual type that involve the state,
adjoint and variational inequality in the necessary optimality condition and a
separate marking strategy. We prove the convergence of the adaptive algorithm
in the following sense: the sequence of discrete solutions contains a
subsequence convergent to a solution of the continuous necessary optimality
system. Several numerical examples are presented to illustrate the convergence
behavior of the algorithm.Comment: 26 pages, 12 figure
A new approach to nonlinear constrained Tikhonov regularization
We present a novel approach to nonlinear constrained Tikhonov regularization
from the viewpoint of optimization theory. A second-order sufficient optimality
condition is suggested as a nonlinearity condition to handle the nonlinearity
of the forward operator. The approach is exploited to derive convergence rates
results for a priori as well as a posteriori choice rules, e.g., discrepancy
principle and balancing principle, for selecting the regularization parameter.
The idea is further illustrated on a general class of parameter identification
problems, for which (new) source and nonlinearity conditions are derived and
the structural property of the nonlinearity term is revealed. A number of
examples including identifying distributed parameters in elliptic differential
equations are presented.Comment: 21 pages, to appear in Inverse Problem
On the Regularizing Property of Stochastic Gradient Descent
Stochastic gradient descent is one of the most successful approaches for
solving large-scale problems, especially in machine learning and statistics. At
each iteration, it employs an unbiased estimator of the full gradient computed
from one single randomly selected data point. Hence, it scales well with
problem size and is very attractive for truly massive dataset, and holds
significant potentials for solving large-scale inverse problems. In the recent
literature of machine learning, it was empirically observed that when equipped
with early stopping, it has regularizing property. In this work, we rigorously
establish its regularizing property (under \textit{a priori} early stopping
rule), and also prove convergence rates under the canonical sourcewise
condition, for minimizing the quadratic functional for linear inverse problems.
This is achieved by combining tools from classical regularization theory and
stochastic analysis. Further, we analyze the preasymptotic weak and strong
convergence behavior of the algorithm. The theoretical findings shed insights
into the performance of the algorithm, and are complemented with illustrative
numerical experiments.Comment: 22 pages, better presentatio
Expectation Propagation for Nonlinear Inverse Problems -- with an Application to Electrical Impedance Tomography
In this paper, we study a fast approximate inference method based on
expectation propagation for exploring the posterior probability distribution
arising from the Bayesian formulation of nonlinear inverse problems. It is
capable of efficiently delivering reliable estimates of the posterior mean and
covariance, thereby providing an inverse solution together with quantified
uncertainties. Some theoretical properties of the iterative algorithm are
discussed, and the efficient implementation for an important class of problems
of projection type is described. The method is illustrated with one typical
nonlinear inverse problem, electrical impedance tomography with complete
electrode model, under sparsity constraints. Numerical results for real
experimental data are presented, and compared with that by Markov chain Monte
Carlo. The results indicate that the method is accurate and computationally
very efficient.Comment: Journal of Computational Physics, to appea
Lagrange optimality system for a class of nonsmooth convex optimization
In this paper, we revisit the augmented Lagrangian method for a class of
nonsmooth convex optimization. We present the Lagrange optimality system of the
augmented Lagrangian associated with the problems, and establish its
connections with the standard optimality condition and the saddle point
condition of the augmented Lagrangian, which provides a powerful tool for
developing numerical algorithms. We apply a linear Newton method to the
Lagrange optimality system to obtain a novel algorithm applicable to a variety
of nonsmooth convex optimization problems arising in practical applications.
Under suitable conditions, we prove the nonsingularity of the Newton system and
the local convergence of the algorithm.Comment: 19 page
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
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