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Register a new userPointwise-in-time error estimates for an optimal control problem with subdiffusion constraint
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In this work, we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation which involves a fractional derivative of order $alphain(0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational type discretization. With a space mesh size $h$ and time stepsize $tau$, we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(tau^{min({1}/{2}+alpha-epsilon,1)}+h^2)$ in the discrete $L^2(0,T;L^2(Omega))$ norm and $O(tau^{alpha-epsilon}+ell_h^2h^2)$ in the discrete $L^infty(0,T;L^2(Omega))$ norm, with any small $epsilon>0$ and $ell_h=ln(2+1/h)$. The analysis relies essentially on the maximal $L^p$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.
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This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The state equation is discretized by a space-time finite element method. The controls are not discretized. Under suitable assumptions optimal convergence rates for the error in the state and control variable are proven. Based on a conditional gradient method the solution of the semi-discretized optimal control problem is computed. The theoretical convergence rates are confirmed in a numerical example.
Given a function $uin L^2=L^2(D,mu)$, where $Dsubset mathbb R^d$ and $mu$ is a measure on $D$, and a linear subspace $V_nsubset L^2$ of dimension $n$, we show that near-best approximation of $u$ in $V_n$ can be computed from a near-optimal budget of $Cn$ pointwise evaluations of $u$, with $C>1$ a universal constant. The sampling points are drawn according to some random distribution, the approximation is computed by a weighted least-squares method, and the error is assessed in expected $L^2$ norm. This result improves on the results in [6,8] which require a sampling budget that is sub-optimal by a logarithmic factor, thanks to a sparsification strategy introduced in [17,18]. As a consequence, we obtain for any compact class $mathcal Ksubset L^2$ that the sampling number $rho_{Cn}^{rm rand}(mathcal K)_{L^2}$ in the randomized setting is dominated by the Kolmogorov $n$-width $d_n(mathcal K)_{L^2}$. While our result shows the existence of a randomized sampling with such near-optimal properties, we discuss remaining issues concerning its generation by a computationally efficient algorithm.
We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) $L^infty$-estimates and $L^2$-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for $p$ and $hp$ methods for singular problems, and answer some open questions posed in some recent literature.
A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1({Omega})-conforming Taylor-Hood finite element combination, consisting of polynomial degrees k + 1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori error estimator is derived on the basis of H(div)-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can be performed locally on a set of vertex patches and lead to a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. Particular emphasis is given to nearly incompressible materials and the error estimates hold uniformly in the incompressible limit. Numerical results on the L-shaped domain confirm the theory and the suitable use of the error estimator in adaptive strategies.
In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.
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