Stochastic Spline-Collocation Method for Constrained Optimal Control Problem Governed by Random Elliptic PDE

In this paper, we investigate a stochastic spline-collocation approximation scheme for an optimal control problem governed by an elliptic PDE with random field coefficients. We obtain the necessary and sufficient optimality conditions for the optimal control problem and establish a scheme to approximate the optimality system through the discretization with respect to the spatial space by finite elements method and the probability space by stochastic splinecollocation method. We further investigate Smolyak approximation schemes, which are effective collocation strategies for smooth problems that depend on a moderately large number of random variables. For more general control problems where the state may be non-smooth with respect to the random variables in some areas, we adopt a domain decomposition strategy to partition the random space into smooth and non-smooth parts and then apply Smolyak scheme and spline approximation respectively. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results

Tungsten Oxide and Polyaniline Composite Fabricated by Surfactant-Templated Electrodeposition and Its Use in Supercapacitors

Composite nanostructures of tungsten oxide and polyaniline (PANI) were fabricated on carbon electrode by electrocodeposition using sodium dodecylbenzene sulfonate (SDBS) as the template. The morphology of the composite can be controlled by changing SDBS surfactant and aniline monomer concentrations in solution. With increasing concentration of aniline in surfactant solution, the morphological change from nanoparticles to nanofibers was observed. The nanostructured WO3/PANI composite exhibited enhanced capacitive charge storage with the specific capacitance of 201 F g−1 at 1.28 mA cm−2 in large potential window of -0.5~ 0.65 V versus SCE compared to the bulk composite film. The capacitance retained about 78% when the sweeping potential rate increased from 10 to 150 mV/s

A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Random Parabolic PDE with Constrained Control

A stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random perturbation in its coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We obtain the necessary and sufficient optimality conditions and establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results

A Generalized Iterated Tikhonov Method in the Fourier Domain for Determining the Unknown Source of the Time-Fractional Diffusion Equation

In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A modified implicit iteration method is proposed as a regularization technique to stabilize the solution process. The convergence rates are derived when a discrepancy principle serves as the principle for choosing the regularization parameters. Numerical tests are provided to further verify the efficacy of the proposed method