A Posteriori Error Estimates for a Semidiscrete Parabolic Integrodifferential Control on Multimeshes

We extend the existing techniques to study semidiscrete adaptive finite element approximation schemes for a constrained optimal control problem governed by parabolic integrodifferential equations. The control problem involves time accumulation and the control constrain is given in an integral obstacle sense. We first prove the uniqueness and existence of the solution of this optimal control problem. We then derive the upper a posteriori error estimators for both the state and the control approximation, which are useful indicators in adaptive multimesh finite element approximation schemes

MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling

We propose, Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which is a generalization of MC-fPINNs in \cite{guo2022monte}, for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar as in MC-fPINNs, our MC-Nonlocal-PINNs handle the nonlocal operators in a Monte Carlo way, resulting in a very stable approach for high dimensional problems. We present a variety of test problems, including high dimensional Volterra type integral equations, hypersingular integral equations and nonlocal PDEs, to demonstrate the effectiveness of our approach.Comment: 23pages, 13figure

DEA models with Russell measures

In real applications, data envelopment analysis (DEA) models with Russell measures are widely used although their theoretical studies are scattered over the literature. They often have seemingly similar structures but play very different roles in performance evaluation. In this work, we systematically examine some of the models from the viewpoint of preferences used in their Production Possibility Sets (PPS). We identify their key differences through the convexity and free-disposability of their PPS. We believe that this study will provide guidelines for the correct use of these models. Two empirical cases are used to compare their differences

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

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

Evaluation and Empirical Research on Eco-Efficiency of Financial Industry Based on Carbon Footprint in China

Since finance is the core of economic development, the green development of the financial industry is an essential driving force not only for achieving the dual “carbon” goal of China but also for economic and social sustainable development. An accurate understanding of the ecological efficiency of the financial industry is of great importance for guiding sustainable economic development. In this paper, we first calculate the carbon footprint of China’s financial industry in 2012 and 2017 based on the life cycle theory and the input–output analysis method. Second, we analyze the primary sources and final flows of the carbon footprint of the financial industry in each province from the perspectives of the industrial chain and final demand. Finally, we estimate the ecological efficiency, emission reduction, and value-added potential of the financial industry by using the radially adjusted slack variable DEA model (SRAM-DEA) under two assumptions, natural disposability and managerial disposability. The results show that (1) the ecological efficiency of the financial industry in most provinces is low, and the regional differences are significant; (2) the overall ecological efficiency of the financial industry in 2017 was better than that in 2012; (3) technological innovation of financial products and the upgrading of capital supervision play an essential role in promoting the improvement of ecological efficiency. Especially, under managerial disposability, the ecological efficiency of the financial industry in each province has a greater potential for emission reduction and added value

Stochastic Galerkin Method for Optimal Control Problem Governed by Random Elliptic PDE with State Constraints

In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The optimal control minimizes the expectation of a cost functional with mean-state constraints. We first represent the stochastic elliptic PDE in terms of the generalized polynomial chaos expansion and obtain the parameterized optimal control problems. By applying the Slater condition in the subdifferential calculus, we obtain the necessary and sufficient optimality conditions for the state-constrained stochastic optimal control problem for the first time in the literature. We then establish a stochastic Galerkin scheme to approximate the optimality system in the spatial space and the probability space. Then the a priori error estimates are derived for the state, the co-state and the control variables. A projection algorithm is proposed and analyzed. Numerical examples are presented to illustrate our theoretical results

Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense: Uad={u∈X;∫ΩUu⩾0, t∈[0,T]}. Then the a posteriori error estimates in L∞(0,T;H1(Ω))-norm and L2(0,T;L2(Ω))-norm for both the state and the control approximation are given