This paper is a survey on some recent results in optimal control and stochastic filtering. The goal is not to cover all recent developments in control and filtering, instead we focus on maximum principle for optimality of partial information backward or forward-backward stochastic differential equations and branching particle approximation of nonlinear filtering.

In this article, we consider the derivation of $hp$-optimal spectral bounds for a class of domain decomposition preconditioners based on the Schwarz framework for discontinuous Galerkin finite element approximations of second-order elliptic partial differential equations. In particular, we improve the bounds derived in our earlier article [P.F. Antonietti and P. Houston, J. Sci. Comput., 46(1):124-149, 2011] in the sense that the resulting bound on the condition number of the preconditioned system is not only explicit with respect to the coarse and fine mesh sizes $H$ and $h$, respectively, and the fine mesh polynomial degree $p$, but now also explicit with respect to the polynomial degree $q$ employed for the coarse grid solver. More precisely, we show that the resulting spectral bounds are of order $p^{2}H/(qh)$ for the $hp$-version of the discontinuous Galerkin method.

We apply in this paper the weak Galerkin method to the second order parabolic differential equations based on a discrete weak gradient operator. We establish both the continuous time and the discrete time weak Galerkin finite element schemes, which allow using the totally discrete functions in approximation space and the finite element partitions of arbitrary polygons with certain shape regularity. We show as well that the continuous time weak Galerkin finite element method preserves the energy conservation law. The optimal convergence order estimates in both $H^1$ and $L^2$ norms are obtained. Numerical experiments are performed to confirm the theoretical results.

In this article, we discuss and analyze discontinuous finite volume approximations of the distributed optimal control problems governed by a class of semilinear parabolic partial differential equations with control constraints. For the spatial discretization of the state and costate variables, piecewise linear elements are used and an implicit finite difference scheme is used for time derivatives; whereas, for the approximation of the control variable, three different strategies are used: variational discretization, piecewise constant and piecewise linear discretization. A priori error estimates (for these three approaches) in suitable $L^2$-norm are derived for state, co-state and control variables. Numerical experiments are presented in order to assure the accuracy and rate of the convergence of the proposed scheme.

The quanto options pricing model is a typical two-dimensional Black-Scholes equation with a mixed derivative term, and it has been increasingly attracting interest over the last decade. A kind of improved alternating direction implicit methods, which is based on the Douglas-Rachford (D-R ADI) and Craig-Sneyd (C-S ADI) split forms, is given in this paper for solving the quanto options pricing model. The improved ADI methods first split the original problem into two separate one-dimensional problems, and then solve the tri-diagonal matrix equations at each time-step. There are several advantages in this method such as: parallel property, unconditional stability, convergency and better accuracy. The numerical experiments show that this kind of methods is very efficient compared to the existent explicit finite difference method. In addition, because of the natural parallel property of the improved ADI methods, the parallel computing is very easy, and about 50% computational cost can been saved. Thus the improved ADI methods can be used to solve the multi-asset option pricing problems effectively.

The goal of this paper is to compare two computational models for the inverse problem of electroencephalography: the localization of brain activity from measurements of the electric potential on the surface of the head. The source current is modeled as a dipole whose localization and polarization has to be determined. Two methods are considered for solving the corresponding forward problems: the so called subtraction approach and direct approach. The former is based on subtracting a fundamental solution, which has the same singular character of the actual solution, and solving computationally the resulting non-singular problem. Instead, the latter consists in solving directly the problem with singular data by means of an adaptive process based on an a posteriori error estimator, which allows creating meshes appropriately refined around the singularity. A set of experimental tests for both, the forward and the inverse problems, are reported. The main conclusion of these tests is that the direct approach combined with adaptivity is preferable when the localization of the dipole is close to an interface between brain tissues with different conductivities.

In this paper, we propose a splitting least-squares mixed finite element method for the approximation of elliptic optimal control problem with the control constrained by pointwise inequality. By selecting a properly least-squares minimization functional, we derive equivalent two independent, symmetric and positive definite weak formulation for the primal state variable and its flux. Then, using the first order necessary and also sufficient optimality condition, we deduce another two corresponding adjoint state equations, which are both independent, symmetric and positive definite. Also, a variational inequality for the control variable is involved. For the discretization of the state and adjoint state equations, either RT mixed finite element or standard $C^0$ finite element can be used, which is not necessary subject to the Ladyzhenkaya-Babuska-Brezzi condition. Optimal a priori error estimates in corresponding norms are derived for the control, the states and adjoint states, respectively. Finally, we use some numerical examples to validate the theoretical analysis.

We consider a mathematical model for a static process of frictionless unilateral contact between a piezoelectric body and a conductive foundation. A variational formulation of the model, in the form of a coupled system for the displacements and the electric potential, is derived. The existence of a unique weak solution for the problem is established. We use the penalty method applied to the frictionless unilateral contact model to replace the Signorini contact condition, we show the existence of a unique solution, and derive error estimates. Moreover, under appropriate regularity assumptions of the solution, we have the convergence of the continuous penalty solution as the penalty parameter ϵ vanishes. Then, the numerical approximation of a penalty problem by using the finite element method is introduced. The error estimates are derived and convergence of the scheme is deduced under suitable regularity conditions.

In order to effectively use books and reasonably distribute purchasing funds, the Support Vector Machines (SVM) method is used in this paper to establish a mathematics model for the related historical data of the library at Beijing University of Civil Engineering and Architecture. The book circulation and the allocation proportion of purchasing funds in the future are predicted based on the model. It is shown that the SVM method is feasible in predicting the book circulation and allocation proportion of purchasing funds with high non-linearity even if the size of a sample is small.