Solution with Singularity of Order One for Singular Integral Equation with Hilbert Kernal

The basic sets of solutions in classH(orH *) for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively. Thus the expressions of solutions and its solvable conditions are simplified. On this basis the solutions and the solvable conditions in classH 1 * as well as the generalized Noether theorem for the complete equation are obtained. Key words Hilbert kernel - solution with singularity of order one - basic set of solutions - Noether theorem - characteristic equation and its adjoint equation CLC number O 175.5 Foundation item: Supported by the National Natural Science Foundation of China (19971064) and Ziqiang Invention Foundation of Wuhan University (201990336)Biography: Zhong Shou-guo(1941-), male, Professor, research direction: singular integral equations and their applications.

Conservation Laws of Differential Equations in Finance

Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.

The Nonlinear Interaction Process in the Wave Assimilation Model and Its Experiments

This paper presents a composite interaction formula based on the discrete-interaction operator of wave-wave nonlinear interaction for deriving its adjoint source function in the wave assimilation model. Assimilation experiments were performed using the significant wave heights observed by the TOPES/POSEIDON satellite, and the gradient distribution in the physical space was also analyzed preliminarily.

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the Orr- Somrnerfeld and Squire （OSS） equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a re- orthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes （N-S） equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is pro- posed. The algorithm is then applied to a one-dimensional base flow （the plane Poiseuille flow） and a two-dimensional base flow （the plane Poiseuille flow with a low-speed streak） in a channel. For one-dimensional cases, the effects of the spanwise width of the chan- nel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is character- ized by high- and low-speed streaks and the corresponding streamwise vortical structures. The lift-up mechanism that induces the transient growth of perturbations is discussed. The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one- and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the cross- check method.

WAVE ASSIMILATION AND NUMERICAL PREDICTION

An adjoint variational method for wave data assimilation in the LAGFD-WAM wave model is proposed. The adjoint equation of the wavenumber energy spectrum balance equation is derived. And fortunately, its characteristic equations are the same as those in the LAGFD-WAM wave model. Simple experiments on the functional optimization and assimilation effectiveness during the prediction period indicated that the adjoint variational method is effective for wave assimilation and that the initial optimization of the wave model is important for the short-range wave prediction. All of this is under the assumption that the wind field is accurate, the method is the important first step for combined wind and wave data assimilation systems. Key words variational data assimilation - adjoint equation - short-range wave prediction Project 97701 and Q99E05 of the Youth Science Fund supported by State Oceanic Administration and Shandong Province, and also funded by Project 863-818-06-02 and Project 921-2.

A Global Optimality Principle for Fully Coupled Mean-field Control Systems

This paper concerns a global optimality principle for fully coupled mean-field control systems.Both the first-order and the second-order variational equations are fully coupled mean-field linear FBSDEs. A new linear relation is introduced, with which we successfully decouple the fully coupled first-order variational equations. We give a new second-order expansion of Y^(ε) that can work well in mean-field framework. Based on this result, the stochastic maximum principle is proved. The comparison with the stochastic maximum principle for controlled mean-field stochastic differential equations is supplied.

A Maximum Principle for General Backward Stochastic Differential Equation

In this paper,the authors consider a stochastic control problem where the system is governed by a general backward stochastic differential equation.The control domain need not be convex,and the diffusion coefficient can contain a control variable.The authors obtain a stochastic maximum principle for the optimal control of this problem by virtue of the second-order duality method.

Identification of hydraulic conductivity distributions in density dependent flow fields of submarine groundwater discharge modeling using adjoint-state sensitivities

A mathematical optimal control method is developed to identify a hydraulic conductivity distribution in a density dependent flow field. Using a variational method, the adjoint partial differential equations are obtained for the density-dependent state equations used for the saline aquifer water flow. The adjoint equations are numerically solved in through a finite difference method. The developed method is applied to identify the hydraulic conductivity distribution through the numerical solution of an optimal control problem. To demonstrate the effectiveness of the optimal control method, three numerical experiments are conducted with artificial observation data. The results indicate that the developed method has the potential to accurately identify the hydraulic conductivity distribution in a saline water aquifer flow system.

Necessary and sufficient conditions for optimal control of stochastic systems associated with Lvy processes

The paper is concerned with a stochastic optimal control problem where the controlled systems are driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, Necessary and sufficient conditions for an optimal control of the control problem with the control domain being convex are proved by the classical method of convex variation, and the coefficients appearing in the systems are allowed to depend on the control variables, As an application, the linear quadratic stochastic optimal control problem is studied.

Necessary and Sufficient Optimality Conditions for Relaxed and Strict Control of Forward-Backward Doubly SDEs with Jumps Under Full and Partial Information

This paper derives necessary and sufficient conditions of optimality in the form of a stochastic maximum principle for relaxed and strict optimal control problems with jumps.These problems are governed by multi-dimensional forward-backward doubly stochastic differential equations(FBDSDEs)with Poisson jumps and has firstly relaxed controls,which are measure-valued processes,and secondly,as an application,the authors allow them to have strict controls.The FBDSDEs with jumps are fully-coupled,the forward and backward equations work in different Euclidean spaces in general,the backward equation is Markovian,and the control problems are considered under full information or partial information in terms ofσ-algebras that provide such information.The formulation of these equations as well as performance functionals are given in abstract forms to allow the possibility to cover most of the applications available in the literature.Moreover,coefficients of such equations are allowed to depend on control variables.

A Mean-Field Optimal Control for Fully Coupled Forward-Backward Stochastic Control Systems with Lévy Processes

This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes.In these systems,the coefficients contain not only the state processes but also their marginal distribution,and the cost function is of mean-field type as well.The necessary and sufficient conditions for such optimal problems are obtained.Furthermore,the applications to the linear quadratic stochastic optimization control problem are investigated.

Near-Relaxed Control Problem of Fully Coupled Forward-Backward Doubly System

In this paper,we are concerned with an optimal control problem where the system is driven by a fully coupled forward-backward doubly stochastic differential equation.We study the relaxed model for which an optimal solution exists.This is an extension of initial control problem,where admissible controls are measure valued processes.We establish necessary as well as sufficient optimality conditions to the relaxed one.

A Branching Random Walk Method for Many-Body Wigner Quantum Dynamics

A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and ana-lyzed in this paper.Using an auxiliary function,the truncated Wigner equation and its adjoint form are cast into integral formulations,which can be then reformulated into renewal-type equations with probabilistic interpretations.We prove that the first mo-ment of a branching random walk is the solution for the adjoint equation.With the help of the additional degree of freedom offered by the auxiliary function,we are able to produce a weighted-particle implementation of the branching random walk.In contrast to existing signed-particle implementations,this weighted-particle one shows a key ca-pacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors.Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings,and demonstrate the accuracy,the efficiency,and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.

Higher Order Harmonics and its Application for Flux Mapping

This paper proposes a flux mapping method directly using the higher order harmonics (HOH) of the neutronics equation of the nominal core. The bi-orthogonality and completeness of the HOH set are studied. and they are the theoretical basis for the flux mapping method. Using the bi-orthogonality of HOH and the strict formula for eigenvalue estimation. the process and formulas for HOH calculation called as the source iteration method with source correction are derived. The analysis can predict any order of harmonics for 2-or 3-dimensional geometries.Preliminary verification of the capability for flux mapping is also given. and other applications of HOH for reactor operation analysis and failure diagnosis are underway.