Optimal control of damped Klein-Gordon equations with state constraints

In this paper, we study the maximum principles for optimal control problems governed by the damped Klein-Gordon equations with state constraints. And we prove the existence of the optimal parameter and deduce the necessary conditions on the optimal parameter.     

Infinite Dimensional Parametric Optimal Control Problems

In this paper we study parametric optimal control problems monitored by nonlinear evolution equations. The parameter appears in all the data, including the nonlinear operator. First we show that for every value of the parameter, the optimal control problem has a solution. Then we study how these solutions as well as the value of the problem respond to changes in the parameter. Finally, we work out in detail two examples of nonlinear parabolic optimal control systems.     

再论机动车辆保险的精算模型及其应用

通过对非参数混合泊松模型的分析 ,我们发现用此类模型建立无赔款优待系统是不合适的 .在文中我们使用 Hofmann分布为我国一家保险公司的索赠数据进行拟合 ,效果令人满意 ,然后导出最优无赔款优待系统和零效用原理下的无赔款优待系统     

The selection of the optimal parameter in the modulus-based matrix splitting algorithm for linear complementarity problems

The modulus-based matrix splitting (MMS) algorithm is effective to solve linear complementarity problems (Bai in Numer Linear Algebra Appl 17: 917–933, 2010). This algorithm is parameter dependent, and previous studies mainly focus on giving the convergence interval of the iteration parameter. Yet the specific selection approach of the optimal parameter has not been systematically studied due to the nonlinearity of the algorithm. In this work, we first propose a novel and simple strategy for obtaining the optimal parameter of the MMS algorithm by merely solving two quadratic equations in each iteration. Further, we figure out the interval of optimal parameter which is iteration independent and give a practical choice of optimal parameter to avoid iteration-based computations. Compared with the experimental optimal parameter, the numerical results from three problems, including the Signorini problem of the Laplacian, show the feasibility, effectiveness and efficiency of the proposed strategy.

     

Renormalization group method and Julia sets

The renormalization group (RG) method has been used successfully in treating a variety of phase change and critical-point problems (Wilson KG, Kogut J. Phys Rev C 1974;12:75; Wilson KG. Rev Mod Phys 1975;773; Wilson KG. Phys Rev B 1971;3174). A relatively simple system is considered at the smallest scale; the problem is then renormalized in order to utilize the same system at next larger scale. The process is repeated at larger and larger scales. In the following we consider a model for the flow of a fluid through a porous-medium. The RG transformations for the flow of a fluid through a porous-medium in two and three dimensions are derived and generalized to the complex plane, and the types of the corresponding Julia sets are found and generated. Also, the RG transformation for Ising model on a square lattice is derived and the corresponding Julia set is found.     

On estimation of the optimal parameter of the modulus-based matrix splitting algorithm for linear complementarity problems on second-order cones

There are many studies on the well-known modulus-based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second-order cones (SOCLCPs). It also works on single second-order cone and the non-negative orthant. On this basis, we further propose an iteration-independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state-of-the-art solvers SCS and SuperSCS that solve the equivalent SOC programming.     

应用指数函数展开法求解非线性发展方程

利用指数函数展开法,研究BBM方程与KG方程,在一个特定的变换下,借助Maple软件的符号运算功能,获得BBM方程与KG方程指数函数型新的孤立波解与周期解.这种方法用于求解非线性发展方程是简单而有效的.     

Regularization Using QR Factorization and the Estimation of the Optimal Parameter

In this paper we propose a direct regularization method using QR factorization for solving linear discrete ill-posed problems. The decomposition of the coefficient matrix requires less computational cost than the singular value decomposition which is usually used for Tikhonov regularization. This method requires a parameter which is similar to the regularization parameter of Tikhonov's method. In order to estimate the optimal parameter, we apply three well-known parameter choice methods for Tikhonov regularization.This revised version was published online in October 2005 with corrections to the Cover Date.     

Convergence and Error Bound for Perturbation of Linear Programs

In various penalty/smoothing approaches to solving a linear program, one regularizes the problem by adding to the linear cost function a separable nonlinear function multiplied by a small positive parameter. Popular choices of this nonlinear function include the quadratic function, the logarithm function, and the x ln(x)-entropy function. Furthermore, the solutions generated by such approaches may satisfy the linear constraints only inexactly and thus are optimal solutions of the regularized problem with a perturbed right-hand side. We give a general condition for such an optimal solution to converge to an optimal solution of the original problem as the perturbation parameter tends to zero. In the case where the nonlinear function is strictly convex, we further derive a local (error) bound on the distance from such an optimal solution to the limiting optimal solution of the original problem, expressed in terms of the perturbation parameter.     

Active Constraint Set Invariancy Sensitivity Analysis in Linear Optimization

Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant.For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: “what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?”. The differences of the results between the linear and convex quadratic optimization problems are highlighted too.     

A Vanka-based parameter-robust multigrid relaxation for the Stokes–Darcy Brinkman problems

We consider a block-structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element-based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka-based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two-grid and V(1,1)-cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter.     

Stability of optimal filter higher-order derivatives

In many scenarios, a state-space model depends on a parameter which needs to be inferred from data. Using stochastic gradient search and the optimal filter first-order derivatives, the parameter can be estimated online. To analyze the asymptotic behavior of such methods, it is necessary to establish results on the existence and stability of the optimal filter higher-order derivatives. These properties are studied here. Under regularity conditions, we show that the optimal filter higher-order derivatives exist and forget initial conditions exponentially fast. We also show that the same derivatives are geometrically ergodic.     

Restricted Maximum Likelihood Estimates in Finite Mixture Models

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Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

We analyze rigourously error estimates and compare numerically temporal/spatial resolution of various numerical methods for solving the Klein–Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter 0 < e << 1{0 < {\varepsilon}\ll 1} which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time, i.e. there are propagating waves with wavelength of O(e²){O({\varepsilon}^2)} and O(1) in time and space, respectively. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter e{{\varepsilon}}. Based on the error bounds, in order to compute ‘correct’ solutions when 0 < e << 1{0 < {\varepsilon}\ll1}, the four FDTD methods share the same e{{\varepsilon}}-scalability: t = O(e³){\tau=O({\varepsilon}^3)}. Then we propose new numerical methods by using either Fourier pseudospectral or finite difference approximation for spatial derivatives combined with the Gautschi-type exponential integrator for temporal derivatives to discretize the KG equation. The new methods are unconditionally stable and their e{{\varepsilon}} -scalability is improved to τ = O(1) and t = O(e²){\tau=O({\varepsilon}^2)} for linear and nonlinear KG equations, respectively, when 0 < e << 1{0 < {\varepsilon}\ll1}. Numerical results are reported to support our error estimates.     

On a Class of Optimal Control Problems with State Jumps

In this paper, we consider a class of optimal control problems in which the dynamical system involves a finite number of switching times together with a state jump at each of these switching times. The locations of these switching times and a parameter vector representing the state jumps are taken as decision variables. We show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems. Gradient formulas for the cost functional and the constraint functional are derived. On this basis, a computational algorithm is proposed. For illustration, a numerical example is included.     

基于BP神经网络的动态稳健参数设计

传统的动态稳健参数设计方法(田口方法)虽然在工业生产实践中展现了极大的方便,但是其本身也存在较大的改进空间.当调节变量不存在时,传统的田口方法难以实现;此外,田口方法只能根据所选取的参数水平得到最优参数组合,而这种所谓的最优结果有时并不符合实际的需要.首先构建BP神经网络模型,利用训练后的BP神经网络获得参数设计中质量特性、噪声因子以及各参数间的动态关系;然后,利用超拉丁方抽样,计算信号与特性参数间的斜率,并由此将动态稳健参数设计的寻优问题转化为相应的非线性规划问题;最后,利用次序二次规划(SQP)算法解决并优化动态稳健参数的设计。此外,我们选取了一个简单的数据案例对本文提出的方法的有效性进行了说明.     

Portfolio selection problem with liquidity constraints under non-extensive statistical mechanics

In this study, we consider the optimal portfolio selection problem with liquidity limits. A portfolio selection model is proposed in which the risky asset price is driven by the process based on non-extensive statistical mechanics instead of the classic Wiener process. Using dynamic programming and Lagrange multiplier methods, we obtain the optimal policy and value function. Moreover, the numerical results indicate that this model is considerably different from the model based on the classic Wiener process, the optimal strategy is affected by the non-extensive parameter q, the increase in the investment in the risky asset is faster at a larger parameter q and the increase in wealth is similar.     

An Optimal Method for Adjusting the Centering Parameter in the Wide-Neighborhood Primal-Dual Interior-Point Algorithm for Linear Programming

In this paper we present a dynamic optimal method for adjusting the centering parameter in the wide-neighborhood primal-dual interior-point algorithms for linear programming, while the centering pararheter is generally a constant in the classical wideneighborhood primal-dual interior-point algorithms. The computational results show that the new method is more efficient.     

Properties and numerical testing of a parallel global optimization algorithm

In the present paper, we have considered three methods with which to control the error in the homotopy analysis of elliptic differential equations and related boundary value problems, namely, control of residual errors, minimization of error functionals, and optimal homotopy selection through appropriate choice of auxiliary function H(x). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman?CForchheimer equation. Finally, we carefully selected the auxiliary function H(x) in order to obtain an optimal homotopy solution for Liouville??s equation.     

Robust error estimates for the finite element approximation of elliptic optimal control problems

In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang-bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results.