Reconstruction of boundary regimes in the inverse problem of thermal convection of a high-viscosity fluid

A problem of reconstruction of boundary regimes in a model for free convection of a high-viscosity fluid is considered. A variational method and a quasi-inversion method are suggested for solving the problem in question. The variational method is based on the reduction of the original inverse problem to some equivalent variational minimum problem for an appropriate objective functional and solving this problem by a gradient method. When realizing the gradient method for finding a minimizing element of the objective functional, an iterative process actually reducing the original problem to a series of direct well-posed problems is organized. For the quasi-inversion method, the original differential model is modified by means of introducing special additional differential terms of higher order with small parameters as coefficients. The new perturbed problem is well-posed; this allows one to solve this problem by standard methods. An appropriate choice of small parameters gives an opportunity to obtain acceptable qualitative and quantitative results in solving the inverse problem. A comparison of the methods suggested for solving the inverse problem is made with the use of model examples.     

Nonclassical pseudospectral method for the solution of brachistochrone problem

In this paper, nonclassical pseudospectral method is proposed for solving the classic brachistochrone problem. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Properties of nonclassical pseudospectral method are presented, these properties are then utilized to reduce the computation of brachistochrone problem to the solution of algebraic equations. Using this method, the solution to the brachistochrone problem is compared with those in the literature.     

A dynamical method of DAEs for the smallest eigenvalue problem

This article gives a new method based on the dynamical system of differential-algebraic equations for the smallest eigenvalue problem of a symmetric matrix. First, the smallest eigenvalue problem is converted into an equivalent constrained optimization problem. Second, from the Karush–Kuhn–Tucker conditions for this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Lastly, based on the implicit Euler method and an analogous trust-region technique, we obtain a prediction-correction method to compute a steady-state solution of this special system of differential-algebraic equations, and consequently obtain the smallest eigenvalue of the original problem. We also analyze the local superlinear property for this new method, and present the promising numerical results, in comparison with other methods.     

An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem

In this paper, we are concerned with finding the least solution to the tensor complementarity problem. When the involved tensor is strongly monotone, we present a way to estimate the nonzero elements of the solution in a successive manner. The procedure for identifying the nonzero elements of the solution gives rise to an iterative method of solving the tensor complementarity problem. In each iteration, we obtain an iterate by solving a lower-dimensional tensor equation. After finitely many iterations, the method terminates with a solution to the problem. Moreover, the sequence generated by the method is monotonically convergent to the least solution to the problem. We then extend this idea for general case and propose a sequential mathematical programming method for finding the least solution to the problem. Since the least solution to the tensor complementarity problem is the sparsest solution to the problem, the method can be regarded as an extension of a recent result by Luo et al. (Optim Lett 11:471–482, 2017). Our limited numerical results show that the method can be used to solve the tensor complementarity problem efficiently.     

A general Hungarian method for the algebraic transportation problem

In this paper the algebraic transportation problem is introduced which covers besides the Hitchcock and the time transportation problem several other types of transportation problems of practical relevance. To solve this algebraic transportation problem admissible transformations are considered and characterized. Thereupon a transformation algorithm is described which is a generalization of the Hungarian method for the classical transportation problem as well as of a threshold method for time transportation problems.     

A comparison of some inverse methods for estimating the initial condition of the heat equation

In this work we analyze two explicit methods for the solution of an inverse heat conduction problem and we confront them with the least-squares method, using for the solution of the associated direct problem a classical finite difference method and a method based on an integral formulation. Finally, the Tikhonov regularization connected to the least-squares criterion is examined. We show that the explicit approaches to this inverse heat conduction problem will present disastrous results unless some kind of regularization is used.     

On the general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations with singularities

A nonlinear self-adjoint eigenvalue problem for the general linear system of ordinary differential equations is examined on an unbounded interval. A method is proposed for the approximate reduction of this problem to the corresponding problem on a finite interval. Under the assumption that the initial data are monotone functions of the spectral parameter, a method is given for determining the number of eigenvalues lying on a prescribed interval of this parameter. No direct calculation of eigenvalues is required in this method.     

Central difference regularization method for inverse source problem on the Poisson equation

In this paper, we consider an inverse problem of determining an unknown source for the Poisson equation. Since this problem is mildly ill-posed, we apply a central difference regularization method to solve this problem. Furthermore, the convergence estimate is established under a priori choice of the regularization parameter. Some numerical results verify that the proposed method is stable and effective.     

Algebraic multilevel method AMG: Comparison with the method BICGSTAB + ILU and its use in the method CPR

The algebraic multilevel method AMG is compared with the method BICGSTAB+ILU on a model problem with a strong anisotropy. The method AMG was used as a part of the method CPR for solution of the filtration problems of a viscous compressible fluid flow in porous media. The algorithm CPR+AMG is compared with CPR+ILU on the base of this problem. The work of the CPR+AMG is analyzed by solving the problem for a model oil field.     

The Homotopy Perturbation Renormalization Group Method to Solve the WKB Problem with Turn Points

In this paper, we give the homotopy perturbation renormalization group method, this is a new method for turning point problem. Using this method, the independent variables are introduced by transformation without introducing new related variables and no matching is needed. The WKB approximation method problem can be solved.     

最短时限最少耗费的缺省指派问题及决策求解

文章指出了存在于军事决策与管理科学中最短时限最少耗费的缺省指派问题，并对其进行了深入的理论研究。论证了逼近最短时限的一个重要的定理及联系最短时限、最少耗费缺省指派最优解与经典指派问题最优解之间的相关性定理。据此首次建立了求解最短时限、最少耗费缺省指派的决策方法。这一方法可被广泛地应用于军事决策中进攻目标最优缺省选择与经济建设中工程最优缺省立项尽快见效等方面的一类新的科学决策。     

Solving the optimal control problem of the parabolic PDEs in exploitation of oil

In this paper, the optimal control problem is governed by weak coupled parabolic PDEs and involves pointwise state and control constraints. We use measure theory method for solving this problem. In order to use the weak solution of problem, first problem has been transformed into measure form. This problem is reduced to a linear programming problem. Then we obtain an optimal measure which is approximated by a finite combination of atomic measures. We find piecewise-constant optimal control functions which are an approximate control for the original optimal control problem.     

On the tour planning problem

Increasingly, tourists are planning trips by themselves using the vast amount of information available on the Web. However, they still expect and want trip plan advisory services. In this paper, we study the tour planning problem in which our goal is to design a tour trip with the most desirable sites, subject to various budget and time constraints. We first establish a framework for this problem, and then formulate it as a mixed integer linear programming problem. However, except when the size of the problem is small, say, with less than 20–30 sites, it is computationally infeasible to solve the mixed-integer linear programming problem. Therefore, we propose a heuristic method based on local search ideas. The method is efficient and provides good approximation solutions. Numerical results are provided to validate the method. We also apply our method to the team orienteering problem, a special case of the tour planning problem which has been considered in the literature, and compare our method with other existing methods. Our numerical results show that our method produces very good approximation solutions with relatively small computational efforts comparing with other existing methods.     

On constructing the solution to the exterior Dirichlet problem for the Helmholtz equation

In this paper a method is given for constructing the solution to the exterior Dirichlet problem for the Helmholtz equation in three dimensions. This method is modeled after the procedure of Colton and Kleinman (Proc. Roy. Soc. Edin. 86A(1980), 29–42) for solving the corresponding two-dimensional problem. The scattering problem is reformulated as an integral equation and it is shown that its solution can be represented as a convergent Neumann series for small values of the wave number. Comparisons are made between the present method and known results. Examples are given which illustrate the method.     

Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem

The null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound.     

The inverse solution of the atomic mixing equations by an operator-splitting method

The quantification problem of recovering the original material distribution from secondary ion mass spectrometry (SIMS) data is considered in this paper. It is an inverse problem, is ill-posed and hence it requires a special technique for its solution. The quantification problem is essentially an inverse diffusion or (classically) a backward heat conduction problem. In this paper an operator-splitting method (that is proposed in a previous paper by the first author for the solution of inverse diffusion problems) is developed for the solution of the problem of recovering the original structure from the SIMS data. A detailed development of the quantification method is given and it is applied to typical data to demonstrate its effectiveness.     

An algorithm for the capacitated vehicle routing problem with route balancing

In this paper, we present a multi-objective evolutionary algorithm for the capacitated vehicle routing problem with route balancing. The algorithm is based on a formerly developed multi-objective algorithm using an explicit collective memory method, namely the extended virtual loser (EVL). We adapted and improved the algorithm and the EVL method for this problem. We achieved good results with this simple technique. In case of this problem the quality of the results of the algorithm is similar to that of other evolutionary algorithms.     

Numerical solution method for the electric impedance tomography problem in the case of piecewise constant conductivity and several unknown boundaries

We study the electrical impedance tomography problem with piecewise constant electric conductivity coefficient, whose values are assumed to be known. The problem is to find the unknown boundaries of domains with distinct conductivities. The input information for the solution of this problem includes several pairs of Dirichlet and Neumann data on the known external boundary of the domain, i.e., several cases of specification of the potential and its normal derivative. We suggest a numerical solution method for this problem on the basis of the derivation of a nonlinear operator equation for the functions that define the unknown boundaries and an iterative solution method for this equation with the use of the Tikhonov regularization method. The results of numerical experiments are presented.     

Identification of source term for the Rayleigh‐Stokes problem with Gaussian random noise

In this paper, we study an inverse source problem for the Rayleigh‐Stokes problem for a generalized second‐grade fluid with a fractional derivative model. The problem is severely ill‐posed in the sense of Hadamard. To regularize the unstable solution, we apply a general filter method for constructing regularized solution, and the convergence rate of this method also has been investigated.