Numerical solution of forward and backward problem for 2-D heat conduction equation

For a two-dimensional heat conduction problem, we consider its initial boundary value problem and the related inverse problem of determining the initial temperature distribution from transient temperature measurements. The conditional stability for this inverse problem and the error analysis for the Tikhonov regularization are presented. An implicit inversion method, which is based on the regularization technique and the successive over-relaxation (SOR) iteration process, is established. Due to the explicit difference scheme for a direct heat problem developed in this paper, the inversion process is very efficient, while the application of SOR technique makes our inversion convergent rapidly. Numerical results illustrating our method are also given.     

The Initial Stages of Cooling of a Non-homogeneous Sphere

An asymptotic representation is obtained, valid for small valuesof the time variable, of the temperature distribution in a solidsphere cooling from a prescribed (non-symmetrical) initial distributionof temperature by radiative exchange with its surroundings.The thermal properties of the solid are assumed to be known(sufficiently smooth) functions of the radial co-ordinate r.The method of analysis applied to this problem can be used toinvestigate the initial stages of other time-dependent processesin non-homogeneous media.     

Distribution algorithm in finite difference method and its application to a 2D simulation of temperature inversion

In this article we present a numerical algorithm in a finite difference method coupled with the Schwartz distribution on a surface and its application to 2D simulation of a temperature inversion in meteorology. This phenomenon has been observed in Kyoto in Japan frequently and one of the reasons why the inversion occurs is the radiative cooling on the ground. In order to describe it in a mathematical model, we introduce transmission conditions on the ground and show its distribution formula. The characteristics of our numerical method are based on the direct method and the finite difference approximation of the Schwartz distribution defined on the ground. Through our discussion, a numerical experiment for the inversion of the temperature gradient is presented.     

A binary algorithm with low divergence for modular inversion on SIMD architectures

When performing modular arithmetic the most computationally expensive operation is the modular inversion of an integer. Its cost might be a problem for cryptanalytical applications (like the transformation from projective to affine coordinates within a Pollard rho algorithm implementation to solve the discrete logarithm problem on an elliptic curve) where performances constitute a key aspect. Good platforms for such operations are single-instruction multiple-data architectures, like graphic processing units (GPUs) because of their extremely competitive performance/price ratio. Unfortunately, when a single thread computes a single inversion, the whole computation on GPUs can be significantly slowed down in the presence of divergent threads. In this paper we describe a new algorithm to compute modular inversion on GPUs based on Stein’s Binary GCD. By exploiting the De Bruijn sequences and the Montgomery arithmetic, our version of Stein’s algorithm better fits GPUs, since it reduces the divergence among threads of the original algorithm. The paper includes a brief report on tests of our algorithm in six prime fields with characteristics of size ranging from 109 to 359 bits and in the two prime fields associated to the Mersenne primes \(2^{521} - 1\) and \(2^{607} - 1\).     

Inversion of electroencephalography data for a 2‐D current distribution

It is known from the fundamental work of Albanese and Monk that, the recovery of the support of a three dimensional current, within a conducting medium, from measurements of the generated exterior electric potential, is not possible. However, it is possible to recover the support of any other current, which is supported on a set of dimension lower than three. Nevertheless, no algorithm for such an inversion is known. Here, we propose such an algorithm for a two dimensional current distribution, and in particular, we apply this algorithm to the inverse problem of electroencephalography in the case where the neuronal current is restricted to a small disk of arbitrary location and orientation within the brain. The solution of this inverse problem is reduced to the solution of a nonlinear algebraic system, and numerical tests show that the there exists a unique real solution to this system.Copyright © 2014 John Wiley & Sons, Ltd.     

基于部分基变量的LP问题矩阵算法

基于部分基变量提出了LP问题的矩阵算法. 该算法以最优基矩阵的一个充分必要条件为基础,首先将一个初始矩阵转化为右端项和检验数均满足要求的矩阵,再转为检验数满足要求的基矩阵,最后转化为最优基矩阵.该算法具有使用范围广、计算规模小、计算过程简化、计算机易于实现的优势.矩阵算法的核心运算是求逆矩阵的运算,提出了矩阵算法的求逆问题,讨论并给出了求逆快速算法,该算法充分利用了矩阵算法迭代过程中提供的原来的逆矩阵的信息经过简单的变换得到新的逆矩阵,该算法比直接求逆法计算效率更高.     

两个矩阵问题的并行算法

本文讨论在阵列机上两个矩阵问题的并行算法.一个是用高斯-约当法求逆矩阵的并行实现;另一个是确定矩阵特征值的个数的并行送代法.这些算法已在IBM PC/XT微型机上模拟实现.     

Construction of Invertible Input-Output Mappings and Parameter Identification

The problem of continuation of an input-output mapping to a right invertible mapping is solved. The proposed solution is based on transforming the system to a normal form and solving the problem for such systems. The well-known Singh inversion algorithm is modified to calculate the normal forms. It is proved that each step of the modified algorithm can be realized and the result of the algorithm application is a normal form. A new approach to the parameter identification problem based on the inversion of the input-output mapping is proposed to illustrate the application of the results.     

MM Algorithms for Some Discrete Multivariate Distributions

The MM (minorization–maximization) principle is a versatile tool for constructing optimization algorithms. Every EM algorithm is an MM algorithm but not vice versa. This article derives MM algorithms for maximum likelihood estimation with discrete multivariate distributions such as the Dirichlet-multinomial and Connor–Mosimann distributions, the Neerchal–Morel distribution, the negative-multinomial distribution, certain distributions on partitions, and zero-truncated and zero-inflated distributions. These MM algorithms increase the likelihood at each iteration and reliably converge to the maximum from well-chosen initial values. Because they involve no matrix inversion, the algorithms are especially pertinent to high-dimensional problems. To illustrate the performance of the MM algorithms, we compare them to Newton’s method on data used to classify handwritten digits.     

Numerical solution of positive control problem via linear programming

The reachability problem for linear time-invariant discrete-time control systems with sign-restricted input is considered. The time-optimal control is constructed by an iterative procedure. Each step of the iteration is defined as a linear programming problem. This problem is solved by the simplex algorithm. The initial feasible solution for the simplex algorithm is provided by the preceding step of the iteration. The inversion of the basis matrix is reduced to a bordering procedure. The structural stability of the solution is investigated.     

Managing pharmaceuticals delivery service using a hybrid particle swarm intelligence approach

Medicines or drugs have unique characteristics of short life cycle, small size, light weight, restrictive distribution time and the need of temperature and humidity control (selected items only). Thus, logistics companies often use different types of vehicles with different carrying capacities, and considering fixed and variable costs in service delivery, which make the vehicle assignment and route optimization more complicated. In this study, we formulate the problem to a multi-type vehicle assignment and mixed integer programming route optimization model with fixed fleet size under the constraints of distribution time and carrying capacity. Given non-deterministic polynomial hard and optimal algorithm can only be used to solve small-size problem, a hybrid particle swarm intelligence (PSI) heuristic approach, which adopts the crossover and mutation operators from genetic algorithm and 2-opt local search strategy, is proposed to solve the problem. We also adapt a principle based on cost network and Dijkstra’s algorithm for vehicle scheduling to balance the distribution time limit and the high loading rate. We verify the relative performance of the proposed method against several known optimal or heuristic solutions using a standard data set for heterogeneous fleet vehicle routing problem. Additionally, we compare the relative performance of our proposed Hybrid PSI algorithm with two intelligent-based algorithms, Hybrid Population Heuristic algorithm and Improved Genetic Algorithm, using a real-world data set to illustrate the practical and validity of the model and algorithm.

     

Are linear algorithms always good for linear problems?

We exhibit linear problems for which every linear algorithm has infinite error, and show a (mildly) nonlinear algorithm with finite error. The error of this nonlinear algorithm can be arbitrarily small if appropriate information is used. We illustrate these examples by the inversion of a finite Laplace transform, a problem arising in remote sensing.     

Simultaneous identification of diffusion coefficient,spacewise dependent source and initial value for one‐dimensional heat equation

This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one‐dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross‐validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

The balanced linear programming problem

The Balanced Linear Programming Problem (BLPP) arises in situations which require equitable distribution of a scarce resource. The BLPP can be transformed to the standard form of the linear programming problem by introducing 2∥N∥ + 2 additional variables and 2∥N∥ additional constraints. This transformation is not desirable from the computational point of view for larger values of ∥N∥ as it increases the problem size substantially. It is also undesirable from a theoretical perspective as it might affect the special structure of the constraint matrix. In this paper, we develop an algorithm for the BLPP which does not require problem enlargement. The algorithm is based on the relationship between the BLPP and the minimax linear programming problem, and solving the latter problem parametrically. Our algorithm, in essence, performs steps that are similar to those performed in the parametric simplex method with parametric right hand side. We then adapt our algorithm for the network flow problem and this specialized algorithm can be applied on the network directly without maintaining the simplex tableau.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

Meta-control of an interacting-particle algorithm for global optimization

A common issue for stochastic global optimization algorithms is how to set the parameters of the sampling distribution (e.g. temperature, mutation/cross-over rates, selection rate, etc.) so that the samplings converge to the optimum effectively and efficiently. We consider an interacting-particle algorithm and develop a meta-control methodology which analytically guides the inverse temperature parameter of the algorithm to achieve desired performance characteristics (e.g. quality of the final outcome, algorithm running time, etc.). The main aspect of our meta-control methodology is to formulate an optimal control problem where the fractional change in the inverse temperature parameter is the control variable. The objectives of the optimal control problem are set according to the desired behavior of the interacting-particle algorithm. The control problem considers particles’ average behavior, rather than treating the behavior of individual particles. The solution to the control problem provides feedback on the inverse temperature parameter of the algorithm.     

First exit time probability for multidimensional diffusions: A PDE-based approach

First exit time distributions for multidimensional processes are key quantities in many areas of risk management and option pricing. The aim of this paper is to provide a flexible, fast and accurate algorithm for computing the probability of the first exit time from a bounded domain for multidimensional diffusions. First, we show that the probability distribution of this stopping time is the unique (weak) solution of a parabolic initial and boundary value problem. Then, we describe the algorithm which is based on a combination of the sparse tensor product finite element spaces and an hp-discontinuous Galerkin method. We illustrate our approach with several examples. We also compare the numerical results to classical Monte Carlo methods.     

Determination of nonstationary temperature fields and stresses in piecewise homogeneous circular plates on the basis of a numerical-analytic Laplace inversion formula

We investigate nonstationary temperature fields and stresses generated by them in piecewise homogeneous annular plates. An algorithm for the solution of the problem is based on the direct calculation of the Laplace transform and a modified Prudnikov formula for its inversion.     

A generalization of Cooke's integral inversion formula with application to remote-sensing theory

The remote sensing of environmental particulate pollutants, particularly their size distribution, frequently leads to the solution of first-kind Fredholm integral equations. The corresponding physical kernel tends to smooth the behavior of the required function for all values of the dependent variable. Thus, the problem is ill posed and needs regularization by the introduction of constraints on the solution (closure condition). However, under physically realistic conditions, the original problem can be transformed so that it presents a unique and stable solution. One such condition is the so-called anomalous-diffraction approximation, for which we provide two alternate inversion formulae. We derive a new inversion formula (see our theorem) which generalizes that of Cooke and which also provides, as a special case, one of Titchmarsh's formulae. We propose a unifying viewpoint for a number of known integral inversion formulae, including those of Fox (his first theorem), Hardy, Hankel, Titchmarsh, Cooke, and our own, along with the mutual interrelationships that exist between them (Fig. 1 and Table 1). One solution to the particulate sounding problem is then obtained from a direct application of our formula [Eq. (25)]. An alternate solution is likewise obtained by applying Titchmarsh's formula (II) [Eq. (27)]. Both solutions can be independently recovered from Fox's first theorem, although under somewhat more restrictive conditions. They are shown to be identical, and to provide the unique solution to the remote sensing problem considered.     

On an algorithm for tracking the motion of the reference system with aftereffect when only part of the coordinates is measured

We consider the problem of tracking a given trajectory by a system described by an equation with aftereffect. We suggest an algorithm, stable to information noise and numerical errors, for solving this problem in the case of incomplete information on the phase trajectory (measurement of part of the coordinates). The algorithm is based on the dynamic inversion and guaranteed control method.