60th Anniversary of the Seminal Paper on Interval Analysis and Computations by T. Sunaga

Sunaga considered all computational procedures, which had been traditionally defined on real numbers, as being too ideal and proposed to replace them by the procedures on real intervals in order to make everything "more realistic". Sunaga studied many different kinds of numerical procedures including the Taylor-series interval solution of the initial-value problem of ordinary differential equations.The purpose of the present note is to mark the 60-th anniversary of the publication of the seminal work by the Japan mathematician Teruo Sunaga. The paper summarizes the results of his Master Thesis \cite{Sunaga1956}. Sunaga's work sets the foundation of the contemporary interval analysis and reliable computing. This is an interdisciplinary field, combining abstract mathematical theories and practical applications related to computer science, numerical analysis and mathematical modeling in the natural, engineering and social-economic sciences.

Application of BSDE in Standard Inventory Financing Loan

This paper examines the issue of loans obtained by the small and medium-sized enterprises (SMEs) from banks through the mortgage inventory of goods. And the loan-to-value (LTV) ratio which affects the loan business is a very critical factor. In this paper, we provide a general framework to determine a bank’s optimal loan-to-value (LTV) ratio when we consider the collateral value in the financial market with Knightian uncertainty. We assume that the short-term prices of the collateral follow a geometric Brownian motion. We use a set of equivalent martingale measures to build the models about a bank’s maximum and minimum levels of risk tolerance in an environment with Knightian uncertainty. The models about the LTV ratios are established with the bank’s maximum and minimum risk preferences. Applying backward stochastic differential equations (BSDEs), we get the explicit solutions of the models. Applying the explicit solutions, we can obtain an interval solution for the optimal LTV ratio. Our numerical analysis shows that the LTV ratio in the Knightian uncertainty-neutral environment belongs to the interval solutions derived from the models.

On multi-objective linear programming problems with inexact rough interval-fuzzy coefficients

This paper deals with a multi-objective linear programming problem with an inexact rough interval fuzzy coefficients IRFMOLP. This problem is considered by incorporating an inexact rough interval fuzzy number in both the objective function and constrains. The concept of "Rough interval" is introduced in the modeling framework to represent dualuncertain parameters. A suggested solution procedure is given to obtain rough interval solution for IRFLP(w) problem. Finally,two numerical example is given to clarify the obtained results in this paper.

LARGE INTERVAL SOLUTION OF THE EMDEN–FOWLER EQUATION USING A MODIFIED ADOMIAN DECOMPOSITION METHOD WITH AN INTEGRATING FACTOR

We propose a new Adomian decomposition method (ADM) using an integrating factor for the Emden–Fowler equation. With this method, we are able to solve certain Emden–Fowler equations for which the traditional ADM fails. Numerical results obtained from testing our linear and nonlinear models are far more reliable and efficient than those from existing methods. We also present a complete error analysis and a convergence criterion for this method. One drawback of the traditional ADM is that the interval of convergence of the Adomian truncated series is very small. Some techniques, such as Pade approximants, can enlarge this interval, but they are too complicated. Here, we use a continuation technique to extend our method to a larger interval.

Influence of the Parameterization in the Interval Solution of Elastic Beams

We are going to analyze the interval solution of an elastic beam under uncertain boundary conditions. Boundary conditions are defined as rotational springs presenting interval stiffness. Developments occur according to the interval analysis theory, which is affected, at the same time, by the overestimation of interval limits (also known as overbounding, because of the propagation of the uncertainty in the model). We suggest a method which aims to reduce such an overestimation in the uncertain solution. This method consists in a reparameterization of the closed form Euler-Bernoulli solution and set intersection.

A Method for Static Interval Analysis of Uncertain Structures with Interval Parameters

By representing the uncertain parameters as interval numbers, the static linear interval equations about the structural system were obtained in this paper by means of the finite element method. These equations are linear interval equations, for which some solution methods were discussed and a step-dividing method was presented. In this method, the independent uncertain parameters were given the discrete values within each interval, and the linear interval equations were changed into the corresponding certain ones. And then the boundaries of every interval solution components are determined by searching for the maximum and minimum values of the equation solutions. Some mathematical examples were used to examine the correctness and efficiency of the algorithm and which was applied to static interval analysis of engineering problems. Compared with other methods, the calculation results show that the algorithm of this paper is efficient and accurate.