Japan Journal of Industrial and Applied Mathematics最新文献

Abstract

In this paper, we study the structure of the least square solutions to overdetermined systems with no solution. In the main theorem, we prove that if an overdetermined system with no solution is deformed into a system of linear equations by the semi-equivalent deformations defined in this paper, then an approximate solution to the original overdetermined system with no solution can be given as the unique least square solution to the deformed system of linear equations. We also introduce some applications of our main theorem to practical inverse problems.

In a previous article, we have introduced a model to study life expectancy based on a sequence of Weibull distributions. Each of these distributions characterizes the living expectancy within a certain age interval. In the second step, we estimate the parameters for this model by combining the moment estimations and censoring methods. We have named this combination “the local parametric estimation method”. In this article, we present a different model that requires less number of random variable characteristics which help alleviate the estimation procedure. In fact, in comparison with the previous model, we are able to obtain an explicit formula for the variance of life expectancy. This is particularly useful in obtaining a normal approximation to life expectancy. Extensive computations with real-world datasets show that the local parametric method provides less biased estimation with lower variance in comparison to the Chiang method. This fact allows one to use statistical tests to detect life expectancy estimation differences as shown in the data where the Chiang method does not perform well. Additionally, the new life expectancy estimation method is also useful in the assessment of the health inequality in small area/small population settings by conducting statistical tests.

Abstract

Recently Oishi published a paper allowing lower bounds for the minimum singular value of coefficient matrices of linearized Galerkin equations, which in turn arise in the computation of periodic solutions of nonlinear delay differential equations with some smooth nonlinearity. The coefficient matrix of linearized Galerkin equations may be large, so the computation of a valid lower bound of the smallest singular value may be costly. Oishi’s method is based on the inverse of a small upper left principal submatrix, and subsequent computations use a Schur complement with small computational cost. In this note some assumptions are removed and the bounds improved. Furthermore a technique is derived to reduce the total computationally cost significantly allowing to treat infinite dimensional matrices.

The conjugate gradient (CG) method is the most basic iterative solver for large sparse symmetric positive definite linear systems. In finite precision arithmetic, the residual and error norms of the CG method often stagnate owing to rounding errors. The groupwise update is a strategy to reduce the residual gap (the difference between the recursively updated and true residuals) and improve the attainable accuracy of approximations. However, when there is a severe loss of information in updating approximations, it is difficult to sufficiently reduce the true residual and error norms. To overcome this problem, we propose a mixed-precision algorithm of the CG method using the groupwise update strategy. In particular, we perform the underlying CG iterations with the standard double-precision arithmetic and compute the groupwise update with high-precision arithmetic. This approach prevents a loss of information and efficiently avoids stagnation. Numerical experiments using double-double arithmetic demonstrate that the proposed algorithm significantly improves the accuracy of the approximate solutions with a small overhead of computation time. The presented approach can be used in other related solvers as well.

This paper develops a continuous approximation model for analyzing the effect of road pricing on the spatial distribution of traffic flow. The traffic flow density, which describes traffic flow as a function of position, is derived for a rectangular city with a grid network. The analytical expression for the traffic flow density demonstrates how the toll level, the size and shape of the toll area, and the shape of the city affect the spatial distribution of traffic flow. As the size of the toll area increases, reducing the traffic flow density at the city center becomes difficult. As the aspect ratio of the toll area increases, the traffic flow density at the city center increases. The shape of the city has less impact on the traffic flow density than the shape of the toll area.

This paper presents a computer-assisted proof of the existence and unimodality of steady-state solutions for the Proudman–Johnson equation which is representative of two-dimensional fluid flow. The proposed approach is based on an infinite-dimensional fixed-point theorem with interval arithmetic, and is another proof by Miyaji and Okamoto (Jpn J Ind Appl Math 36:287–298, 2019). Verification results show the validity of both proofs.

The fixed point iteration method is an effective method for solving absolute value equation via equivalent two-by-two block form. To further improve the computational efficiency of the fixed point iteration method, by using the preconditioned shift-splitting strategy, we propose an inexact fixed point iteration method for solving absolute value equation in this paper. We obtain some convergence conditions for the proposed method. The effectiveness of the proposed method are shown by three examples.

Optical fibers are among the most widely used tools in information communication today, and they have had an active development trajectory. Film interconnection is among the methods applied to optical fibers. However, due to its characteristics, the interconnection design must be done well so that each fiber satisfies several design requirements. In this study, we developed a mathematical model for automatic wiring design using film optical fiber interconnections. To design optical fiber routing from the top to the bottom of the film, we propose an exact solution method using a mixed-integer programming problem and a heuristic method based on the exact solution method. In this paper, we compare the results of our methods with rule-based methods and confirm that our methods are superior. For the experiments, we created sample data based on a previous joint research project with Sumitomo Electric Industries, Ltd. and used it. Additionally, we investigated conditions for wiring design, and this paper discusses conditions for obtaining more efficient wiring methods.

We discuss how to compute the Brauer group of the product of two elliptic curves over a finite field. Specifically, we apply the Artin–Tate formula for abelian surfaces to give a simple formula for computing the order of the Brauer group of the product of two elliptic curves. Our formula enables to compute the order of the Brauer group using traces of Frobenius maps on elliptic curves and the discriminant of the group of homomorphisms between two elliptic curves. In addition, for the product of non-isogenous elliptic curves, we give an algorithm for computing torsion subgroups of a certain Galois cohomology that can be embedded as subgroups of the Brauer group.

In this paper, a new stochastic volatility model for pricing European call option is proposed, which introduces the regime-switching mechanism controlled by Markov chain in stochastic volatility and stochastic long-term mean. The advantage of adopting this new model is that it can deduce a colsed-form solution of European call option based on characteristic function of the underlying price, which can save a lot of time and effort when applied in the real market. The effect of introducing regime-switching into European call option pricing model is investigated through numerical experiments, and the results show that the regime-switching has a significant effect on the model. The empirical results further demonstrate that our model has some advantages over the other two models.