Error Analysis for Designed Test and Numerical Integral by Using UED in Material Research

: In this article, the error analysis for designed test and definite integral in employing uniform experimental design is analogically developed on basis of midpoint rule in rectangle method for assessing definite integral through conducting discretized sampling approximately. It concluded that the discrete sampling - point by means of good lattice point in the evaluations of definite integral and maximum value of a function is promised with higher accuracy, and the predicted entire error of this uniform sampling point method for above problems decreases with the number of sampling points significantly.


INTRODUCTION
Experiment design is an essential topic to scientific and industrial developments.How do we arrange and design experiments to get effective results with less number of trials?Some approaches are proposed to answer this encountered question frequently, such as Response Surface Methodology, Orthogonal Experimental Design, and Uniform Experimental Design, etc. Better design could lead to results that are more effective.
In 1978, due to the need for missile designs, a special demand for total trial number being no greater than 50 with 18 factor levels of a five-factor experiment was faced to Prof. K. T. Fang [1], He and Prof. Y. Wang thus proposed a novel solution by employing number-theoretical methods [2].They created a brand new experimental design methodology to conduct the design of the problem in that time, which was named as Uniform Experimental Design (UED).UED is attributed to number -theoretical method or quasi-Monte Carlo method.This novel methodology was employed to the design of missiles successfully, and got series of meaningful achievements in China due to its wide applications [1,2].
Early in 1950s, Ulam and Von Neumann developed Monte Carlo method, which is a statistical stimulation.The main idea of their method is to convert an analysis problem into a probability problem and then employ a statistical simulation to deal with the problem to get a same solution.This seems an effective solution for some difficult analysis problems, including the approximate estimation of complicated definite integrals.The general idea of Monte Carlo method is the need of a set of stochastic numbers to enable to perform this statistical simulation.The precision of this method strongly depends on the independence and uniformity of stochastic numbers.
Almost in the same period of 1950s, deterministic methods were also proposed to deal with some difficult analysis problems by some mathematicians, which aimed to give solution by using uniformly distributed points in space instead of random numbers like that in Monte Carlo method [3], such as, Korobov put forward the concept of a point set, which is uniformly distributed.Since then Hua et al developed the good lattice point (GLP) method in 1960s for evaluation of definite integral approximately, which is with low-discrepancy on basis of number theory [2,3].Therefore, this kind of method is called a number-theoretical method or quasi-Monte Carlo method naturally thereafter.UED can be seen as one of the successful applications of the numbertheoretical method [1,2].This methodology was subsequently used in evaluations for approximation of multiple-integral successfully.
There are many beneficial features of GLP [4][5][6], besides the uniformity of distribution of sampling-point over the specific domain and good space-filling characteristic.
Currently, the UED finds its wide applications over the world, it spreads in designs of Chinese medicine, chemical reaction and missile, as well as Ford Motor Co. Ltd for its design of standard exercises and automotive as computer experiments for providing a support of the preliminary design of production [7].

Essential Characteristics of UED
The essential characteristics of UED involves [1,2,7]: A) Homogenization The distribution of specimen point is homogeneously scattered in the variable space; therefore it gains a surname of "space filling design" occasionally [1,2,7].
A number of "Uniform Design Table " (UDT) was specifically developed by Fang to arrange the distribution of the specimen point for UED [8], which is fully deterministic.B) Entire Mean Model UED intends to get an outcome consequence, which is with minimum deviation of the entire averaged value from the actual total averaged value through uniform distribution of specimen points.

C) Robustness
UED is expected to be used in many cases with robustness regardless of variation of model.

Basic Principle of UED
The fundamental principle of UED is as followings:

1) Entire Mean Model
The fundamental hypothesis is the existence of a deterministic relationship of the response g vs. the independent input variables x 1 , x 2 , x 3 , ..., x s , the formula of the response can be expressed as, ( , , , ..., ), { , , , ..., } .
The further hypothesis is that the entire averaged value of the response g on C s = [0, 1] s is, Moreover, if one takes m sampling points q 1 , q 2 , q 3 , …, q m on C s to conduct an average value of g, then the averaged value of g over these m specimen points is, In Eq. ( 3), D m = {q 1 , q 2 , q 3 , …, q m } represents a design of such m specien points.
Fang et al indicated that the deviation ( ) ( ) − of the specimen point set on C s and D m will be quite small provided the specimen points q 1 , q 2 , q 3 , …, q m are uniformly distributed in the domain C s .

2) Uniform Design Table
In order to provide an appropriate application for UED, a number of UDT as well as the "Utility Table " were conducted [8], with which the location of specimen points can be specifically determined with convenience.

3) Regression
In general, under condition of discretization with sampling points, an approximate expression for response r' = R'(x1, x 2 , x 3 , ..., x m ) vs. the independent input variables can be regressed to reveal the resemble formation.

Aim of This Study
Until now, the estimation of the accuracy of applying each UDT to conduct actual problem is unclear though the discrepancy of each point set of UDT is provided by Fang in his book [5].
In this paper, the study of entire error of definite integral and maximum value by using sampling points from UDT is preliminarily conducted in the point of view of practical application.
Obviously, the homogeneously spreading of the specimen point in the above design for one independent variable case is the same as that of the Midpoint Rule in rectangle method of definite integral [9][10][11].
Following midpoint rule, the definite integral around position x 0 with a subsection δ is approximated by I = g(x 0 )•δ, which is with the local error of ε M =δ 3 ⋅g″(x 0 )/24 [9][10][11], where g″(x 0 ) indicates the second derivative at location x 0 , in general ε M is negligible as δ is sufficiently small.According to mean value theorem of integral, one could always find a proper position such that the value of the function g(ζ) meets the demand of However, above discussion indicates that the value of the integral approximates to I = g(x 0 )•δ with the small local error of ε M =δ 3 ⋅g″(x 0 )/24 [9][10][11], therefore, g(ζ) approximately equals to g(x 0 ), i.e., g(ζ) ≈ g(x 0 ).Furthermore, the local error of the function g(x) around x 0 within area of [x 0 -δ/2, x 0 + δ/2] is about g′(x 0 )⋅δ/2.
Moreover, considering the maximum value of the function g(x) within its range of [a, b], it supposes that if the function g(x) within its range [a, b] derives its maximum value g(x l ) at a discrete point x l = a + (2l -1)(b -a))/2n, l ∈ [1, 2, 3, …, n], then the realistic error of the actual maximum of this function g max (x) from the nominal maximum g(x l ) at a discrete point x l is E actual = g max (x) -g(x l ), and it could be approximately estimated by, est.
1 As an example, lets' conduct the error analysis for e x within range of [0, 1.5] by using 11 discrete uniform sampling points [12,13].As to such problem, following the procedure of UED [1,2,8], the 11 sampling points are uniformly distributed within range of [0, 1.5], see Tab. 1.
The actual value of function e x is 4.4817 at x = 1.5, while the maximum value of discretized function e x in Tab. 1 is 4.1863.The actual error is E actual = 0.2954, while the estimated value by using Eq. ( 4) is E est.= 0.2668, which is close to E actual .

Error Analysis of Applying UED for Maximum Value
In general, as to a s-dimensional problem, it assumes that the discretization is conducted for a function p x  within its In Eq. ( 5), γ is the number of nearest neighbour of p x  .
Thus, the error of the maximum ( )  corresponding to the discretization can be estimated by using Eq. ( 5) in principle.

Error Analysis of Applying UED for Integral
As to midpoint rule [9][10][11], the definite integral can be approximated by a summation, i.e., [ ] where g″(a ( 1 2) ( ) tiny; or else the summarizing ε n remains for some monotone function like e x .However even in latter case good precision could be obtained with 11 specimen points which are homogeneously distributed [12,13].Take the integral ./ i .g D .
the actual total error is 0.0027, while the predicted value of total error is of 0.0025, which is not far from the actual error.
For higher dimensions, the entire error could be estimated analogically by, In Eq. ( 7), the term M(l) indicates max|g″(x)|in l th independent variable, Until now, the error estimation of applying UED for integral is estimated in principle.

APPLICATIONS 3.1 Error Analysis of Maximum Value of Functions in Their Domains by Using Discrete Uniform Sampling Points
Lets' study the function f1(x, y) = ln(x + 2y) in the domain of [1.4,2.0] × [1.0, 1.5] by using discrete uniform sampling points first.Assume that the actual maximum value of the function f 1 (x, y) within the domain is f 1max (x, y), and the maximum value of the function f 1 (x, y) due to the discretization is at a discrete point p x  and denoted by 1 ( ) Fig. 1 shows the variations of the actual error the number of sampling points n, together with the estimated error E est. .It can be seen that the tendency of the variation of the actual error E actual. is the same as the those of the estimated error E est , which decreases significantly with the increase of the number of sampling points n.Next, Lets' study the function f 2 (x, y) = 1 + 2x 2 + 2y 3 in the domain of [1.4,2.0] × [1.0, 1.5] by using discrete uniform sampling points.Fig. 2 shows the comparison of the variations of the actual error E actual and the estimated error E est. vs. the number of sampling points n for f 2 (x, y) .It can be seen again that the tendency of the variations of the actual error E actual and the estimated error E est. is the same, which The UDT U 17 (17 8 ) is tried to be used to conduct the definite integral, 17 sampling points are included [12,13].
The accurate data of this integration is 0.429560 [14].

Comparative Assessment of Discrete Specimen Points by
Means of GLP with Monte Carlo Simulation for Integral The accurate value of the definite integral of 6 0 ( Han and Ren once assessed this integral by employing Monte Carlo algorithm [15].Now, lets' try to re-study this integral by means of GLP with 11 discrete specimen points for comparison [15].The locations of the specimen points and the discrete values of the function f(x) are given in Tab. 3.  , which is exactly and luckily same as that of the accurate value of this integral.However, the use of Monte Carlo simulation results in a varying error, which changes even up to 1000 random specimen points [15].As was shown in [15], the Monte Carlo simulation gives an error of 0.1214 at 1000 specimen points, while the error of the simulated result reaches to 0.2908 with 100 specimen points [15]!This result reflects the merit of the assessment with discrete specimen points by means of GLP in evaluating definite integral and maximum value of a function once more.

CONCLUSION
Error analysis of designed test and definite integral by employing UED to conduct discrete specimen is analyzed in this paper, the estimation of error is performed with the aid of midpoint rule in rectangle method for predicting definite integral analogically.The study indicates the decreasing tendency of the entire error of specimen point in predictions of definite integral and maximum value of a function vs. the number of specimen points, and the proper number of specimen points can be determined by the promised requirement of accuracy conversely.

∫
as an example, which is with the precise value of

Figure 1
Figure 1 Comparison of the variations of the actual error E actual and the estimated error E est .vs. number of sampling points n for f 1 (x, y) = ln(x + 2y) [1,2,8]h n.According to number theory, it assumes that there is a unique prime indicate different primes.Furthermore, the number of independent factors at most is t = ϕ(n)/2 + 1, i.e., s ≤ t for each n, s is the actual number of the independent variables of the studied problem[1,2,8].

Table 1
Distribution of 11 sampling points within domain [0,1.5] and the maximum ( )

3.2 Error Analysis of Definite Integral in 2-D
with the increase of the number of specimen points n obviously.The accuracy of the estimation varies with the exact detail of the function f(x, y).Comparison of the variations of the actual error E actual and the estimated error E est. vs. number of sampling points n for f 2 (x, y) = 1 + 2x 2 + 2y 3 decreases

Table 2
Positions of the specimen points within domain[1.

Table 3
Loc6]ions of the specimen points and the discrete data of the functionFollowing the procedure of UED, the definite integral F now becomes a summation of the 11 discrete specimen points within the integral domain[0,6].The summation gives a