Dinh Van Tiep, Tran Thi Hue
Faculty of International Training, Thai Nguyen University of Technology, TNU, Thai Nguyen, Vietnam
Correspondence to: Dinh Van Tiep, Faculty of International Training, Thai Nguyen University of Technology, TNU, Thai Nguyen, Vietnam.
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Copyright © 2020 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
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Abstract
Cubature methods has been used and well developed to approximate integrals in high dimension for a long time. However, the number of functions evaluations increasing enormously large make a weak point for such methods. In that situation, adaptive cubature is often preferred choice because of a high efficiency and a low cost of calculation it brings back for the approximation problem. However, the data of the integrated regions and of values of the integrand must be continuous due to the theory of integration. It is infeasible to store in computer memory. To deal with this, the discretization of data for both of the region and the function values are used by constructing the net of the potential mesh points. This technique is acceptable since the result we want to extract is only an approximation within a requisite error. The paper aims to present that technique and some remarkable results.
Keywords:
Adaptive Cubature Program, Approximation Techniques, Discretization, Continuous Data
Cite this paper: Dinh Van Tiep, Tran Thi Hue, A Technique of Discretizing Continuous Data for Programming Adaptive Deterministic Cubature Methods in Moderate Dimensions, Applied Mathematics, Vol. 10 No. 1, 2020, pp. 16. doi: 10.5923/j.am.20201001.01.
1. Introduction
1.1. Background and Problem Statement
The algorithm of the Adaptive cubature as well as other methods of numerical integration in high dimension are developed and have been used for a very long time. It definitely has the advantage of accuracy, but also has the drawback of the cost for computation such as the consummation of time and memory since the increasing complexity in calculation. With the help of computers, the implementations of such algorithm in high dimension need to be adjusted. Moreover, in the general case of high dimension, a program for the algorithm have not been provided yet. A primary obstacle of programming the algorithm is the fact that the integrated domain in high dimension with the smooth boundary are described by hypersurfaces which are produced by continuous data of points. Unfortunately, it is impossible to find enough memorized space of hardware to store the data. This makes the algorithm somewhat theoretical and impractical with a tremendous mass of computation. The algorithm itself amends this by a prescribed error, called the tolerance, which requires the accepted estimate error of the approximation must not exceed this tolerance. This reduces quite a lot the number of calculation, but this itself is not enough to make the algorithm feasible to implement because we still need a strategy to store the information we need in the computation process about the values of the integrand on the boundary of the integrated domain.
1.2. Background and Problem Statement
The cubature is the term introduced by Krommer and Ueberhuber [1,2] to indicate the numerical computation of multiple integral. It includes many techniques such as the Monte Carlo and QuasiMonte Carlo cubature, Bayesian cubature, adaptive cubature. To adaptive cubature, in 2003, Genz and Cools published an algorithm of adaptive cubature for simplices [3] as well as CUBPACK in FORTRAN90. We knew, in high dimensions, the Monte Carlo cubature is the most preferable choice for a numerical integration because of the advantage in dialing with the curse of dimensionality. But this method only yields, in general, a rate of convergence which is quite slow for the number of sample points. Another shortcoming of this method, especially in the case of no permission for using the probability error, is that the order of convergence is only represented in the randomized terms. That is, the estimate error produced by the Monte Carlo method is not deterministic [4] and it is unsuitable if the cubature problem needs a guaranteed error. That is an indirect reason why we still need to invoke the deterministic cubature, especially in a moderate number of dimensions (say, less than 7). The authors Genz and Malik, Berntsen and Espelid and Genz, Dooren and Ridder presented in [5,6,7] their works on the adaptive cubature, however, these were developed on a hypercube, which is the simplest region in high dimensions. A natural development for an adaptive cubature on more general region in high dimensions is discussed in this paper.
1.3. Contribution
In this paper, we derive an algorithm for the numerical integration over more general regions which are enclosed by measurable surfaces. In some sense, the storage of data describing the region is another challenge. We cannot store all data of this boundary because it composes continuous surfaces. With the need of a strategy to store the data required for calculation in each iterated step, we establish a net of the stored data with the mesh points become denser on the region over which the estimate error is not less than the corresponding tolerance distributed over this region. The technique is the way of discretizing the continuous data of the integrated region. The extracted phase to get the data is organized in an adaptive way in which only the values needed for calculation in the next iteration are invoked. The introduction of this technique for the general domain of dimension n is the main contribution of the paper. In addition, the creation of data points and well organize them in the form of a net so that we can easily access the data also reveals another challenge to program the general algorithm. In the paper, we propose a solution to this problem by a simple approach using the bisection in not too high dimensions. This approach in such dimensions is useful to simplify the structure of the program and this definitely contributes to speed up the computation process.
1.4. Organization of the Paper
The paper is divided into 4 section. The second section is intended to briefly describe the basis of the adaptive cubature for approximating the multiple integrals over the quite general region, so called the nonrectangular hyperboxes. The idea is developed from two specified methods, the Simpson’s rule and the Composite Simpson’s rule with 4 subintervals developed for the multiple integrals. The challenges are not only from the curse of dimensionality, but also from the complicated of the integrated region. For this point, as the aforementioned discussion, the complication in depicting the general region can be overcome in a somewhat temporarily acceptable approach in which we store discretely a set of data points that establish the skeleton of the region. We are not going to use these points all at once, but with the accommodation to fit the requirement each checking time. This does not makes the memory of computer working without overloading. Another benefit of this approach is the same as that of a usual adaptive cubature with the flexibility in subdividing the integrated region by adjusting the size of the subregions for the next iteration incorporating with the estimate of the error obtained from the use of the two methods applied at this step.The third section shows off the algorithm of the method which is organized and presented generally in the style of the Matlab language. An illustration for the proper operation of the algorithm is also presented in the case of the dimension 2, the approximation of double integrals. In the last section, we briefly conclude the feature of the algorithm discussed including its advantages and shortcoming.
2. Adaptive Cubature
2.1. Approximation of the Multiple Integrals over Nonrectangular Hyperboxes
Consider the problem of approximating multiple integral I of a function which is continuously differentiable up to order 4 on a region a nonrectangular hyperbox are continuous functions on the rectangular hyperbox Our aim now is to formulate an approximation for  (1) 
with by using the Simpson’s rule and the Composite Simpson’s rule with Simpson’s rule. Firstly, the inner integral of (1) is that of one variable, and treated as  (2) 
where the notation indicates that depends on the fixed point Theorem 1 (Simpson’s rule)  (3) 
where and for some and and some in the interval PROOF: Integrate both sides of (2) over with respect to with the use of Mean Value Theorem for the multiple integral of the term over to get  (4) 
Reapply Simpson’s rule for one variable on for terms in the sum of the first term of (4), denoting that to get the first term of (4) to be  (5) 
where and for some  (6) 
Applying Intermediate Value Theorem for (5), and then Mean Value Theorem for the second term of (4), we obtainfor some Similarly, reapplying the Simpson’s rule consecutively to other dimensions of we complete the proof.Composite Simpson’s rule with n=4. Set Similar to the above derivation of Simpson’s rule, the following result is obtained for the Composite Simpson’s rule.Theorem 2. (Composite Simpson’s rule) Let be the coefficients of the Composite Simpson’s rule with We have,  (7) 
for some and some
2.2. Adaptive Cubature in High Dimensions
Let denote the first term in the righthand side of (3), (7), respectively. Assume that So, Since we have Hence,  (8) 
This means that we can use the difference between two estimates to approximate the error in the approximation revealed by (7). Therefore, we can design the size of the error to be less than a given tolerance Concretely, if then we could believe that approximates to within Otherwise, if we mostly get wrong when using to approximate with the error less than In the latter case, we may get a reasonable approximation by reapplying the abovementioned procedure on smaller regions (each of such a subregion has only a size of nearly of that of the original region and the expected tolerance for approximation of the integral over that subregion is only Now in such smaller region, we search for Since the size of such subregions becomes smaller and smaller, we can eventually reach the target if continuing the procedure. Theoretically, the procedure always succeeds in finding an approximation of lying to within the given tolerance. However, a computer program cannot execute the procedure in the infinite number of times. Therefore, we set up a limitation for the search by requiring that the level of subdivision (or the number of times in which the procedure is repeated) does not exceed a prior number So, the program will reveal the status of failure if is exceeded. Otherwise, we obtain a desired approximation.
3. Algorithm for the Adaptive Cubature
3.1. Pseudo Matlab Code for the Implementation
We describe the algorithm for the aforementioned procedure in the form of a pseudocode with the use of Matlab functions. However, the notations and the structure of the repeat loops or the ifcondition, the assignment operator are not the same as those in Matlab. They are changed in order to make the program familiar with the mathematic notations, so it may be simpler to analyse.INPUT region (including and functions function tolerance limited level OUTPUT approximation or a message announces that the level is exceeded (that is, the procedure fails!).  Figure 1. The region B and its 4 subregions in the first iteration step 
3.2. Numerical Example
Consider the double integral where the nonrectangular region The exact result is An implementation use the above algorithm yields an approximation of to within the given tolerance The limit level is The whole procedure is described in the Table 1.Table 1. The result produced by the implementation 
 

4. Conclusions
The algorithm discussed in the paper dials with a basis problem of numerical analysis with a technique which enables to optimize the implementation of the science computers even when the setting of the multiple integration is quite general, a general iterated regions with a continuous multivariable function refer to as the integrand. There are the repeat loops presented in the implementation which are nested loops. To realize each such loop in the program we need to makes the verification with respect to the dimension of the multiple integral. That means, each dimension has a particular program to the corresponding algorithm. Therefore, it is preferable to use the implementation for not too high dimension, say less than 7. For these sizes of dimension, the algorithm take much more advantages of a high speed of convergence comparing to the Monte Carlo and QuasiMonte Carlo cubature which are preferable choices in very high dimensions.
ACKNOWLEGEMENTS
The authors are working in Thai Nguyen University of Technology. This work are supported by the university. We are very grateful for this help and other assistance in reaching the references we need to complete this work.
References
[1]  Krommer, A. R. and Ueberhuber, C. W. (1998). “Construction of Cubature Formulas”, Computational Integration. Philadelphia, PA: SIAM, pp. 155165. 
[2]  Ueberhuber, C. W. (1997). Numerical Computation 2: Methods, Software, and Analysis. Berlin: SpringerVerlag. 
[3]  Genz, A. C., Cools, R. (2003), “An Adaptive Cubature Algorithm for Simplices”, ACM Trans. Math. Soft., Vol. 26, No. 3, pp. 297308. 
[4]  Atanassov, E., Dimov, I. T. (2008). “What Monte Carlo models can do and cannot do efficiently?”, Applied Mathematical Modelling, Vol. 32, pp. 14771500. 
[5]  Genz, A. C. and Malik, A. A. (1980). “An adaptive algorithm for numeric integration over an Ndimensional rectangular region,” J. Comput. Appl. Math., Vol. 6, No. 4, pp. 295302. 
[6]  Berntsen, J., Espelid, T. O., Genz, A. (1991). “An adaptive algorithm for the approximate calculation of multiple integrals,” ACM Trans. Math. Soft., Vol. 17, No. 4, pp. 437451. 
[7]  Dooren, P., Ridder, L. (1976). “An adaptive algorithm for numerical integration over an ndimensional cube”, J. Comput. Appl. Math., Vol. 2, No. 3, pp. 207217. 
[8]  Burden, R. L., Faires, J. D. (2000). Numerical Analysis. (9th Ed.). Brook/Cole, Cengage Learning, Boston (2000). 
[9]  Chapra, S. C. (2012). Applied Numerical Methods with MATLAB. (3rd Ed.), McGrawHill, New York. 
[10]  Radon, J. (1948). “Zur mechanische Kubatur”. Monatsh. Math. Vol. 42, pp. 286300. 
[11]  Caflisch, R. E. (1998). Monte Carlo and quasiMonte Carlo methods. Acta Numerica, Vol. 7, pp. 149. 
[12]  Sobol, I. M. (1990). “QuasiMonte Carlo methods”. Progress in Nuclear Energy. Vol. 24, Iss.: 13, pp. 5561. 
[13]  Pierre, l’E., Randomized QuasiMonte Carlo: An Introduction for Practitioners. 12th International Conference on Monte Carlo and QuasiMonte Carlo Methods in Scientific Computing (MCQMC 2016), Stanford, U.S. hal01561550. 
[14]  Cools, R. (2003). “An Encyclopaedia of Cubature Formulas”. J. Complexity, Vol. 19, pp. 445453. 