An Improved Taylor-Bernstein Form for Higher Order Convergence

Transcription

1 An Improved Taylor-Bernstein Form for Higher Order Convergence P. S. V. Nataraj, IIT Bombay Talk at the Mini-workshop on Taylor Model Methods and Applications Miami, Florida, USA. Dec. 16th - 20th,

2 1 Outline of talk First, we see essentials of Bernstein form, Taylor form, TB form of Lin and Rokne. Next, we propose a new TB form that is more effective in practice. Then, we test and compare the higher order convergence behavior of The proposed TB form The TB form of Lin and Rokne The Taylor model of Berz et al. For the testing, we consider six benchmark examples dimensions varying fromto. examine higher order convergence for orders up to. Finally, we give the conclusions of the work. 2

3 2 Notation Letbe set of reals, be a box. Let denote the set of all values of an arbitrary function on. Let be the set of all boxes contained in. Let the width of an interval be if and if. A function is an inclusion function for, if for all An inclusion function for is said to have convergence order if for all where and are some positive constants. 3

4 3 Introduction An important problem in interval analysis is : construct inclusion functions with the property of higher order convergence for multidimensional functions. Such inclusion functions have applications in the solutions of equations, optimization, quadrature, and others. Earlier work: 1977: Herzberger showed that higher order convergence can be obtained for a certain class of intervals. However, his requirement on the function is unrealistically strong. 1984: Cornelius and Lohner proposed interpolation and remainder forms for multidimensional functions In theory, any convergence order to be obtained in theory. In practice, convergence order of at mostoris recommended even for unidimensional functions. 1985: Alefeld and Lohner proposed centered forms with higher order convergence for unidimensional functions. 4

5 However, because of strong condition on the functional representation, higher order centered forms have limited practical value. 1990: Neumaier gives improved version of these forms for unidimensional functions. Same restrictions as above. 1996: Lin and Rokne propose higher order convergent forms that combine Taylor and Bernstein (or B-spline) forms for multidimensional functions. However, for small domains these forms become computationally very demanding, even for unidimensional functions. 1990s: Berz et al. propose Taylor models for multidimensional functions. Accuracy of the remainder interval part of the Taylor model increases in a higher order convergent fashion. 5

6 4 Introduction (Contd.) In this talk, we describe a new inclusion function form having the higher order convergence property for multidimensional functions. The new inclusion function form uses Bernstein polynomials for bounding the range of the polynomial obtained from the Taylor form of the function. The Bernstein form is combined with Taylor form to obtain the resulting so-called Taylor-Bernstein (TB) form. The new TB form has some important differences in practical way it is constructed from the TB form of Lin and Rokne. 6

7 5 Useful Features of Bernstein Form Bernstein form is an important tool for finding bounds on range of polynomials. Key features of Bernstein approach are: Computation of bounds conveys information about sharpness of bounds. Approach avoids functional evaluations - which might be costly if degree of polynomial is high. When bisecting a box and applying Bernstein form to one of the two subboxes, we obtain without any extra cost an enclosure for range over other subbox. Bernstein form gives exact range - for sufficiently small boxes. 7

8 6 Multivariate Bernstein Form Let be a dimensional polynomial in. Let be the maximum degree of in, for. Let denote the tuple of maximum degrees, and call as the degree of. Write in power form Without loss of generality, consider domain as the unit box. The mutlivariate Bernstein form for of degree, is The Bernstein coefficients are 8

9 The Bernstein polynomials are 9

10 7 Range and Vertex Properties A key property of the Bernstein form is its range enclosure property. Let denote the range of on. Then, Equality results if and only if the resp. is attained at one of the vertices, i.e., for. This latter property is called the vertex property. In general, the vertex property does not hold, so we get some range overestimation. The amount of overestimation can be reduced by either elevating the degree of the Bernstein form, or successively subdividing and recomputing Bernstein coefficients over the new subdivisions. Typically, subdivision approach is more efficient - preferred to the degree elevation approach. 10

11 8 Taylor Form Consider a function that is times differentiable on. Expand using a Taylor expansion of order (3) Here,is any point in(usually the midpoint), We call the polynomial part and the remainder part of the Taylor expansion. 11

12 9 Taylor form (Contd.) Using Moore s recursive techniques, we can automatically compute The coefficients of the polynomial An enclosure for the remainder part on The Taylor form of order is Lin and Rokne 1996 showed that Taylor form has convergence order 12

13 10 Lin and Rokne s TB Form Instead of finding the range Lin and Rokne proposed to use Bernstein form for of degree given by Compute a (generally non-sharp) enclosure of the range as the to over all the Bernstein coefficients. Interestingly, their new form constructed as retains the th convergence order property. We call the form as the TB form of Lin and Rokne. We observe from above that as domain width becomes smaller, required degree of Bernstein polynomials becomes large - and quite quickly. Consequently, the Bernstein step in computation of becomes very computationally intensive as domain intervals shrink in widths. 13

14 11 Proposed TB form We propose an improved TB form It avoids need for high degrees!! of the Bernstein form, as in. We compute the range using Bernstein form of minimum required degree Subdivision Vertex condition checks The overall method to construct is as follows:. Step 1: Expand given function in Taylor form of order, obtaining the Taylor polynomial coefficients and an enclosure of the remainder part. The automated Taylor model technique of Berz et al. for this step, in view of its known computational efficiency. Step 2: Transform the given domainto the unit box. For transformed polynomial, compute Bernstein coefficients of Bernstein form of degree. Step 3: Successively subdivide and recompute Bernstein coefficients on each new subdivision, till vertex property is satisfied on every subdivision. Step 4: Compute the range as the min- 14

15 imum to maximum over all obtained Bernstein coefficients, Step 5: Construct an enclosure for the range of on " as is a Taylor form of order. So, hasth convergence order property. 15

16 12 Numerical Testing We test higher order convergence property of various inclusion function forms on some examples. Examples are of low to medium dimensions ( through dimensions). We use A PC/Pentium III MHz MB RAM machine, A FORTRAN compiler, Version of the COSY-INFINITY package of Berz et al. In each example, we compute the intervals # - using Taylor model of Berz et al. - using TB form of Lin and Rokne. - using the proposed TB form. $ - using inner estimates of range computed with Moore-Skelboe optimization algorithm. 16

17 13 Measure of Overestimation Let be any two intervals. As a measure of overestimation, we use the Hausdorff metric Consider a sequence of nested intervals For each form, we examine reduction in overestimation with decrease in widths of domain intervals. Consider first the form #. Let # # As a measure of the reduction in overestimation obtained with form # over successive nested intervals and we use the ratio # # # # # 17

18 14 Measure of Overestimation (Contd.) Define If # is an inclusion function form having convergence order then # (where the tending is from above) for small enough. The overestimation is found relative to some inner estimate $ of the range. If theth convergence order property holds relative to $ then it also holds relative to the exact range. So, it is sufficient if we can show theth convergence order property with $. 18

19 15 List of Examples Example 1 : Gritton s second problem in Chemical Engineering : The-dim function is a polynomial ofth degree. The domain is. Example 2 : Jennrich and Sampson function. The -dim function is The domain is. Example 3 : Levy function. The-dim function is % % The domain is. Example 4 : Trigonometric function. The -dim function 19

20 is The domain is. Example 5 : Griewank function. The-dim function is The domain is. Example 6 : Trigonometric function. The -dim function is The domain is. 20

21 16 Summary of Results With the Taylor model as an inclusion function form: We obtain only quadratic convergence in all problems, irrespective of the chosen Taylor order. With the Lin and Rokne s TB form as an inclusion function form: In all cases (except for Taylor orders ) we are unable to proceed after just one subdivision, i.e., with!, due to the excessive memory requirements arising from high degrees of the associated Bernstein polynomials. For Taylor orders we are unable to proceed after just two subdivisions, i.e., with!, for the same reason. Therefore, as an inclusion form for obtaining higher order convergence, the practical utility of is found to be severely restricted. With the proposed TB form as an inclusion function form: In problems of up to-dimensions, we quite easily obtain higher order convergence of orders up to. In problems ofand-dimensions, we obtain 21

22 higher order convergence of orders up to. But, computational demands are somewhat large for -dimensional problem, And become excessive for the -dimensional one. 22

23 17 Conclusions A new TB form as an inclusion form for multidimensional functions. With new TB form, we quite easily obtain higher order convergence (of orders up to) for low to medium dimensional problems. To our knowledge, it is perhaps for the first time that higher order convergence of such high orders has actually been demonstrated on multidimensional problems. The new higher order convergent form can be constructed on a computer with the fully automated algorithm presented, without any need for hand computations. For a problem of higher dimensions ( ), the proposed TB form was found to be computationally inefficient. This strongly suggests the need for further improvements in the proposed algorithm for dealing with higher dimensional ( ) problems. 23

24 18 Acknowledgments We thank Drs. Berz, Makino, and Hoefkens for providing the COSY-INFINITY software and extending a lot of help regarding the usage of the software. We would also like to thank Dr. N. D. Jotwani of GCET for motivating, encouraging and providing the required facilities for this work. 24

25 Table 1: Overestimations and their reduction ratios for various Taylor orders obtained with various forms in Example 1 Gritton (1-dimensional). & ' ' ' ' ' ' ' ' & ' ' ' & ' ' ' ' ' ' ' ' - & ' ' ' ' ' ' ' ' & ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 25

26 Table 2: Overestimations and their reduction ratios for various Taylor orders obtained with various forms in Example 6 Trignometric (-dimensional). : ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' : ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' - - ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' : ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 26

27 Table 3: Average execution times with various algorithms. The time is in seconds, unless otherwise stated. The average is taken over all, and all Taylor orders. Note that with TB form of LR, mostly only one subdivision () is found possible in the problems. Average Execution Time Example Name Taylor model Algorithm LR Algorithm TB 1 Gritton 2 Jennrich & Sampson 3 Levy 4 Trignometric 5 Griewank 6 Trignometric hours 27