FMM CMSC 878R/AMSC 698R. Lecture 13

Transcription

1 FMM CMSC 878R/AMSC 698R Lecture 13

2 Outline Results of the MLFMM tests Itemized Asymptotic Complexity of the MLFMM; Optimization of the Grouping (Clustering) Parameter; Regular mesh; Random distributions. Neighborhoods and Dimensionality in MLFMM Domains of Expansion Validity; Domains of Translation Validity; Neighborhood Increase Technique. Evaluation of the FMM Error

3 Itemized Cost of MLFMM Regular mesh: Assume that all translation costs are the same, CostTranslation(P) Powers of E 4 and E 2 neighborhoods

4 Optimization of the Grouping Parameter CostMLFMM s opt s

5 Optimization of the Grouping Parameter (Example) In this example optimization results in about 10 times savings!

6 Some Numerical Experiments with MLFMM

7 Regular Mesh, N = M.

8 Error Test. FMM vs Middleman. 1.E-07 1.E-08 Absolute Maximum Error. 1.E-09 1.E-10 1.E-11 1.E-12 Mediator FMM 1.E-13 1.E Number of Points

9 Test with Varying Grouping Parameter. 100 Max Level= CPU Time (s) N= Regular Mesh, d= Number of Points in the Smallest Box

10 Test with Varying N Straightforward 100 y=cx 2 CPU Time (s) 10 1 y=bx FMM (s=4) Setting FMM FMM/(a*log(N)) 0.1 Mediator Number of Points Regular Mesh, d=2

11 Looks like the MLFMM complexity is O(NlogN)?

12 Looks like the MLFMM complexity is O(NlogN)? Explanation of log behavior in the numerical example: Translation was very cheap, PureCostTranslation ~ 5, while β was also small (say ), so some influence of log dependence was observable.

13 Test at different translation costs Single Translation Cost= CPU Time (s) 10 1 y=ax y=bx Single Translation Cost=1 0.1 Regular Mesh, d=2, s= Number of Points

14 Test with Varying TranslationCost(P) s=1 CPU Time (s) y=bx y=ax 1/2 1 Regular Mesh, N=65536, d= Single Translation Cost

15 Test with Varying TranslationCost(P) 1000 Optimum Number of Points in the Smallest Box y=ax 1/2 Regular Mesh, N=65536, d= Single Translation Cost

16 Comparisons for different dimensionalities d=1, s= d=2, s= d=3, s= d=4, s= CPU Time (s) y=ax 0.1 d=1 Regular mesh, optimal s Number of Points

17 Test at different dimensionalities Optimal s for each d - s=4 y=b x CPU Time (s) 10 1 y=a x/2 0.1 Regular mesh, N= Space Dimensionality

18 Random Distributions

19 Dependence of CPU Time on the Grouping Parameter, s Uniform Random Distribution 4 CPU Time (s) Regular Mesh 6 MaxLevel=5 4 N=4096, d= Number of Points in the Smallest Box

20 Dependence of CPU Time on the Maximum Space Subdivision Level CPU Time (s) 10 1 Regular Mesh N= Uniform Random Distribution d= Maximum Level

21 Dependence of CPU Time on M CPU Time (s) 1 Max Level=6 Optimum Max Level Uniform Random Distributions N=4096, d= Number of Evaluation Points

22 Dependence of Optimum Max Level on M 8 7 Optimum Max Level Uniform Random Distributions N=4096, d= Number of Evaluation Points

23 Example of A Non-Uniform Random Distribution 30 d=2, N=M= CPU Time (s) Uniform Random 5 Non-Uniform Random Number of Points in the Smallest Box

24 Domains of Expansion Validity (1) r min (l) size(l)

25 Domains of Expansion Validity (2) R-expansion S-expansion x * x * closest x i closest y the most far y The most far x i

26 Domains of Expansion Validity (3). R-expansion.

27 Domains of Expansion Validity (4) Picture from Lecture 4 S R x * x i What frequently happens, is that that both S and R expansions converge even on the sphere of radius x * - x i, except of point y = x i. Singular Point is located at the Boundary of regions for the R- and S-expansions! If this is the case, d = 9 is OK for R-expansion with the E 3 neighborhood.

28 Domains of Expansion Validity (5). S-expansion. strict Original Reduced weak

29 Domains of Expansion Validity (6). S-expansion. In case of original FMM scheme the limitation for dimensionality coming from S-expansion validity is the same as for R-expansion validity: d < 9 strict, d 9 weak.

30 Domains of Expansion Validity (7). S-expansion. In case of reduced FMM scheme the limitation for dimensionality coming from S-expansion validity is the same as for R-expansion validity: d < 4 strict, d 4 weak. closest y x * The most far x i

31 Domains of Expansion Validity (8). R R and S S-translations. No additional constraints S S R R y x i y x *1 x *2 x *1 x *2 x i

32 Domains of Expansion Validity (9). S R-translation. Original x *1 x *2 y x i d < 4 strict, d 4 weak.

33 Domains of Expansion Validity (9). Reduced S R-translation. d < 3 strict, x i d 3 weak. x *2 x y *1 d < 4 strict, d 4 weak.

34 What to do in larger dimensions? Neighborhood Increase Technique

35 Neighborhood Increase Technique Recipe: Build a fractal structure with larger neighborhood Everything works by the same way, but E 2 and E 4 are larger We need only: E 3 = E 2, E 3 (Parent(n),l-1)UE 4 (n,l)=e 3 (n,l). 5-neighborhood Ε 1 Ε 2 Ε 3 Ε 4 3-neighborhood

36 Neighborhood Increase Technique 5-neighborhood, Translation Reduction Trick x i x *2 x *1 y d < 6 strict, d 6 weak.

37 Neighborhood Increase Technique 5-neighborhood The same as for nonreduced scheme with 3-neighborhoods! x *2 y x i x *1 Number of Translations:

38 Neighborhood Increase Technique Advantages: 1) Enables larger dimensionality of the problem; 2) Smaller error (or smaller truncation number). Unfortunately, cannot Increase Indefinitely

39 Evaluation of the Error of FMM Each Translation introduces an error; This includes error of expansion/evaluation;

40 Example For p=10, about 100 times higher accuracy! For the same error, about 1.5 times smaller p

41 Problem: Let the error is specified. What method is faster Smaller neighborhood with larger p or Larger neighborhood with smaller p?