Example Problems - Advances Topics for Numerical Methods - Notes | CMSC 878R, Study notes of Computer Science

Download Example Problems - Advances Topics for Numerical Methods - Notes | CMSC 878R and more Study notes Computer Science in PDF only on Docsity! FMM CMSC 878R/AMSC 698R Lecture 18 Outline • Example problem – S-expansion error; – S|S-translation error; – S|R-translation error; – R|R-translation error. • Error and Neighborhoods • Optimization of MLFMM within error bounds • Effects of machine precision R|R-operator Hn Hn RaQ) = +0" = 5) — AE __prrym 2S mt "RB n()). = mith — my! 0, m>n CRIB rm(t) = nl poe 1 0 (RIR\E) = (AIR),,2)= | 0 0 1 3f 0 S|S-operator (-1)"" yn! attm — i)! SoH) G4 are f) AD Sn), 0, m<n ED wale) = (-1)" mn! m eee men nl (mt)! 1 0 0 0 t+ 100.. (SIHEQ) = m= | f ze 1 0 2 32 36 1 S|R-operator Sn GHt) = fs t+yyrl = =f ot (1 +3 y a 7 > Ct +) em lym ait - re 1)"(n +n)! emt, Cy). mint — ("in t+}! fl f2 ae (SIRI) = a? 2 3. f2 364 of. S|S-Translation Error Translation from level α+1 to α: p first coefficients at level α can be exactly computed from p first coefficients at level α+1. Τhis is exact translation of first p coefficients! S|S-Translation Error(2) Translation from level α+1 to α: For any level α! This factor shows that we are on level α S|S-Translation Error(3) x*1 x*2 y xi In this example S|S-translations do not cause any additional error! S|R-Translation Error (3) | Pey,xi) = PO Cy x) | Long one! pol pol VY Mem - EV en m=O n=O m=0 n=O P-1 oo pol pol Y Mem t PV om YY om m=0 n=O map #=0 m=0 =O) Pol pol Polo pol pl Y Mem + VY emt Vem YD en m=O x=O m=O nap map n=O m=O x=0 Pol oo pol « Y Vem + EV on] < Den +O Dom m=O np wep #=0 m=O nap map n=O oo o o oo YVemls E Vlew| = EP eml +E Ven m=O ne, map n=O nap mm=0 map #=0 ao oo o YD ewl+ DS Dem | = LED Kem tle) er, mp mo mp mo continued —> S|R-Translation Error (4) It is really long! continued we used this S|R-Translation Error (5) That’s it! d = 1: ρ /r = 2, d = 2: ρ /r = 2(2 - √2)/√2 = 2(√2 -1) > 0.8 R|R-Translation Error(3) Indeed, in our case the regular basis functions are polynomials up to order p-1, which are obviously can be expressed via other polynomial basis up to order p-1 near arbitrary expansion center. Zero error is provided due to domains of validity are included hierarchically to larger validity domains. Total Error AbsSingleSourceError < MaxfxpansionError(p,R,r) + AdaxSRTranstationError(p, r, p) = (a) wept ole) AbsTotalError = N+» AbsSingleSourceError _ yd 2 ¢ P -M(R) pip alos) | since R > rtp, AbsTotalError < ay r+? y. Total Error(2) d=1: p=2*, r=05-2%, AbsTotalError < 32. d=2: p=(2-92)24, r=0s2-2%, QL AbsTotalError < 3 N J2 ‘ 2— /2 \ 2(/2/2+2- 2) = 3 2N (25) < 5.2(0.6 +22. 2-72 \4- 2 Both formulae can be described as AbsTotalError < CoP2'N = e(p,NL,d). Total Error(5). 2-Neighborhood. AbsTotalError < Ca?2'*N = e(p,N,L,d). Example: V = 107, £=10,d=1: pep 10 15 20 23 €,< 4 21077 4-107? 2-107! Example: V = 104, £=10,d=2: p 15 20 25 30 €,< 5+107' 2-10-7 4-10- 2-10°% Error for Different Neighborhoods k is the size of the neighborhood, m is the cell consolidation order (in our case m = 0) Optimization of MLFMM within error bounds In the example considered, the FMM error depends on: • Truncation number, p; • Max level of space subdivision, L; • Size of the neighborhood (1,2, or maybe other); • Number of sources, N; • Problem dimensionality, d. For fixed (given) N and d parameters p,L,and Neighborhood Size can be optimized. Effects of Machine Precision (1) 1.E-15 1.E-12 1.E-09 1.E-06 1.E-03 1.E-12 1.E-09 1.E-06 1.E-03 Prescribed Max Abs Error, ε M ax A bs E rr or M = N , q = q (ε), p = p (ε, l max) N=16 64 256 1024 4096 16384 Double Precision Complex Arithmetics Theoretical Error Bound Actual Max Abs Error Machine Precision Thresholds Effects of Machine Error (2) Two different computational problems: 1) Compute with a given prescribed accuracy; 2) Compute with a machine precision for a given type of float numbers. Effects of Machine Error (3) 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Number of Source Points, N C P U T im e (s ) Straightforward Setting FFIA Data Structure FFIA Max Precision C t ti ε = 10-4 ε = 10-6 M = N , q = q (ε), p = p (ε, l max) y = ax 2 y = bx FFT MLFMM MLFMM