Präzise statistische Analyse von Rundungsfehlern bei inversen Problemen

Transcription

1 . Präzise statistische Analyse von Rundungsfehlern bei inversen Problemen Geodätisches Integrationsseminar 1 Andreas Ernst 25. Juni 2009

2 Overview 1 Introduction 2 Norm-based rounding error analysis 2 3 Stochastic rounding error analysis 4 Realization of a precise analysis 5 Numerical experiments

3 Introduction Solutions of geodetic problems require an accuracy analysis Generally the measurement accuracy and adjustment process are considered Errors caused by the computation process are not included Knowledge of rounding errors is important for the evaluation of results 3

4 Solving equation systems Solving a linear equation system of the form Nx = y with a symmetric and positive definite matrix N. The Cholesky-factorization N = R T R leads to 4 which can be substituted to R T }{{} Rx = y, z R T z = y Rx = z forward and backward substitution process

5 Cholesky Algorithm Cholesky factorization of matrix N into the triangular matrices R, R T i 1 r ij = (n ij r ki r kj )/r ii r ii = k=1 i 1 nii k=1 r 2 ki for i j for i = j 5

6 Forward and Backward Substitution Solving R T z = y with forward substitution: i 1 z i = (y i r ki z k )/r ii k=1 Solving Rx = z with backward substitution 6 k=n x i = (z i r ik x k )/r ii i+1 Elements of R are used in both equations Recursive computations It is necessary to consider the errors of the Cholesky factors and their effect on the solution

7 Error Analysis 7 Input Algorithm Output Input Algorithm Output x f (x) y Input errors Input errors Errors during Errors the algorithm during the algorithm Output Output errors errors x x f (x) f (x) y y

8 Error sources Floating point representation: e.g. IEEE binary64 This standard defines: arithmetic formats interchange formats rounding algorithms operations exception handling v 53 bit 10 bit sign significand exponent ( 1) s c b q Rounding to nearest x ξ eps ξ eps(1) = 2 52 = eps(realmax) = = eps(0) = =

9 Error Analysis f (x)... mathematical function f (x)... rounding errors during the algorithm f (x) f (x) = 9

10 Error Analysis f (x)... mathematical function f (x)... rounding errors during the algorithm f (x) f (x) = f (x) f (x) f (x) + f (x) 9

11 Error Analysis f (x)... mathematical function f (x)... rounding errors during the algorithm f (x) f (x) = f (x) f (x) f (x) + f (x) f (x) f (x) } {{ } + f (x) f (x) } {{ } stability of the problem stability of the algorithm 9

12 Error Analysis f (x)... mathematical function f (x)... rounding errors during the algorithm f (x) f (x) = f (x) f (x) f (x) + f (x) f (x) f (x) } {{ } + f (x) f (x) } {{ } stability of the problem stability of the algorithm 9 Forward Analysis: f (x) f (x) σκε κ... condition of the problem Nx = y x = N 1 y N 1 y N 1 y σκε x x x σκ ε

13 Error Analysis Backward Analysis: Wilkinson (1963) Nx = y (N + N)x = y N σε x x x σκ ε 10 For the Cholesky solution: e.g. Stewart (1973) n ij (c 1 n + 2c 2 n 2 + c 2 2 n 3 ε) max n ij g ε x x x κ 2 (N)(c 1 n + 2c 2 n 2 + c 2 2 n 3 ε) g ε Refinements by: Sun (1992), Stewart (1996)

14 Stochastical approach Interpreting rounding errors as uncorrelated random variables [Meissl (1980)] Rounding error ε can be described by We need: Expectation E {ε} Variance σ {ε} the mean and standard deviation of the basic computations addition, subtraction, multiplication and division the mean and standard deviation of the square root Using IEEE Standard with double-precision 11

15 Stochastical rounding errors On true rounding machines: E {ε} = 0 σ {ε} = c Summation / Subtraction a ± b: c = 2 γ > max{ a, b, a ± b } 12 Multiplication a b : c = 2 γ > a b Division a/b: c = 2 γ > a/b Square root a, Assumption: Only a few square roots are needed c = 2 γ > a

16 Precise error analysis Idea: Provide estimated elemental error for every computation step. Necessary steps: Analyse every part of the calculation Estimate the stochastic rounding error of the basic operations Deduct the dependencies of all recursively determined Cholesky terms Calculate the full variance / covariance information of the unknown parameters with variance propagation 13

17 The factorization process Example: Basic calculation of r 34 r 34 = (n 34 r 13 r 14 r 23 r 24 ) /r 33 Modified equation considering all elemental rounding errors 14 r 34 = ((n 34 ((r 13 r 14 ) + ε 1 )) + ε 2 ) ((r 23 r 24 ) + ε 3 ) + ε 4 r 33 + ε 5 ε 1 and ε 3 are caused by the multiplications ε 2 and ε 4 are caused by the subtraction ε 5 is caused by the division These errors can be combined in one value ε εr34

18 Expressing all dependencies The cumulative rounding error does not only consist of the basic rounding errors. r ij are substitutions for r ij = r ij + ε rij Past rounding errors are needed to get the actual rounding errors. Recursive problem: 15 r 34 + ε r34 = ((n 34 + ε n34 ) ((r 13 + ε r13 )(r 14 + ε r14 ) + ε 1 ) + ε 2 ) +... r 33 + ε r33 = r ε n34 r 14 ε r13 r 13 ε r ε r33 + ε εr34 r 33 r 33 r 33 2r 33

19 Equationsystem of the rounding errors Implicit formulation for example ε r34 ε r34 1 r 33 ε n34 r 14 r 33 ε r13 r 13 r 33 ε r r 33 ε r33 + ε εr34 = 0 Structure of the complete equation system: 16 r z x n y r z x r z x

20 Reduced Equation System Reduction via Gauss-Jordan-Algorithm: r z x n y r z x r 17 z x Functional Matrix F Every rounding error of R, z and x is formulated as a linear combination of known elemental errors!

21 Variance propagation Rounding errors of x With Σ {ε basic } consisting of: σε 2 nij,σε 2 yi, σε 2 εrij, σε 2 εzi and σε 2 εxi Σ {ε x } = F Σ {ε basic } F T Calculated with formulas by Meissl Regarding the computation process of Cholesky-factorization and forward/backward substitution 18

22 ~... '" I './ IJO UO UO " ISO 11 ALTENBURG ",..xl,'~,,~ lull: 'f 41' 1/0410 Gravity anomalies austria: Applications: Kriging 10 "tl.,.!,,"!!" n 1, 1/ 49" Übersichtskarte 41 desschweregrundnetzes vonösterreich ~ -.. Legende: Punkta0. Ordnung(AbSlilulsl:hwerepunkle) Punkre1. Ordnung Punkta2. Ordnung( 0 geplant) '.. Melk..,.. :.,-- 1" 0.,' 19 A\' 41 1,,,". "\,...',/ ", j,',-'. r.' \ (,,.... "., ". "... '.. " "-.' " Wejflklrchen.. Dechantskirchen.. GRAl ' 1' 1Itl.'.!n"ml n 12 13" 1, 11 Ion""I'"'.. """,., '" (1'1',,'!"."mlm.., W!,n 'Lli_._~I- '!_--'! '.' '.' '.'.. 11 '.' '!'k' Figure: Overview gravity network of austria 1986

23 Kriging gravity anomalies austria Results of Wiener-Kolmogorov-Filter Polynom(2,2) + Wiener Kolmogorov sigma posteriori = mgal latitude longitude 14.85

24 Results I: Rounding errors Kriging 2.5 x size of rounding errors σ εx parameters (a) Standard deviations of ε x (b) Correlations of ε x Figure: Kriging results are well conditioned

25 Kriging gravity anomalies austria Results of Wiener-Kolmogorov-Filter reproduction Wiener Kolmogorov reproduction sigma posteriori = e 13mgal latitude longitude 14.85

26 Results II: Rounding errors Kriging size of rounding errors σ εx parameters (a) Standard deviations of ε x (b) Correlations of ε x Figure: Kriging results are ill conditioned

27 Conclusions and Outlook Conclusions: Rounding errors can be analysed with all variance/covariance information Outlook: Ill-conditioned problems have highly correlated rounding errors The analysis produces a very large sparse equation system A-priori information about previous rounding errors could be taken into account The effect of reordering strategies on the rounding errors can be analysed Local effects on rounding errors must be investigated A comparison with norm-based error analyses is necessary 24