Algebraic Computations on Noisy, Measured Data
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1 Algebraic Computations on Noisy, Measured Data 3/1/06 File Title Copyright: Shell Exploration & Production Ltd. Daniel Heldt, Sebastian Pokutta, Hennie Poulisse Based on Joint On-Going Work: Daniel Heldt,, Martin Kreuzer,, Sebastian Pokutta, Hennie Poulisse
2 Contents Industrial Application of Computer Algebra ApVI Calculated
3 Going out into the Field! The Champions Field, South-Chinese Sea, offshore Brunei
4 transportation tubing well heads headers (test- and bulk) separators
5 Production Relation for an Oil Well WELL MEASUREMENTS Surface and Sub-surface Gas Selected Well on Test THP, DHP, THT, FLP, LGF etc. Test Separator (3 Phase) TEST SEPARATOR MEASUREMENTS imitating production circumstances: deliberate disturbances Oil Water inputs outputs variables of polynomial function output = function of inputs output = polynomial function input physical relation between inputs
6 example of a polynomial well production low degree, large number of indeterminates
7 Motivating Example Gas production well A production well B production well C transportation Line Bulk Separator (3 Phase) total production production well D Oil Water measured constructed polynomials construct polynomial polynomials: interactions between productions
8 construction of well production polynomial indeterminates production polynomial f has to be fitted to a set of points: evaluations of indeterminates at first data point number of points in the order of thousands not uncommon
9 important observation: different polynomials (supports, degrees) evaluations close together
10 relations in the data, among the indeterminates transitivity fails
11 Small Polynomials: Vanishing Ideal
12 example univariate polynomial perturb points polynomial of degree 5 vanishing on points remove the purturbations first before constructing the polynomials construct polynomial allowing it to pass close by prescribed points
13 delta-approximate Vanishing Ideal of course we still have:
14 Best thing one can do: Empirical Ring: Empirical Polynomial:
15 HOW TO CALCULATE THE ApVI
16
17 Classical Version
18 Drawbacks: The found relations are far away from vanishing on the points: numerical error prevent accurate calculation exact solution would result in an O with #O = #P (1000 here!!) The exact relations (for the perturbed data) are usually of very high degree How to overcome these problems: Do not force the relations to pass through every point, but demand passing close by Allow points which lie close together to be melt together Divide the good ones from the bad ones Process blocks rather than single elements to (hopefully) prevent sub-optimal solutions and speed-up computations
19
20
21
22 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.001
23 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.01
24 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.1
25 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.5
26 Another (short) example...
27 Approximate Vanishing Ideal with Singular Value Decomposition Classical Version Truncate below eps = 0.7
28
29 Timings (GB-Version) (2024 points,, 9 indets)
30 Timings (BB-Version) (2024 points,, 9 indets)
31 Application within Shell Calculate relations for the productions... Typical datasize: points in up to 15 indets
32 Application within steel industries Application fields: predict production quality automated quality assurance Typical datasize: points in indets solution time < s #G = 464, #O = 391
33 Acknowledgement: C. Fassino and J. Abbott worked in parallel on an algorithm which also computes almost vanishing ideals, but with a different approach... See: C. Fassino. An Approximation to the Gröbner Basis of Ideals of Perturbed Point. Preprint (2006)
34 Thank you!!
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