Andrzej Pownuk Silesian University of Technology, Poland

1. Efficient Method of Solution of Large Scale Engineering Problems with Interval Parameters Based on Sensitivity Analysis Andrzej Pownuk Silesian University of Technology, Poland

2. Slightly compressible flow- 2D case

3. … Measurements

4. Example:inexact ruler, … Accuracy of measurements We can calculate this number in controllable environments (in laboratory). This error is not connected with probability.

5. Inexact measurements - accuracy of measurements

6. Set-valued random variable …

7. Characteristics of discrete random variable Mean value Variance

8. This formula is true only for Gauss PDF. Usually we don’t know probability density function (PDF) Probabilistic methods require assumptions about the probability density function.

9. Confidence interval

10. Interval estimation of probability

11. Probability for X: Probability for Y: 1 1 2 x

13. Updating results using latest information Old data New data

14. Properties of confidence intervals • Definition of confidence intervals is not based • on the probability density function. 2) Confidence intervals can be defined using set-valued random variables (uncertain measurements).

15. Input membership function x

17. Output membership function

18. Interval solutions of the slightly compressible flow equation Similar treatment for saturation.

19. Example Injection well Production well

20. Interval solution (time step 1)p_upper(t) - p_lower(t)

21. “Single-region problems”

22. Solution of single-region problem Solution of multi-region problem “Multi-region problems”

23. constraints: Result with constraints (single-region) Results without constraints (multi-region) More constraints – less uncertainty

24. Multi-region case

25. Data file alpha_c 5.614583 /* volume conversion factor */ beta_c 1.127 /* transmissibility conversion factor */ /* size of the block */ dx 100 dy 100 h 100 /* time steps */ time_step 15 number_of_timesteps 10 reservoir_size 20 20

26. Interval solution (time step 5)

27. Comparison Single region - Multi-region [0,55] [psi] [0, 390] [psi]

28. Exact solution of equationswith interval parameters

30. Monotone functions

31. - calculations of y(x) Extreme value of monotone functions

32. Sensitivity analysis If , then If , then

33. … - n derivatives 1 We have to calculate the value of n+3 functions. n 2 Complexity of the algorithm, which is based on sensitivity analysis

34. Vector-valued functions … In this case we have to repeat previous algorithm m times. We have to calculate the value of m*(n+2) functions.

35. Implicit function

36. Sensitivity matrix

37. Sign vector matrix

38. Number of independent sign vectors: Independent sign vectors

39. n - derivatives 2*p – solutions (p times upper and lower bound). Complexity of the whole algorithm. 1 - solution

40. Complexity of the algorithm: All sensitivity vector can be calculated in one system of equations

41. Sensitivity analysis method give us the extreme combination of the parameters • We know which combination of upper bound or lower bound generate the exact solution.We can use these values in the design process.

42. Example

43. Sensitivity matrix

44. Sign vectors

45. Independent sign vectors

46. Lower bound- first sign vector

47. Upper bound- first sign vector

48. Lower bound – second sign vector

49. Upper bound – second sign vector

50. Interval solution