Lecture 03

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Faisal Mehmood

Discrete Mathematics

Education Spiritual

CS-708 Conditional Statements or Implications - 7

CS-708 Conditional Statements OR Implications – 7a

CS-708 Alternative Ways of Expressing Implications – 10

CS-708 Alternative Ways of Expressing Implications – 10a

CS-708 Translating English Sentences to Symbols - 12

CS-708 Translating English Sentences to Symbols – 12a

CS-708 Translating English Sentences to Symbols – 12b

CS-708 Translating English Sentences to Symbols – (3 – 12c)

CS-708 Translating Symbolic Propositions to English – 13

CS-708 Translating Symbolic Propositions to English – 13a

CS-708 Translating Symbolic Propositions to English – 13b

CS-708 Hierarchy of Operations for Logical Connectives - 14

CS-708 Truth Table for p v ~ q  ~ p – 20a

CS-708 Truth Table for p v ~ q  ~ p – 20b

CS-708 Negation of a Conditional Statement - 15

CS-708 Inverse of a Conditional Statement - 24

CS-708 p  q is not equivalent to ~p  ~q – (3 – 25)

CS-708 Converse of a Conditional Statement - 26

CS-708 Converse of a conditional Statement – (3 – 27)

CS-708 Contrapositive of a Conditional Statement– 28

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Poster to be presented at Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, Kaust, Saudi Arabia, https://cemse.kaust.edu.sa/stochnum/events/event/snsl-workshop-2024. In this work we have considered a setting that mimics the Henry problem \cite{Simpson2003,Simpson04_Henry}, modeling seawater intrusion into a 2D coastal aquifer. The pure water recharge from the ``land side'' resists the salinisation of the aquifer due to the influx of saline water through the ``sea side'', thereby achieving some equilibrium in the salt concentration. In our setting, following \cite{GRILLO2010}, we consider a fracture on the sea side that significantly increases the permeability of the porous medium. The flow and transport essentially depend on the geological parameters of the porous medium, including the fracture. We investigated the effects of various uncertainties on saltwater intrusion. We assumed uncertainties in the fracture width, the porosity of the bulk medium, its permeability and the pure water recharge from the land side. The porosity and permeability were modeled by random fields, the recharge by a random but periodic intensity and the thickness by a random variable. We calculated the mean and variance of the salt mass fraction, which is also uncertain. The main question we investigated in this work was how well the MLMC method can be used to compute statistics of different QoIs. We found that the answer depends on the choice of the QoI. First, not every QoI requires a hierarchy of meshes and MLMC. Second, MLMC requires stable convergence rates for $\EXP{g_{\ell} - g_{\ell-1}}$ and $\Var{g_{\ell} - g_{\ell-1}}$. These rates should be independent of $\ell$. If these convergence rates vary for different $\ell$, then it will be hard to estimate $L$ and $m_{\ell}$, and MLMC will either not work or be suboptimal. We were not able to get stable convergence rates for all levels $\ell=1,\ldots,5$ when the QoI was an integral as in \eqref{eq:integral_box}. We found that for $\ell=1,\ldots 4$ and $\ell=5$ the rate $\alpha$ was different. Further investigation is needed to find the reason for this. Another difficulty is the dependence on time, i.e. the number of levels $L$ and the number of sums $m_{\ell}$ depend on $t$. At the beginning the variability is small, then it increases, and after the process of mixing salt and fresh water has stopped, the variance decreases again. The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. These estimates depend on the minimisation function in the MLMC algorithm. To achieve the efficiency of the MLMC approach presented in this work, it is essential that the complexity of the numerical solution of each random realisation is proportional to the number of grid vertices on the grid levels.

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Lecture 03

1. CS-708 Application - 1

2. CS-708 Example - 2

3. CS-708 Simplifying a Statement – 3

4. CS-708 Distributive Law in Reverse – 4

5. CS-708 Exercise – 5

6. CS-708 Exercise - 5a

7. CS-708 Conditional Statements - 6

8. CS-708 Conditional Statements – 6a

9. CS-708 Truth Table for p  q - 8

10. CS-708 Conditional Statements or Implications - 7

11. CS-708 Conditional Statements OR Implications – 7a

12. CS-708 Example – 9

13. CS-708 Alternative Ways of Expressing Implications – 10

14. CS-708 Alternative Ways of Expressing Implications – 10a

15. CS-708 Exercise - 11

16. CS-708 Exercise – 11a

17. CS-708 Translating English Sentences to Symbols - 12

18. CS-708 Translating English Sentences to Symbols – 12a

19. CS-708 Translating English Sentences to Symbols – 12b

20. CS-708 Translating English Sentences to Symbols – (3 – 12c)

21. CS-708 Translating Symbolic Propositions to English – 13

22. CS-708 Translating Symbolic Propositions to English – 13a

23. CS-708 Translating Symbolic Propositions to English – 13b

24. CS-708 Hierarchy of Operations for Logical Connectives - 14

25. CS-708 Truth Table for p v ~ q  ~ p – 20a

26. CS-708 Truth Table for p v ~ q  ~ p – 20b

27. CS-708 (p  q) (~p  r) - 21

28. CS-708 (p  q) (~p  r) – 21a

29. CS-708 (p  q) (~p  r) – 21c

30. CS-708 p  q ≡ ~q  ~p- 22

31. CS-708 Implication Law – 23

32. CS-708 Negation of a Conditional Statement - 15

33. CS-708 Examples – (3 - 16)

34. CS-708 Example – 16a

35. CS-708 Inverse of a Conditional Statement - 24

36. CS-708 p  q is not equivalent to ~p  ~q – (3 – 25)

37. CS-708 Writing Inverse – 17

38. CS-708 Writing Inverse – 17a

39. CS-708 Converse of a Conditional Statement - 26

40. CS-708 Converse of a conditional Statement – (3 – 27)

41. CS-708 Writing Converse – 18

42. CS-708 Writing Converse – 18a

43. CS-708 Contrapositive of a Conditional Statement– 28

44. CS-708 Writing Contrapositivity – 19

45. CS-708 Writing Contrapositivity – 19a

Lecture 03

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