ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ

1. ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ Εισαγωγή

2. 1. ΕΙΣΑΓΩΓΗ • Η ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ και ΤΑ ΥΠΟΛΟΓΙΣΤΙΚΑ ΜΑΘΗΜΑΤΙΚΑ είναι σχεδόν ταυτόσημες έννοιες. • Αποτελούν τον κλάδο των Εφαρμοσμένων Μαθηματικών που ασχολείται με τη «διακριτοποίηση» και την εύρεση προσεγγιστικών λύσεων Μαθηματικών προβλημάτων των οποίων η αναλυτική λύση είναι αδύνατον να βρεθεί αναλυτικά ή σχεδόν ακατόρθωτη. • Το διακριτό πρόβλημα που προκύπτει ονομάζεται Αριθμητική Μέθοδος.

3. Computer Science Applied Math Engineering Scientific Computing Scientific computing The interface of Computer Science, Engineering and Applied Mathematics.

4. INDUSTRY 4.0

5. Exponential Technologies

6. Νόμος του Moore

8. Exponential rate of change is creating new challenges and opportunities: Talent is key in closing the gap and increasingthe rate of change

9. Industry 4.0

11. Exponential Technologies in Manufacturing The pace of change is exponential, and manufacturers are not immune

12. A snapshot of exponential technologies

13. THE INDUSTRIAL INTERNET OF THINGS • Industry 4.0 means that more devices—sometimes including unfinished products—will be enriched with embedded computing. This will allow field devices to communicate and interact both with one another and with more centralized controllers, as necessary. It will also decentralize analytics and decision making, enabling real-time responses.

14. CYBERSECURITY • With the increased connectivity and use of standard communications protocols that come with Industry 4.0, the need to protect critical industrial systems and manufacturing lines from cybersecurity threats increases dramatically. As a result, secure, reliable communications as well as sophisticated identity and access management of machines and users are essential.

15. Current applications • Hiring of a Chief Information Security Officer and dedicated security team; performing penetration testing and war-gaming scenarios within IT and OT groups; in industrial environments, inventory and security testing of all assets and continuous monitoring for potential security events, unauthorized access, or changes; implementing ICS Security programs to remediate any weaknesses. • Comprehensive and layered security solutions for connected cars that protect privacy, ensure safety, and thwart unauthorized access. • Designing, developing, and implementing Product Security Programs to secure connected products both at an enterprise level (e.g., security incident response) and product level (e.g., technical security testing). There could be 3.5 million cybersecurity jobs unfilled, globally, by 2021.

16. ADDITIVE MANUFACTURING • Companies have just begun to adopt additive manufacturing, such as 3-D printing, which they use mostly to prototype and produce individual components. With Industry 4.0, these additive-manufacturing methods will be widely used to produce small batches of customized products that offer construction advantages, such as complex, lightweight designs.

17. https://www.3dhubs.com/knowledge-base/additive-manufacturing-technologies-overviewhttps://www.3dhubs.com/knowledge-base/additive-manufacturing-technologies-overview

18. Digital twin • The digital twin refers to a digital replicaof physical assets, processes and systems that can be used in real-time for control and decision purposes • Computerized mathematical model (what we have done over years) • Real-time, thanks to IoT • In contrast to a physical asset, the digital twin can immediately respond to what-if inquiries

19. Why are manufacturers slow to adopt exponential technologies? 1. Structural and cultural challenges 2. Regulatory burdens 3. Talent constraints 4. Leadership mind-set

20. 3. Talent constraints • Lack of availability of talent, STEM or otherwise, throughout the manufacturing industry • Retirement of Baby Boomers and the potential loss of decades of intrinsic knowledge • Perception of the manufacturing industry not being seen as innovative and/or high-tech • High exodus of potential tech-savvy employees to other industries and sectors

24. Industry 4.0 – Main Application Fields Algorithms for smart assembly & disassembly through AR Mobile Production Plant – Smart factory Production monitoring & visualization • Algorithms for parametric design, and smart assembly and disassembly • Constant access to assembly/ disassembly instructions • Reconfigurable and adaptive environment • Autonomous systems • Mobile robots and resources • Production Line monitoring through comprehensible visualizations • Smart sensing and use of industrial protocols

25. Industry 4.0 – Main Application Fields Augmented reality applications for human operators support New methods for human-robot collaboration • Reduction of physical workload in heavy-load tasks • Automation of ergonomically inconvenient work steps • Efficient and safe human –robot collaboration • Augmented reality technology to support assembly, design , and maintenance tasks, among others • Allows operators to work on multiple workstations

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29. Top 10 Skills to be relevant in Industry 4.0

30. Ο ρόλος της Αριθμητικής Ανάλυσης και του Scientific Computing

31. Τι είναι η Αριθμητική Ανάλυση; • Είναι Επιστήμη: • Ασχολείται με μεθόδους επίλυσης μαθηματικών προβλημάτων με χρήση αριθμητικών πράξεων (με Η/Υ) καθώς και με την ανάλυση των σφαλμάτων στην προσέγγιση των λύσεων. • Είναι Τέχνη: • Αφορά στην επιλογή εκείνης της μεθόδου που είναι πιο «κατάλληλη» για την επίλυση ενός συγκεκριμένου προβλήματος.

32. Το θεωρητικό μέρος της Αριθμητικής Ανάλυσης περιλαμβάνει την κατασκευή αλγορίθμων - ανάλυση, μελέτη της ακρίβειας και της ευστάθειας - , δηλαδή, την ανάλυση και εύρεση των πιθανών σφαλμάτων τους. • Το εφαρμοσμένο μέρος αφορά τον προγραμματισμό των αλγορίθμων σε μια γλώσσα προγραμματισμού με το βέλτιστο τρόπο, δηλαδή, με όσο το δυνατό λιγότερο υπολογιστικό χρόνο (CPU) και απαιτούμενο χώρο μνήμης (RAM). Το θεωρητικό και το εφαρμοσμένο μέρος είναι, συνήθως, αλληλένδετα. • Η ανάπτυξη των υπολογιστικών συστημάτων καθιστά απαραίτητη και επιτακτική την εκμάθηση αριθμητικών μεθόδων για την επίλυση προβλημάτων επιστημονικών εφαρμογών.

33. Συνεχείς διαδικασίες  Διακριτές διαδικασίες • Άπειρες διαδικασίες  Πεπερασμένες διαδικασίες • Στόχος : Η προσεγγιστική επίλυση προβλημάτων που συναντώνται στις επιστήμες και την τεχνολογία, σε εφικτό υπολογιστικό χρόνο και με το μικρότερο σφάλμα.

34. Numerical methods & applications • Why are numerical methods needed? To accurately approximate the solutions of problems thatcannot be solved exactly. • What kind of applications can benefit from numerical studies? Engineering, physics, chemistry, computer, biological and social sciences.Image processing / computer vision, computer graphics (rendering, animation), climate modeling, weather predictions,“virtual” crash-testing of cars, medical imaging (CT = Computed Tomography), AIDS research (virus decay vs. medication), financial math.

35. Computing Efficiency • Computers are getting faster, but the computer’s speed is only one part of the overall performance for a computation... • Computing speed depends on FLOPS (floating-point operations or number of additions and multiplications) and memory accesses.

36. A Small Example • The Difference in Numerical Computing is the numbers • A computation of π

37. Simple iteration:

38. 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 Result of 15 digit computation Red digits are correct Black and green digits are incorrect

39. 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 . . . Result of 15 digit computation Red digits are correct Black and green digits are incorrect π = 0 ?

40. Where’s the problem? is calculated as zero

41. Let’s replace with the algebraically identical expression

42. New iteration: results in …

43. 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545940 6 3.140331156954753 7 3.141277250932773 8 3.141513801144301 9 3.141572940367091 10 3.141587725277160 11 3.141591421511200 12 3.141592345570118 13 3.141592576584872 14 3.141592634338563 15 3.141592648776985 16 3.141592652386591 17 3.141592653288992 18 3.141592653514593 19 3.141592653570993 20 3.141592653585093 21 3.141592653588618 22 3.141592653589499 23 3.141592653589719 24 3.141592653589774 25 3.141592653589788 26 3.141592653589792 27 3.141592653589793 28 3.141592653589793 29 3.141592653589793 30 3.141592653589793 31 3.141592653589793 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 π correct to all digits

44. The result of this computation affects • The ability of the next plane you fly to stay in the air • The integrity of the next bridge you cross • The path of a missile that isn’t intended to strike you

45. NumericalDisasters • Patriot system hit by SCUD missile • position predicted from time and velocity • the system up-time in 1/10 of a secondwas converted to seconds using 24bit precision (by multiplying with 1/10) • 1/10 has non-terminating binary expansion • after 100h, the error accumulated to 0.34s • the SCUD travels 1600 m/s so it travels >500m in this time • Ariane 5 • a 64bit FP number containing the horizontal velocity was converted to 16bit signed integer • range overflow followed

46. Application areas of numerical analysis • Petroleum modeling • Atomic energy – including weapons • Weather modeling • Other modeling such as aircraft and automobile • Computer graphics & computer vision • Simulation for prototyping • Circuit design • Mechanical design • CAD/CAM

47. Algorithm areas of numerical analysis • Linear Equations • Nonlinear equations - single and systems • Optimization • Data Fitting - interpolation and approximation • Integration • Differential Equations - ordinary and partial

48. Why You Need to Learn Numerical Methods? • Numerical methods are extremely powerful problem-solving tools. • During your career, you may often need to use commercial computer programs (canned programs) that involve numerical methods. You need to know the basic theory of numerical methods in order to be a better user. • You will often encounter problems that cannot be solved by existing canned programs; you must write your own program of numerical methods. • Numerical methods are an efficient vehicle for learning to use computers. • Numerical methods provide a good opportunity for you to reinforce your understanding of mathematics. • You need that in your life as an engineer or a scientist.

49. Why use Numerical Methods? • To solve problems that cannot be solved exactly

50. How high does a rocket go?