Physics 2660: Fundamentals of Scientific Computing. Lecture 14 Instructor: Prof. Chris Neu

Transcription

1 Physics 2660: Fundamentals of Scientific Computing Lecture 14 Instructor: Prof. Chris Neu

2 Notes Last lecture! Upcoming homeworks: HW13 due Wednesday 2 May at 11:59pm HW14 due Wednesday 2 May at 11:59pm a course evaluation alt3zq Tuma, Andrew ces2fa Stephens, Catherine ecj4wb Jones, Ethan jls6gw Selby, Jonathan raj9qv Jennings, Robert rm3xw Mora, Raymundo sam3bf Murri, Samuel vh2[ Huang, Vicki Why have you not completed this? Solutions to all labs and hw s are in the process of being posted sorry for recent delays 2

3 Reminder Grading of HWs almost caught up! Register for piazza! Only 50 out of 60 so far All questions about final exam will only be answered via piazza Office hours: all held in our computer lab, room 022-C of this bldg Me: After lecture 3:30-4:30 every Tuesday TAs: Mondays: 3-5pm and 6-8pm Wednesdays: 5-9pm 3

4 Notes Final exam is coming: Take-home projects 3 problems Like a more involved, longer multipart homework assignment Assigned last week of semester on Tuesday 2 May will be posted by 3:30pm today Due Thursday May 10: electronic copies by 9:00am hard-copies must be submiied Thursday 10 May between 08:00-10:00 in room 022-C, our computer lab.or earlier by personal arrangement with me. 4

5 Final exam is coming: Take-home projects 3 problems Notes Like a more involved, longer multipart homework assignment Assigned last week of semester on Tuesday 1 May Due Thursday May 10: electronic copies by 9:00am hard-copies must be submiied Thursday 10 May between 08:00-10:00 in room 022-C, our computer lab.or earlier by personal arrangement with me. Due Thursday May 10, electronically by 9:00am and hard copies between 08:00 and 10:00 in 022-C 5

6 Last time: Comparing data to a prediction Review and Outline Today: Tuning a model to best represent the data More tests of compatibility 6

7 Summary so far Compare some data to a model, account for uncertainties Calculate reduced χ 2 If good agreement should see different points sometimes high/low if k large, reduced χ 2 ~ 1.0 7

8 Tuning a Model to Best Match Some Data 8

9 Tuning a Model 9

10 Tuning a Model Can we figure out which model which values of a and b the data most favors? 10

11 Probability of Some Observation 11

12 Probability of Multiple Observations The probability of the collection of data 3 observations is just the product of the three individual probabilities 12

13 Probability of Multiple Observations The probability of the collection of data k observations is just the product of the k individual probabilities 13

14 P: The χ 2 Likelihood Function 14

15 Minimizing the χ 2 15

16 Minimizing the χ 2 16

17 Minimizing the χ 2 17

18 Minimizing the χ 2 18

19 More Powerful Application: An Arbitrary Theory 19

20 FiPing with Gnuplot P(x;a, b,c) = 1 2πc 2 e (ax b) 2 2c 2 20

21 Assessing the Quality of a Fit 21

22 Assessing the Quality of a Fit Trivial case Consequence: If the number of fit parameters is greater than or equal to the number of data points the χ 2 is undefined. 22

23 Assessing the Quality of a Fit P(x;a, b,c) = 1 2πc 2 e (ax b) 2 2c 2 23

24 Assessing the Quality of a Fit So there is a 90% probability that, if the data were consistent with the model (here a Gaussian-like thing with 3 params), the data would have a higher chi2 value. Too good to be true? Why are the points so close to the model? Did the fit procedure cheat in some way? Are the uncertainties over-estimated? 24

25 FiPing is Done EVERYWHERE 25

26 Aside: Gravitational Waves hips:// 26

27 Curve FiPing 27

28 Deviations from the Model 28

29 The Pull Distribution 29

30 Bias Is the Prediction In Accord with the Data? 30

31 Clusters of Data Above/Below 31

32 Clusters of Data Above/Below 32

33 What Have We Learned 33

34 What Have We Learned We learned a whole lot this semester! Computers Programming Language C For many of you this was your first entrypoint into the use of a computer as a tool for data analysis Many of you started from very liile experience Generated a set of tools you can rely on in the future for a whole host of problems in your chosen field, science or otherwise Data Presentation Data Analysis and Assessment Some Statistics some very important statistical concepts 34

35 C vs. C++ 35

36 C vs. C++ 0 C is much simpler for beginners than C++ but far less powerful 1 C++ can run most C code but not the other way round! 2 Functions are the building blocks of C whereas objects are the building blocks of C++ 3 In C, the program is formulated step-by-step but in C++ the base elements are formulated first and then linked together in larger systems 4 In C++, functions can be overloaded, but not in C 5 C++ has a much larger library than C 6 C++ allows user to create classes (similar to structures) to which methods and functions can be assigned and within which they can operate 7 In C++ you can add your own types and objects inheriting properties and routines from other classes. This allows you to add more features to your own classes and the standard library as well 8 C++ uses Namespace, which is very useful in avoiding multiple declarations 9 C++ has built in exception support 36

37 What is next? What more could you learn? 37

38 What Could You Try Next? You now have a firm foundation in C. What could be next? Another powerful language used in scientific seiings, such as python Some scripting language: Bash scripts, perl, etc., to automate many types of text and data manipulation, etc. Object Oriented features of C++, Java, C#, etc: Powerful (and sometimes dangerous) design tools! A physics/math library, like ROOT, R, CERNLIB, CLHEP, LINPACK, GSL, etc Overview of analytical techniques in classes like PHYS 5630/40(Computational Physics 1/2) Get a job w/ a research group, apply your skills, and learn more! 38

39 What Could You Try Next? Worldwide distributed computing grids hip:// UVa s High-Performance Computing cluster: hip://arcs.virginia.edu/rivanna Massively Parallel Processing Real time controls automation robotics eg. autonomous vehicles,... Quantum Computing hip:// 39

40 Grid Computing We have operated this semester in the following fashion: sit down at some personal or lab computer use this as a portal to a cluster of computers, galileo, featuring large amounts of storage many nodes allowing multiple users to exploit the same resource simultaneously Some drawbacks: galileo can get slow bogged down by other users. What do we do in these cases? Other clusters available to us and where? /home is a shared resource and has goien full a few times, preventing work not an infinite space resource interactive jobs are tempermental: what if you needed to run 1000s of jobs need 1000s of terminal sessions? what if I needed lots of data to be stored but only needed it very occasionally? 40

41 TIER-0 CERN AT LHC, need to store, analyze and distribute 15 Petabytes of data per year (1% of digital information produced in the world) All data is processed at CERN and backed up on tape But CERN provides only ~20% total compute capacity

42 Distributed Computing Tier-1 sites Sufficient storage capacity for large fraction of the data Only store data specific to given task Large scale reprocessing Distribution to Tier-2 sites 11 sites for all LHC experiments Distributes data to ~30 Tier-3 sites

43 Worldwide LHC Computing Grid LHC Grid gives access to more than 7000 scientists in more than 30 countries World s largest computer grid connecting more than 200,000 CPUs at universities and research institutes Runs over 1,000,000 tasks every day

44 TIER-0/1s CMS Data Processing on the Grid CMS observes more than 400 unique users submitting jobs per week Capacity run up to 200,000 jobs per day on the Grid TIER-2s TIER-3s

45 CERN Computing hips:// 45

46 The Final Exam 46

47 Final exam: Take-home projects Notes Like a more involved, longer multipart homework assignment Due Thursday May 10: electronic copies by 9:00am hard-copies must be submiied Thursday 10 May between 08:00-10:00 in room 022-C, our computer lab or earlier by personal arrangement with me Due Thursday May 10, electronically by 9:00am and hard copies between 08:00 and 10:00 in 022-C 47

48 Some Guidelines for Final All students must work individually, no collaboration. You can ask questions via or chat or on Piazza but the insight will be limited to conceptual assistance or clarifications. TA hours will be held on Thursday 3 May xxxxx and Friday 4 May yyyyyyy. These will not be standard lab-ta hours: students can come individually to ask questions to the TA but then you will go and work on the questions on your own. Conceptual and clarifications on content only, no assistance in determining the solutions. No examination of code will be allowed. Pose all questions before Monday 8 May 8pm. Allowed resources: class texts, class notes, class web page (including solutions), and web links provided directly from the class web pages. You can use functions available in the standard C libraries we have used in class/lab/homeworks. /chat/Piazza any requests for problem clarifications by Monday, May 7, 8pm. This may be followed by a single FAQ covering appropriate questions. Watch for this . After this, there will be no more discussion during the exam. 48

49 Some Guidelines for Final All students must work individually, no collaboration. You can ask questions via or chat or on Piazza but the insight will be limited to conceptual assistance or clarifications. Start your work early! TA hours will be held on Thursday 3 May xxxxx and Friday 4 May yyyyyyy. These will not be standard lab-ta hours: students can come individually to ask questions to the TA but then you will go and work on the questions on your own. Conceptual and There is an early deadline on clarifications on content only, no assistance in determining the solutions. No examination of code will be allowed. questions to encourage you to start working on it at least thinking about it right away. Pose all questions before Monday 8 May 8pm. Allowed resources: class texts, class notes, class web page (including solutions), and web links provided directly from the class web pages. You can use functions available in the standard C libraries we have used in class/lab/homeworks. /chat/Piazza any requests for problem clarifications by Monday, May 7, 8pm. This may be followed by a single FAQ covering appropriate questions. Watch for this . After this, there will be no more discussion during the exam. 49

50 The End! 50

51 More Testing of Compatibility 51

52 Cumulative Distribution Function 52

53 Cumulative Distribution Function 53

54 Example: PDF and CDF 54

55 Empirical Distribution Function 55

56 Empirical Distribution Function The ECDF is made from unbinned data not from a binned histogram use raw measured values Do this by: 1. say you have N values, x i 2. sort the N values in order of increasing value 3. plot each of the N values with x i on the x-axis and i/ N on the y-axis Now, compare model s CDF and the data s ECDF 56

57 Testing Compatibility 57

58 Testing Compatibility 58

59 Manipulating Binary Data 59

60 Bits and Bytes 60

61 Flipping Bits 61

62 Bitwise Operators 62

63 Bitwise AND 63

64 Bitwise OR 64

65 Bitwise Left Shift 65

66 Bitwise Right Shift 66

67 Bitwise Inverse 67

68 Bitwise Shift on Constants 68

69 Testing and SePing Bits ( ) 69

70 Testing and SePing Bits: Masks Mask ( ) 70

71 Testing and SePing Bits: Masks ( ) 71

72 Clearing a Bit 72

73 Manipulating Binary Data: Example 73

74 Hexadecimal Representation 74

75 Hexadecimal Representation 75

76 Hexadecimal Representation 76

77 Data Storage 77

78 Example 78

79 Example 79

80 Example 80

81 The Exclusive Or (XOR) Operation 81

82 The Exclusive Or (XOR) Operation 82

83 XOR Application 83

84 XOR Application Say you want to send this plain-text message : You and the recipient have a copy of some predetermined key ^ ^ ^ ^ ^ ^ ^ ^ key ^ message = encrypted message: And you send this encrypted message to your intended recipient. 84

85 XOR Application Your recipient receives your encrypted message: and applies the key: ^ ^ ^ ^ ^ ^ ^ ^ and gets a copy of the plain-text message. Successful secret communication! 85

86 XOR Application One flaw: if someone intercepts both the coded and plain message: ^ ^ ^ ^ ^ ^ ^ ^ They can derive the key! This a problem if the sender/recipient ever re-use a key. 86

87 Review: Bitwise Operators 87

88 Review: Testing and SePing Bits 88

89 Storage of Binary Data 89

90 The fputc Function 90

91 The fputc Function 1 byte 4 bytes 1 byte 4 bytes = 4 chars 91

92 Comparison of ASCII and Binary Storage 10 bytes = 10 chars 4 bytes = 4 chars 92

93 More on Characters 93

94 Signed v. Unsigned Vars 94

95 Example: Say Signed vs. Unsigned Vars a = 1 d = b and b = -1 d = b 95

96 Example: Say Signed vs. Unsigned Vars a = 1 d = b and b = -1 d = b Then a+b = 1 d + (-1 d ) =? 96

97 Example: Say Signed vs. Unsigned Vars a = 1 d = b and b = -1 d = b Then a+b = 1 d + (-1 d ) = b b 97

98 Example: Say Signed vs. Unsigned Vars a = 1 d = b and b = -1 d = b Then a+b = 1 d + (-1 d ) = b b = b 98

99 Example: Say Signed vs. Unsigned Vars a = 1 d = b and b = -1 d = b Then a+b = 1 d + (-1 d ) = b b = b = -2 d 99

100 Example: Say Signed vs. Unsigned Vars a = 1 d = b and b = -1 d = b Then a+b = 1 d + (-1 d ) = b b = b = -2 d That looks wrong to me. 100

101 Trick: Two s Complement Notation 101

102 Trick: Two s Complement Notation To form the two s complement of any negative number: 1. Convert number to N-bit representation 2. Subtract number from 2 N 3. Add one to the number Examples, 8-bit negative numbers: If a = -1 then -1 in 8-bit representation is a 2c = ( ) = If a = -5 then a 2c = =

103 Example: Say Signed vs. Unsigned Vars a = 1 d = b and b = -1 d = b Then (using 2 s complement) a+b = 1 d + (-1 d ) = b b = b = 0 d That s beier! 103

104 More On Binary Data Output 104

105 More On Binary Data Output 105

106 More On Binary Data Output 106

107 More On Binary Data Input 107

108 More On Binary Data Output 108

109 More On Binary Data Output 109

110 More On Binary Data Input 110

111 More On Binary Data Input 111

112 fopen: File I/O Modes 112

113 fopen, fwrite and fread 113

114 rewind 114

115 fseek 115

116 fseek 116

117 ftell 117

118 feof 118

119 Writing Binary for Arbitrary Structs 119

120 Filesize Comparison of Binary v. ASCII Formats 120

121 Solving Differential Equations 121

122 A Trajectory 122

123 A Trajectory We don t know v(t) but we know v(t0) and the time derivative. We seek v(t). 123

124 A Trajectory All we know is the derivative of v! Here v=y and t=x. result: not very accurate!!! one step: 124

125 A BePer Method 125

126 A BePer Method All we know is the derivative! 126

127 A BePer Method All we know is the derivative! 127

128 A BePer Method 128

129 A BePer Method 129

130 Passing Pointers to Structures 130

131 Unintended Consequences 131

132 Unintended Consequences Warning tells you to rewrite this! 132

133 Optional Function Arguments 133

134 Default Argument Values in Functions: C++ 134

135 Default Argument Values in Functions: C++ 135

136 Default Argument Values in Functions: C++ Example: 136

137 Searching in Arrays 137

138 Searching for a Particular Item 138

139 Method: Linear Search 139

140 Method: Binary Search 140

141 Method: Binary Search 141

142 Method: Binary Search 142

143 Method: Binary Search 143

144 Method: Binary Search 144

145 Search Speed: Linear Method 145

146 Search Speed: Binary Method 146

147 Sorting 147

148 Sorting Methods 148

149 Method: Selection Sort 149

150 Method: Selection Sort 150

151 Method: Selection Sort 151

152 Method: Quicksort 152

153 Method: Quicksort 153

154 Method: Quicksort 154

155 Method: Quicksort 155

156 Method: Quicksort 156

157 Method: Quicksort 157

158 Linked Lists 158

159 Reminder About Arrays 159

160 Linked Lists: A Special Data Structure 160

161 Linked Lists: A Special Data Structure mallc calloc realloc 161

162 Linked Lists: A Special Data Structure 162

163 Linked Lists: A Special Data Structure 163

164 Linked Lists: A Special Data Structure If these were stored in an array, this could be a huge job. 164

165 Linked Lists: A Special Data Structure If these were stored in an array, this could be a huge job. 165

166 Linked Lists: A Special Data Structure 166

167 Linked Lists: A Special Data Structure 167

168 Linked Lists: A Special Data Structure 168

169 Linked Lists: A Special Data Structure 169

170 Linked Lists: A Special Data Structure 170

171 So What? These things can actually be very useful! 171

172 Linked Lists Example: FIFOs 172

173 Linked Lists: Summary 173

174 Dynamic Memory Allocation: Memory is typically allocated by the compiler but what if we need to change the requirements on the fly while executing a program? (We did not go through this unit in lecture.) 174

175 The Stack 175

176 The Stack Each running program has its own stack As functions are called, the variables local to that function are loaded to the top of the stack. When the function is complete that memory is freed for other uses One can use up all the available memory in a stack: stack overflow error 176

177 The Heap 177

178 The malloc function 178

179 The calloc function 179

180 The free function 180

181 The realloc function 181

182 Memory Allocation Failure 182

183 Dynamic Memory Allocation: Example I 183

184 Dynamic Memory Allocation: Example II 184

185 Dynamic Memory Allocation: Example II 185

186 Optional Function Arguments 186

187 Default Argument Values in Functions: C++ 187

188 Default Argument Values in Functions: C++ 188

189 Default Argument Values in Functions: C++ Example: 189