A Primer on Mathematical Modelling

✍ Scribed by Quarteroni, Alfio, Gervasio, Paola

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 244
• Series: Unitext volume 121, La Matematica per il 3+2
• Edition: 1
• Category: Library

✦ Synopsis

In this book we describe the magic world of mathematical models: starting from real-life problems, we formulate them in terms of equations, transform equations into algorithms and algorithms into programs to be executed on computers.

A broad variety of examples and exercises illustrate that properly designed models can, e.g.: predict the way the number of dolphins in the Aeolian Sea will change as food availability and fishing activity vary; describe the blood flow in a capillary network; calculate the PageRank of websites.

This book also includes a chapter with an elementary introduction to Octave, an open-source programming language widely used in the scientific community. Octave functions and scripts for dealing with the problems presented in the text can be downloaded from https://paola-gervasio.unibs.it/quarteroni-gervasio
This book is addressed to any student interested in learning how to construct and apply mathematical models.

Paola Gervasio completed her PhD in Mathematics at the University of Milan (Italy) in 1995, and she has been an Associate Professor of Numerical Analysis at the University of Brescia (Italy) since 2005.
She is the author of 4 books and of about 40 papers. Her research focuses on the approximation of partial differential equations by high-order methods and domain decomposition techniques, particularly in the context of multi-physics problems.

Alfio Quarteroni is a Professor of Numerical Analysis at Politecnico of Milan and Professor Emeritus at EPFL of Lausanne. He is the author of 25 books and about 400 papers, and editor of 9 books. He is the recipient of two ERC Advanced Grants; the Galileian Chair from the Scuola Normale Superiore, Pisa; the Galileo international prize for Science; and an honorary doctorate in Naval Engineering from the University of Trieste. He is member of the Italian Academy of Science, the European Academy of Science, the Academia Europaea, and the Lisbon Academy of Science. His research interests include mathematical and numerical modeling for fluid mechanics, geophysics, medicine and the improvement of sports performance. His research group at EPFL carried out the mathematical simulation for the Alinghi sailing boat, which won the America’s Cup in 2003 and 2007.

✦ Table of Contents

Preface......Page 7
Octave Functions and Scripts......Page 8
Contents......Page 10
Main Symbols......Page 13
1 A Warm-Up to Models......Page 14
1.2 Numerical Modelling and Scientific Computing......Page 15
1.3.1 Different Physical Problems, the Same Mathematical Model (a Subtle, Yet Crucial, Difference)......Page 17
1.3.2 One Physical Problem, Many Mathematical Models......Page 19
The Numerical Error......Page 21
The Validation of the Process and the Verification of the Numerical Model......Page 23
What Do Errors Depend On?......Page 24
1.5 There Is Error and Error......Page 25
1.6 What We Have Learnt......Page 27
2 Getting Started......Page 28
2.1 When the Solution to the Mathematical Model Is Not Known......Page 29
How Do We Compute the Area of the Trapezoid T?......Page 30
Divide et impera': Let Us Subdivide the Interval......Page 31<br>Up and Down, Step by Step......Page 32<br>Quadrature Formula at Midpoints, or by Rectangles......Page 33<br>…and Answers......Page 35<br>2.1.3 The Numerical Solution of Problem ``Heights''......Page 36<br>Can We Determine How Many Intervals to Use Without Wasting Time?......Page 37<br>2.2 Vectors and Matrices to Handle Complexity......Page 38<br>Vectors: Orderly Queues......Page 40<br>Matrices, or Close-Order Setting?......Page 41<br>Equality of Vectors......Page 43<br>The Determinant of a 22 Matrix......Page 44<br>2.2.4 The Gauss Elimination Method (GEM) for a 22System......Page 45<br>2.2.5 From 2 to 3......Page 46<br>The Product of an n n Matrix with a Column Vector of Dimension n......Page 47<br>2.2.6 The 3 3 Linear System in Matrix Form......Page 48<br>2.2.7 The Cramer Method for a 33 System......Page 49<br>2.2.8 for Loops in Algorithms......Page 52<br>2.2.9 Variable Assignment in an Algorithm......Page 53<br>2.2.10 The Gauss Elimination Method (GEM) for a 33System......Page 54<br>2.2.11 We Reached 100!......Page 58<br>2.2.12 How Many Operations Must We Do?......Page 59<br>2.2.13 The Computer Comes into Play......Page 60<br>2.2.14 GEM, Too, Adapts to the Computer......Page 61<br>Mathematical Tools......Page 63<br>Numerical Methods......Page 64<br>3 Reckoning with the Computer......Page 65<br>3.1 Programming with Octave......Page 66<br>3.1.1 Arithmetic Operations and Variables......Page 67<br>3.1.2 Mathematical Functions and Their Graphical Representation......Page 69<br>3.1.3 Vectors and Matrices......Page 71<br>Operations with Matrices......Page 74<br>Row Vectors via the Instruction ``:''......Page 76<br>Simple Vector Operations......Page 77<br>Augmenting Arrays (Adding Components to Vectors)......Page 78<br>``Element-by-Element'' Operations......Page 79<br>Printing Instructions......Page 81<br>The Script: A Shopping List of Sorts......Page 83<br>User-Defined Functions or, for Short, Functions......Page 84<br>The for Loop......Page 87<br>Selection Blocks......Page 89<br>Relational Operators......Page 91<br>While Loops......Page 93<br>3.1.6 Tidbits of Graphics......Page 95<br>Graphs in Logarithmic Scale......Page 97<br>3.2 How Computers Represent Numbers......Page 98<br>An Elementary, But Inevitable, Mistake......Page 99<br>3.2.1 Floating-Point Numbers......Page 100<br>3.2.2 Roundoff Errors......Page 102<br>Exact Arithmetic and Floating-Point Arithmetic......Page 104<br>A Recursive Formula for Computing π......Page 105<br>3.4 Script of This Chapter......Page 107<br>4.1 Surfing with Google......Page 108<br>4.2 Understanding the Problem......Page 109<br>Ordering Pages......Page 110<br>A Computer for Searching, Many Computers for Finding......Page 111<br>4.3.1 The Directed Graph as a Model of the Web......Page 112<br>4.3.2 The Adjacency Matrix of a Graph......Page 113<br>The Probability of Clicking on a Link......Page 114<br>``If You Feel You Are in a Black Hole, Don't Give Up. There's a Way Out'' (S. Hawking)......Page 115<br>The Possibility of Changing Path......Page 116<br>It's a Matter of Probability......Page 117<br>Matrices and Vectors to Help Us Model......Page 118<br>4.6 The Numerical Model: How Do We Solve the System p = Gp?......Page 119<br>Is the Milkshake Ready? The Stopping Test......Page 121<br>4.7.1 While Loops in Algorithms......Page 122<br>4.9 A Bit of History and a Glimpse Beyond......Page 125<br>4.10 List of Functions and Scripts of the Chapter......Page 127<br>5 A Network of Capillaries......Page 129<br>5.1 Heart, Arteries, Veins and Capillaries......Page 130<br>It's Easy to SayBlood'......Page 131
5.2.1 The Beat That Keeps Us Alive......Page 133
5.2.2 Fluid Flow and Viscosity......Page 134
Non-Newtonian Fluids......Page 136
5.4 The Model for One Capillary......Page 137
5.5 Blood Flow in a Small Capillary Bed......Page 140
The Adjacency Matrix of a Graph......Page 141
5.5.2 To Compute the Speeds We Need the Pressures......Page 142
5.5.3 To Compute the Pressures We Need the BalanceEquations......Page 143
5.5.4 A System of Balance Equations......Page 144
5.5.5 From the Matrix G of the Graph to the Matrix A of the System......Page 146
5.5.7 From the Mathematical Model to the Solution: GEM......Page 148
5.5.8 Volumetric Flow Rates and Velocities in EachCapillary......Page 149
Computing Pressures......Page 150
After the Volumetric Flow Rates Compute the Average Speeds......Page 151
The Graph......Page 152
The Adjacency Matrix of the Graph......Page 154
Graph → Pressures → Volumetric Flow Rates → Speeds......Page 155
5.8 What Lies Beyond......Page 157
5.9 List of Functions and Scripts of the Chapter......Page 158
6 Predators and Preys in the Maths Ocean......Page 162
6.1 Fishing in the Adriatic Sea......Page 163
6.2 Simplify in Order to Model......Page 164
6.3 Understanding the Problem......Page 166
6.4.1 The Natural Dynamics......Page 167
6.4.2 The Interaction with the Environment: Constant Harvest in Time......Page 171
6.4.3 The Interaction with the Environment: Time-Dependent Harvest (the Model)......Page 173
6.4.4 A Numerical Method to Approximate First Derivatives......Page 174
The Linear Scale and the Logarithmic Scale......Page 178
Reading the Order of Convergence Off the Plot......Page 179
6.4.6 The Euler Method for Approximating a Differential Equation......Page 180
6.4.7 Error of the Euler Method......Page 184
6.4.8 The Interaction With the Environment: Time-Dependent Harvest (the Numerical Solution)......Page 185
6.4.9 An Even More Realistic Model: The Competition Within the Population......Page 188
6.5 The Dynamics of Two Populations......Page 191
6.5.1 The Prey-Predator System (the Model)......Page 192
6.5.2 Numerical Approximation of a System of Differential Equations......Page 193
6.5.3 The Prey-Predator System (the Numerical Solution)......Page 194
6.5.4 An Even More Realistic Model: The Prey-Predator System with Harvest......Page 197
6.5.5 The Prey-Predator Model with Harvest: The Validation......Page 199
The Way to Proceed......Page 200
6.7 Looking Beyond......Page 201
6.8 List of Functions and Scripts of the Chapter......Page 203
7 Take-Home Message......Page 209
8.1 Exercises of Chapter 2......Page 210
8.2 Exercises of Chapter 3......Page 218
8.3 Exercises of Chapter 4......Page 223
8.4 Exercises of Chapter 5......Page 227
8.5 Exercises of Chapter 6......Page 233
Index......Page 241