Slide 1MotivationExampleSlide 4Slide 5ObjectivesA Pipeline of ModelsMathematical Models (Sep. 29, 2010)Example 1Example 2Example 3Example 4Example 5Example 5 (cont.)Slide 15Example 6Example 7Example 8Example 8 (cont.)Simulation ProjectSimulation ProjectNumerical Method - Iteration (Oct. 5, 2010)IterationSlide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Differential Equations (Oct. 7, 2010)Differential EquationsSlide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Differential Equations (continued): Interval and Sub-interval (Oct. 14, 2010)Interval and Sub-intervalSub-intervalSlide 51Slide 52Slide 53The Effects of Sub-intervalSlide 55IntegrationPrimitive IntegrationMathematical notationIntegration: geometric meaningIntegration: Piece-wise ApproachIntegration to IterationExample: f(x)=xSlide 63Slide 64Slide 65Example: f(x)=x2Slide 67IntegrationVerification and Validation (Oct. 19, 2010)Scientific MethodVerification Versus ValidationExampleCharacterization (Step 1)Hypothesis (Step 2)Prediction (Step 3)Slide 76Testing (Step 4)Model 2: Repeat step 2, 3 and 4 with different scientific modelSlide 79Slide 80Slide 81Slide 82Model 3: Repeat step 2, 3 and 4 with different scientific modelSlide 84Slide 85Slide 86Slide 87Slide 88The end (Oct. 21, 2010)CDS 130 - 003Fall, 2010Computing for ScientistsScientific Simulation(Sep. 27, 2010 – Oct. 21, 2010)Jie ZhangCopyright ©Motivation•Scientific simulation allows one to reproduce the details of a complex scientific system, e.g., physical systems, biological systems•It is a new way of scientific research, in additional to traditional experimental and mathematical approaches.ExampleMerger of the Milky Way: http://video.google.com/videoplay?docid=3671579993993423979#The Milky Way and the Andromeda galaxy will likely fall together and merge within a few billion years. In this speculative simulation, the two galaxies flyby one another, exciting tidal tails and bridges and collide on a second pass finally merging after several convulsions. The last remnants of the smashed spirals show up as shells and ripples surrounding a newborn elliptical galaxy.ExampleMany scientific simulations are based on so-called Navier-Stokes Equations (Note: do not worry about the math complexity)http://www.cfd-online.com/Wiki/Navier-Stokes_equationsNavier-Stokes EquationsExampleCellular Blood Flow: http://www.youtube.com/watch?v=o11NDvrZMNs&feature=relatedThe RBCs are modeled as a membrane with hemoglobin inside with a viscosity of 6cP. Each RBC membrane is constructed of linear finite element triangular shells. These shell elements deform due to the fluid-structure interactions with the blood plasma and the hemoglobin.Objectives1. Mathematical Models2. Numerical Methods•Iteration•Differentiation•Integration3. Verification and ValidationA Pipeline of Models1. Domain specialists (e.g., biologists) develop a conceptual representation of the system or a scientific model2. The domain specialists collaborate with mathematicians to develop a mathematical representation, or mathematical model that corresponds to the science model3. The domain specialists and mathematicians work with computational scientists to implement the equations and explore the results using a computer, that is to develop a computational model of the mathematical model.4. The computational methods need to be verified, and the results need to be validated.A common way that science gets doneMathematical Models(Sep. 29, 2010)Example 1scientific model: Every year your bank account balance increases by 20%Mathematical model: B(next-year) = B(this-year) + 0.2 * B(this-year)Or B(next-year) - B(this-year) = 0.2 * B(this-year)Example 2scientific model: Every year your bank account balance increases by 20%. Every year you pay a fee of $100 to the bankMathematical model: B(next-year) = B(this-year) + 0.2 * B(this-year) - 100Example 3scientific model: Every year your bank account balance doublesMathematical model: B(next-year) = 2 * B(this-year)Example 4scientific model: The death rate is 1% and no babies are bornMathematical model: P (next-year) = P(this-year) – 0.01*P(this-year)OR: this year is represented by index I next year is represented by index 2P(2) = P(1) – 0.01*P(1)Example 5scientific model: The rabbit birth rate is 10% and no rabbits dieMathematical model: P (next-year) = P(this-year) + 0.1 * P(this-year)Using index “i”, “i” represents this year. In the context of iteration, “I” represents current yearP(i+1) = P(i) + 0.1*P(i)Example 5 (cont.)Mathematical model: P(i+1) = P(i) + 0.1*P(i)One instance of simulation: Assuming initial population of 100, find the population in the next five year•First year (i=1): P(1) = 100•Second year (i=2): P(2) = P(1) + 0.1*P(1) = 110•Third year (i=3): P(3) = P(2) + 0.1*P(2) = 121•Fourth year (i=4): P(4) = P(3) + 0.1*P(3) = 133•Fifth year (i=5): P(5) = P(4) + 0.1*P(4) = 146Example 5 (cont.)Calculation in Excel>P(1) = 100>P(2) = P(1) + 0.1*P(1) = 110>P(3) = P(2) + 0.1*P(2) = 121>P(4) = P(3) + 0.1*P(3) = 133>P(5) = P(4) + 0.1*P(4) = 146>plot(P,’*’)Calculation in Matlab/OctavExample 6scientific model: The birth rate of rabbits is 10%. The death rate or rabbits is 0.02 times the number of rabbits multiplied by the number of foxes.Mathematical model: R (next-year) = R(this-year) + 0.10*R(this-year) –0.02*R(this-year)*F(this-year)Use “i” represents the current yearR(i+1)=R(i) + 0.10*R(i) – 0.02*R(i)*F(i)Example 7Scientific model: The death rate of foxes is 10%. The birth rate of foxes is 2% of the number of rabbits multiplied by the number of foxes.Mathematical model: F (next-year) = ?Example 8scientific model: •The birth rate of rabbits is 10%. The death rate or rabbits is 0.02 times the number of rabbits multiplied by the number of foxes.•The death rate of foxes is 10%. The birth rate of foxes is 2% of the number of rabbits multiplied by the number of foxes.Mathematical model: R(i+1)=R(i) + 0.10*R(i) – 0.02*R(i)*F(i)F(i+1)=F(i) - 0.10*R(i) + 0.02*R(i)*F(i)Example 8 (cont.)Mathematical model: R(i+1)=R(i) + 0.10*R(i) – 0.02*R(i)*F(i)F(i+1)=F(i) - 0.10*F(i) + 0.02*R(i)*F(i)One instance of simulation: Initial population of rabbits is 100, and that of foxes is 20. What are the population in the second year?>R(1) = 100>F(1) = 50>R(2)=R(1) + 0.10*R(1) – 0.02*R(1)*F(1) = 70>F(2)=F(1) - 0.10*F(1) +