1. Numerical Modeling Ramaz Botchorishvili
Faculty of Exact and Natural Sciences
Tbilisi State University
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2. Questions Given Write down Investigate numerical scheme Write and test
partial differential equations in N dimension
N-1 dimensional surface
Write down
Partial differential equation on N-1 dimensional surface
Numerical scheme on N-1 dimensional surface
Investigate numerical scheme
Write and test
Algorithm
Code
Formulate conclusions
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3. Questions –special cases
Given
Parabolic partial differential equation/ advection equation in 2 space dimension
Curve in a plane , e.g.circle
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4. Questions –special case
Given
Parabolic partial differential equation/ convection equation in 2 space dimension
Curve in a plane, e.g.circle
Parabolic partial differential equation/ convection equation in 3 space dimension
Surface in a 3D space, e.g. sphere
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5. Questions –special case
Given
Parabolic partial differential equation/ convection equation in 2 space dimension
Curve in a plane , e.g.circle
Parabolic partial differential equation/ convection equation in 3 space dimension
Surface in a 3D space , e.g. sphere
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6. Modeling Simplify Represent Translate Simulate Interpret
Real World Problem
Working
Model
Mathematical/
Behavioral
Computational Model
Results/
Conclusions
Simplify
Represent
Translate
Simulate
Interpret
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10/14/2019

7. Modeling Simplify Represent Translate Simulate Interpret
Real World Problem
Working
Model
Mathematical/
Behavioral
Computational Model
Results/
Conclusions
Simplify
Represent
Translate
Simulate
Interpret
Numerical Modeling
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10/14/2019

8. Modeling Simplify Represent Translate Simulate Interpret
Real World Problem
Working
Model
Mathematical/
Behavioral
Computational Model
Results/
Conclusions
Simplify
Represent
Translate
Simulate
Interpret
Numerical Modeling
Numerical “scheme”
Algorithm
GGSWBS12
code
10/14/2019

9. Modeling Simplify Represent Translate Simulate Interpret
Real World Problem
Working
Model
Mathematical/
Behavioral
Computational Model
Results/
Conclusions
Simplify
Represent
Translate
Simulate
Interpret
Numerical Modeling
Run software
Test scheme
Verify algorithm
Numerical “scheme”
Algorithm
GGSWBS12
code
10/14/2019

10. Modeling Simplify Represent Translate Simulate Interpret
Real World Problem
Working
Model
Mathematical/
Behavioral
Computational Model
Results/
Conclusions
Simplify
Represent
Translate
Simulate
Interpret
Numerical Modeling
Run software
Test scheme
Verify algorithm
Numerical “scheme”
Algorithm
GGSWBS12
code
10/14/2019

11. Computer vs Algorithm Powerful computers are important
Fast and accurate algorithms are important
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12. Computer vs Algorithm
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13. Computer vs Algorithm Poisson–like equation Computer power
1980
computers
2000
algorithms
Poisson–like
equation
Computer power
multigrid
200,000 sec
50 sec
adaptivity
coupling
10 sec
0,01 sec
Fast algorithm
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14. Black box numerical models
Software available
Free
Paid
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15. Black box numerical models
Software available
Free
Paid
Are they useful?
Yes, if used with knowledge of key features
No, if used without knowledge of key features
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16. Black box numerical models
Software available
Free
Paid
Are they useful?
Yes, if used with knowledge of key features
No, if used without knowledge of key features
Some useful key features
Number format, computer arithmetic, number of operations
Condition number, sources of errors, accuracy
Stability, approximation, convergence
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17. Numbers Set of integer or real numbers – infinite
Set of numbers in computer – finite
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18. Numbers Set of integer or real numbers – infinite
Set of numbers in computer – finite
Integer
8 bits – 0,1,..,255; or ;
16 bits – 0,1, …, 65535; or ;
32 bits – 0,1, …, ; or ;
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19. Numbers Set of integer or real numbers – infinite
Set of numbers in computer – finite
Integer
8 bits – 0,1,..,255; or ;
16 bits – 0,1, …, 65535; or ;
32 bits – 0,1, …, ; or ;
Float
32 bits
64 bits
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20. Numbers do not fall in theses sets
Set of integer or real numbers – infinite
Set of numbers in computer – finite
Integer
8 bits – 0,1,..,255; or ;
16 bits – 0,1, …, 65535; or ;
32 bits – 0,1, …, ; or ;
Float
32 bits
64 bits
Source of errors:
Numbers do not fall in theses sets
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21. Numerical disaster – Ariane 5
Cost – 7 billion
EXPLOSION – 04/06/1994
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22. Numerical disaster – Ariane 5
Cost – 7 billion
EXPLOSION – 04/06/1994
The failure of the Ariane 501 was caused by the complete loss of guidance and attitude information 37 seconds after start of the main engine ignition sequence (30 seconds after lift-off). This loss of information was due to specification and design errors in the software of the inertial reference system.
The internal SRI* software exception was caused during execution of a data conversion from 64-bit floating point to 16-bit signed integer value. The floating point number which was converted had a value greater than what could be represented by a 16-bit signed integer.
Source: Kees Vuik
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23. A + B = B + A ? – NO!
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24. A + B = B + A ? – NO! + = + Example: Source: R.Hiptmair, R.Jeltsch
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25. Some easy problems are difficult for computer
Ax=b
Axe=be
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26. Some easy problems are difficult for computer
Ax=b
Single precision
e=2-127, n=258,
xe,1= = 2129
Double precision
e=2-1022,n=2049,
xe,1= = 21025
Axe=be
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27. Some easy problems are difficult for computer
Machine zero can cause
Errors equal to machine infinity
Ax=b
Single precision
e=2-127, n=258,
xe,1= = 2129
Double precision
e=2-1022,n=2049,
xe,1= = 21025
Axe=be
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10/14/2019

28. Well-posedness & Condition number
Problem is well posed
if it admits unique solution which continuously depends on input data
Otherwise a problem is ill posed
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29. Well-posedness & Condition number
Problem is well posed
if it admits unique solution which continuously depends on input data
Otherwise a problem is ill posed
Example: linear system Ax=b
Det(A)= 0 – ill posed problem
Otherwise – well posed
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30. Well-posedness & Condition number
Problem is well posed
if it admits unique solution which continuously depends on input data
Otherwise a problem is ill posed
Example: linear system Ax=b
Det(A)= 0 – ill posed problem
Otherwise – well posed
Condition number characterizes continuous dependence of solution on initial data
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31. Well-posedness & Condition number
Problem is well posed
if it admits unique solution which continuously depends on input data
Otherwise a problem is ill posed
Example: linear system Ax=b
Det(A)= 0 – ill posed problem
Otherwise – well posed
Condition number characterizes continuous dependence of solution on initial data
Condition Number of a matrix
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32. Solving linear systems of equations
Ax=b, det(A)/=0
Two large classes of methods
direct methods
iterative methods
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33. Solving linear systems of equations
Ax=b, det(A)/=0
Two large classes of methods
direct methods
iterative methods
Exact solution is obtained in finite number of arithmetic operations
Exact solution is obtained in infinite number of arithmetic operations
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34. Solving linear systems of equations
Ax=b, det(A)/=0
Two large classes of methods
direct methods
Gaussian elimination, Thomas algorithm, square root method..
iterative methods
Jacobi method, Gaus-Seidel method, Relaxation, variational methods…
Exact solution is obtained in finite number of arithmetic operations
Exact solution is obtained in infinite number of arithmetic operations
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35. Iterative solvers One step methods, general form Consistent with Ax=b
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36. Iterative solvers One step methods, general form Number of iteration
Approximate solution on k+1-th iteration
Consistent with Ax=b
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37. Iterative solvers One step methods, general form Number of iteration
Approximate solution on k+1-th iteration
Consistent with Ax=b
Parameter of iteration
τ= τk-nonstationary, otherwise stationary method
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38. Iterative solvers One step methods, general form Number of iteration
Approximate solution on k+1-th iteration
Consistent with Ax=b
P=I – explicit method
Otherwise - implicit method
Parameter of iteration
τ= τk-nonstationary, otherwise stationary method
Preconditioner
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39. Jacobi iterations
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40. Jacobi method with relaxation
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41. Gauss-Seidel iterations
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42. Successive over relaxation
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43. Convergence
Jacobi and Gauss-Seidel methods converge if matrix A has diagonal dominance in rows
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44. Convergence
Jacobi and Gauss-Seidel methods converge if matrix A has diagonal dominance in rows
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45. Convergence
Jacobi and Gauss-Seidel methods converge if matrix A has diagonal dominance in rows
Sufficient condition of convergence
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46. Convergence
Jacobi and Gauss-Seidel methods converge if matrix A has diagonal dominance in rows
Sufficient condition of convergence
Necessary and sufficient condition of convergence
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47. Stopping criteria
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48. Interpolation
Nodal points, function values
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49. Interpolation Basis functions, interpolant
Nodal points, function values
Basis functions, interpolant
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50. ? Interpolation Basis functions, interpolant
Nodal points, function values
Basis functions, interpolant ?
Must be satisfied
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51. Lagrange interpolation
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52. Newton interpolation
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53. Error estimate Convergence and divergence Faber’s Theorem
Marcienkevich’s Theorem
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54. Other approaches Tchebishev Hermite Splines Piecevise Rational Pade
Trigonometric
Least squares …
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55. Numerical differentiation
Simplest approach:
Interpolate function values
Compute derivate in a given point
Finite differences
Pade
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56. Numerical differentiation
Pade 4th and 6th order
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57. First order scalar equation
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58. First order scalar equation
First order accurate
a>0 – stable, a<0 - unstable
Courant-Friedrichs-Levy condition
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59. First order scalar equation
Exact solution in numerical scheme
Truncation error
Lax’s Equivalence theorem: stability + approximation/truncation error = convergence
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60. First order scalar equation
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61. First order scalar equation
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62. First order scalar equation
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63. First order scalar equation
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64. First order scalar equation
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65. First order scalar equation
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66. Questions –special case
Given
Parabolic partial differential equation/ convection equation in 2 space dimension
Curve in a plane, e.g.circle
Parabolic partial differential equation/ convection equation in 3 space dimension
Surface in a 3D space, e.g. sphere
Write down
Partial differential equation on a curve or surface
Numerical scheme for equation on a curve or surface
Investigate numerical scheme:
Approximation
Stability
Convergence
Write and test
Algorithm
Code
Conclusions
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67. Thank you for your attention
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