Andrzej Pownuk The University of Texas at El Paso - - PowerPoint PPT Presentation

Automated Solution of Equations with Uncertain Parameters

Andrzej Pownuk The University of Texas at El Paso http://andrzej.pownuk.com

http://andrzej.pownuk.com 1

Outline of the presentation

 Equations with the uncertain parameters

and their applications

 New approach for the solution of the

equations with the interval parameters

 Generalizations and conclusions

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Mathematical model of a machine

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

      

   

                                             

  

Such simulations are possible since early 1970s O.C. Zienkiewicz, Ivo M. Babuška, P.G. Ciarlet, P. Solin,…

Mathematical models

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physical problem mathematical models experiments predictions experimental results  expensive cheap

Truss structure with uncertain forces

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1 2 3 4 5 6 7 8 9 10 11 12 13

1

P

2

P

3

P

14 15 L L L L L

Perturbated forces

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P P P   

No 1 2 3 4 5 6 7 8 ERROR % 10 9,998586 10,00184 10,00126 60,18381 11,67825 9,998955 31,8762 No 9 10 11 12 13 14 15 ERROR % 10,00126 11,67825 60,18381 9,998955 10,00184 10 9,998586

5% uncertainty

1 2 3 4 5 6 7 8 9 10 11 12 13

1

P

2

P

3

P

14 15 L L L L L

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                                       

  

Vibrations of beams

Vibrations of beams

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vibration.mpeg

Vibrations with uncertain parameters

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vibrations-uncertainty.mpeg

Random vibrations

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vibrations-random.mpeg

Interval displacements

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vibrations-interval.mpeg

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                                                                      

Dynamics of plates

Dynamics of plates

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plate-vibrations.mpeg

Uncertain solution

 Set-valued parameters

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( , , ) { ( , , ): } u x t u x t p p   p p ( , , ) [ ( , ), ( , )] u x t u x t u x t  p

Uncertain solution

 Set-valued parameters

and the optimization methods

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( , ) min{ ( , , ): } u x t u x t p p  p ( , ) max{ ( , , ): } u x t u x t p p  p

Uncertain solution

 Set-valued parameters

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( , ) ( , , )

min

u x t u x t p 

min

argmin ( , , ) argmax ( , , ) , . . . .

p p max

u x t p u x t p p p s t p s t p               p p

max

( , ) ( , , ) u x t u x t p 

Example

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                       

Example

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                 

Probabilistic solution

 Random parameters

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 

: ( , ): ( , , ( ))

n m

p p u x t u x R p R t            

Interval and probabilistic solution

 Probabilistic interpretation

• f the interval solution.

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 

   

Set-valued and probabilistic solution

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 

: ( , , ( )) [ ( , ), ( , )], 1 P u x y p u x t u x t   



  

Interval and probabilistic solution

 Width of the solution

 Interval solution (worst case analysis)  Probabilistic solution

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 

 

Taylor method

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   

    



 

    



 

    



Graphical representation of the solution

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 

 

     

                                    

System of linear interval equations

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    

Solution set in 3D

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                                                                     

Solution set in 3D

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   

                             

Uncertainty

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  

     

Problem with real parameters Problem with interval parameters

         

29

[1,2] [1,4]  x [1,2]  x

because

[1,2] [1,2] [1,4]  

Algebraic Solution

United Solution Set

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[1,2] [1,4] x 

   

1,4 1 ,4 1,2 2         x

because

{ : , [1,2], [1,4]} x ax b a b     x

Comparison of the solution sets

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[1,2] [1,4] x 

   

1,4 1 [1,2] ,4 1,2 2           x x

United Solution Set Algebraic Solution There are many ways how it is possible to extend equations with the real parameters into equations with the interval parameters.

Stochastic differential equations

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      

Interval equation

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           

Practical applications

 Presented approach can be applied for

the solution of practical engineering problems.

 It is possible to solve nonlinear and large

scale problems.

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Post-processing

 PDE  Set-valued solution  Post-processing

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                        

Post-processing

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                                    

            

General interval FEM

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http://andrzej.pownuk.com/php/FEM2/

Automatically generated test problems

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http://webapp.math.utep.edu/Pages/IntervalFEMExamples.htm

Automatically generated test problems

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DSL

(Domain Specific Languages)

Slightly compressible flow equations

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t p B c V q y y p B k A y x x p B k A x

• c

b sc y y c x x c

                                    

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Slightly compressible flow

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Slightly compressible flow

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Slightly compressible flow

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Slightly compressible flow

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Input Parameters

2D elasticity problem with the interval parameters

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Model Solution Mathematical model

Montonicity of the solution

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   

1 2 1 2

, ,..., , , ,...,

m m

u u p p p u u p p p  

p p

 

u u p 

 

u u p 

 

u u p 

Monotonicity of the solution

 Monotone solution  Non-monotone solution

                        

  

 

  

Interval solution for monotone functions (gradient descent method)

  

If then

 

  

If then

 

 

Interval solution Verification of the results by using search method with 3 intermediate points

Interval solution

Interval solution

Interval solution

Interval solution

Interval vibrations and monotonicity

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      

Uncertain solution

 Set-valued parameters

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( , ) ( , , )

min

u x t u x t p 

min

argmin ( , , ) argmax ( , , ) , . . . .

p p max

u x t p u x t p p p s t p s t p               p p

max

( , ) ( , , ) u x t u x t p 

Vibrations of beam

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 

min

u u p 

min

p

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(2) (7)

u u 

Vibrations of beam

Hermitte approximation

First order approximation

Data for the calculations

Adaptivity

Example (step 1 – 1 solution)

Example (step 2 – 3 solutions)

Example (step 3 – 5 solutions)

Adaptive Taylor series

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http://webapp.math.utep.edu/AdaptiveTaylorSeries-1.1/

Adaptive Taylor series

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Epistemic uncertainty

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This is a horse. Is this a horse?

 

H – set of horses

Fuzzy sets

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 

 

 

 

 

 

Fuzzy ≠ Probability H – set of horses

 

 

 

 

Fuzzy concept of safety

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 

  

   

Problems with binary logic

 Is it possible to find in the real world

statements which are absolutely true?

(L. Wittgenstein, Tractatus Logico-Philosophicus, Annalen der Naturphilosophie, 14, 1921)

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

Modus ponens can be applied if and are true.



When modus ponens can be applied?

Tools

 Approach with tools

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• Approach without tools

5 years of training Final result

Mathematical tools

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Mathematica Matlab Octave Etc.

Example: http://www.wolframalpha.com

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It is possible to calculate not only the result but also intermediate steps in the calculations

Mathematics and programming

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Mathematics Programming

mathematical method program results results

Main problem

 At this moment it is not possible perform

general mathematical research automatically without human input.

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mathematical method results YES NO

Science

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Experiment Theory Mathematical model Simulations (predictions) Scientific hypothesis

HPC computing

New tool

 Self adaptive computational methods

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Symbolic calculations

 Interval arithmetic example  2*[1,2]*x-x=1  (2*[1,2]-1)x=1  ([2,4]-1)x=1  [1,3]x=1  x=1/[1,3]

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It is possible to solve more complicated equations and get justification of each step of the calculations.

Realistic example

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• uble( 40 ) double( 500 ) ) ) ) ( x1 + y1 - sin( double( 40 ) double( 500 ) ) ) ) + ( z + y / z ( z + y / z ( x1 + y1 - sin( double( 40 ) double( 500 ) ) ) ) ( x1 + y1 - sin( double( 40 ) double( 500 ) ) ) ) y double( 2 ) x ( z + y / z ( z + y / z ( x1 + y1 - sin( double( 40 ) double( 500 ) ) ) ) ( x1 + y1 - sin( double( 40 )

• sin( double( 40 ) double( 500 ) ) ) ) ( x1 + y1 - sin( double( 40 ) double( 500

Automatically generated equations

• f the truss structures

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

symbolic calculations

 

 

       

Efficiency of rewriting systems

 In average scientist can write less than

10 pages per day.

 Automated systems can generate

1 000 000 pages in 1 hour.

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http://andrzej.pownuk.com/publications/test-0-5000.docx (part of 1 000 000 pages document)

Example output

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appendix-2-truss-11-bar-551p.pdf

Equations with the interval parameters

 Automatically generated examples.  Automatically generated methods of

solution.

 Automatic research on how these

methods are related.

 Automatically generated reports.  How many examples? Thousands,

millions … … as many as you want.

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Automated theorem proving

 The method generate not only the final

result of the calculations but also all intermediate steps of the calculations.

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?

Example application

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

symbolic calculations

 

 

       

Example application

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Example applications

 Automated reasoning

• n Sobolev Spaces, group theory etc.

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Computational creativity

 Depending on the amount of background

information and the context in which this background information is applied, it is possible to get new scientific conclusions with complete justification (proofs).

 How

many proofs/theorems … thousands, millions ... as many as you want.

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Science today

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Science tomorrow

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Control

HPC Computing

 Background algorithms are

embarrassingly parallel and can be significantly speed up by using HPC computing.

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HPC Computing

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Total amount of cores from all Top500 supercomputers (June 2011) is 7 779 924. There are between 900 million and one billion personal computers in the world right now.

Interdisciplinary science

 The method can be applied in any

scientific areas which can be described by abstract mathematical concepts.

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Chemistry Biology Physics

Interdisciplinary science

 The method can be applied in any

scientific areas which can be described by abstract mathematical concepts.

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Chemistry Biology Physics Mathematics is the queen of sciences

“Chain reaction” of knowledge

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Control Self-improvement

“Chain reaction” of knowledge

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Control Self-improvement According to my research very little amount of background knowledge may generate very big data set of conclusions. New conclusions may lead to even bigger amounts of new

• knowledge. This process can be continued practically forever.

What kind of language?

 C, C++, FORTRAN  ALGOL, R, Matlab, Mathematica,

FORTH, Cobol, C#, F#, Scala, Lisp, Java, Assembler, Miranda, OCaml, Perl, Prolog, Objective-C, Pascal, PHP, HTML, ASP.NET etc.

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Mathematics - The Language of Science

 “Philosophy is written in this grand book, the universe which

stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.” Galileo Galilee in Assayer

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What kind of language?

 Mathematics can be treated as a

programming language

 Any text written in a natural or artificial

language can be treated as a programming language

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What kind of problems it is possible to investigate by using this tool?

 Almost all currently known mathematical areas

(at this moment there are problems with geometry but … these problems are solvable).

 Important remark

Reserch of the problem does not always lead to solution of that problem.

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How it is possible to get background knowledge for the system?

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You are providing an input

Mathematics is a science

• f abstract concepts

More specific examples …

 All examples in this presentation …

and almost anything else …

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Platonism

 It is possible to say that in the framework

• f SelfNet system mathematical ideas

exist outside of human brain.

 Abstract ideas may exist as a part of

computer program.

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Limitation of human brain

 It takes 20 years of training to get MS in

Mathematics.

 Computer system can process complex

problems after several minutes

• f

training.

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but …it is hard to compete with the flexibility of human brain

Automated Science

 After retirement many scientific ideas are

forgotten.

 Once the idea is added to the system it

will never be forgotten and it can improve itself without interactions with humans.

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Thank you

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Thank you very much