Guaranteed Bounds for Solution of Parameter Dependent System of - PowerPoint PPT Presentation

Guaranteed Bounds for Solution of Parameter Dependent System of Equations Andrew Pownuk 1 , Iwona Skalna 2 , and Jazmin Quezada 1 1 - The University of Texas at El Paso, El Paso, Texas, USA 2 - AGH University of Science and Technology, Krakow, Poland 22th Joint UTEP/NMSU Workshop on Mathematics, Computer Science, and Computational Sciences 1 / 28

Outline Solution Set 1 Interval Methods 2 Optimization methods 3 New Approach 4 Example 1 5 Example 2 6 Example 3 7 Conclusions 8 2 / 28

Solution of PDE Solution Set Parameter dependent Boundary Value Problem Interval Methods Optimization A ( p ) u = f ( p ) , u ∈ V ( p ) , p ∈ P methods New Approach Exact solution Example 1 Example 2 u = inf p ∈ P u ( p ) , u = sup u ( p ) Example 3 p ∈ P Conclusions u ( x , p ) ∈ [ u ( x ) , u ( x )] Approximate solution u h = inf p ∈ P u h ( p ) , u h = sup u h ( p ) p ∈ P u h ( x , p ) ∈ [ u h ( x ) , u h ( x )] 3 / 28

Mathematical Models in Engineering Solution Set Interval Methods Optimization methods New Approach Example 1 Example 2 Example 3 Conclusions Linear and nonlinear equations. Multiphysics (solid mechanics, fluid mechanics etc.) Ordinary and partial differential equations, variational equations, variational inequalities, numerical methods, programming, visualizations, parallel computing etc. 4 / 28

Two point boundary value problem Solution Set Sample problem Interval � − ( g ( x , p ) u ′ ( x )) ′ = f ( x , p ) Methods Optimization methods u (0) = 0 , u (1) = 0 New Approach and u h ( x ) is finite element approximation given by a weak Example 1 Example 2 formulation Example 3 Conclusions 1 1 � � f ( x , p ) v ( x ) dx , ∀ v ∈ V (0) g ( x , p ) u ′ h ( x ) v ′ ( x ) dx = h 0 0 or a ( u h , v ) = l ( v ) , ∀ v ∈ V (0) ⊂ H 1 0 h n where u h ( x ) = � u i ϕ i ( x ) and ϕ i ( x j ) = δ ij . i =1 5 / 28

The Finite Element Method Solution Set Approximate solution Interval Methods 1 1 Optimization � � methods g ( x , p ) u ′ h ( x ) v ′ ( x ) dx = f ( x , p ) v ( x ) dx New Approach 0 0 Example 1 . Example 2 Example 3 1 1   n n Conclusions � � � �  v j = 0 g ( x , p ) ϕ i ( x ) ϕ j ( x ) dxu i − f ( x , p ) ϕ j ( x ) dx  j =1 i =1 0 0 Final system of equations (for one element) Ku = q where 1 1 � � K i , j = g ( x , p ) ϕ i ( x ) ϕ j ( x ) dx , q i = f ( x , p ) ϕ i ( x ) dx 0 0 6 / 28

Global Stiffness Matrix Solution Set Interval Global stiffness matrix Methods Optimization methods  New Approach n e n e n n n e ∂ϕ e u u g ( x , p ) ∂ϕ e j ( x ) � i ( x ) � � � � � U e dxU e Example 1 i , q u q −  j , p ∂ x ∂ x Example 2 p =1 q =1 e =1 i =1 j =1 Ω e Example 3 Conclusions  n e n e n n e u u � � � � � U e f ( x , p ) ϕ e i ( x ) ϕ e  v p = 0 j ( x ) dx j , p q =1 e =1 i =1 j =1 Ω e Final system of equations K ( p ) u = q ( p ) ⇒ F ( u , p ) = 0 7 / 28

Solution Set Solution Set Interval Methods Nonlinear equation F ( u , p ) = 0 for p ∈ P . Optimization methods F : R n × R m → R n New Approach Example 1 Implicit function u = u ( p ) ⇔ F ( u , p ) = 0 Example 2 Example 3 u ( P ) = { u : F ( u , p ) = 0 , p ∈ P } Conclusions Interval solution u i = min { u : F ( u , p ) = 0 , p ∈ P } u i = max { u : F ( u , p ) = 0 , p ∈ P } 8 / 28

Interval Methods Solution Set Interval Methods A. Neumaier, Interval Methods for Systems of Equations Optimization (Encyclopedia of Mathematics and its Applications, Cambridge methods University Press, 1991. New Approach Example 1 Z. Kulpa, A. Pownuk, and I. Skalna, Analysis of linear Example 2 mechanical structures with uncertainties by means of interval Example 3 methods, Computer Assisted Mechanics and Engineering Conclusions Sciences, 5, 443-477, 1998. V. Kreinovich, A.V.Lakeyev, and S.I. Noskov. Optimal solution of interval linear systems is intractable (NP-hard). Interval Computations, 1993, 1, 6-14. 9 / 28

Interval Methods Solution Set T. Burczynski, J. Skrzypczyk, Fuzzy aspects of the boundary Interval Methods element method, Engineering Analysis with Boundary Optimization Elements, Vol.19, No.3, pp. 209216, 1997 methods New Approach A. Neumaier and A. Pownuk, Linear Systems with Large Example 1 Example 2 Uncertainties, with Applications to Truss Structures, Journal of Example 3 Reliable Computing, 13(2), 149-172, 2007. Conclusions Muhanna, R. L. and R. L. Mullen. Uncertainty in Mechanics ProblemsInterval-Based Approach, Journal of Engineering Mechanics 127(6), 557-566, 2001. I. Skalna, A method for outer interval solution of systems of linear equations depending linearly on interval parameters, Reliable Computing, 12, 2, 107-120, 2006. 10 / 28

Optimization methods Solution Set Interval Methods Optimization Interval solution methods New Approach u i = min { u ( p ) : p ∈ P } = min { u : F ( u , p ) = 0 , p ∈ P } Example 1 Example 2 u i = max { u ( p ) : p ∈ P } = max { u : F ( u , p ) = 0 , p ∈ P } Example 3 Conclusions   min u i max u i   u i = F ( u , p ) = 0 , u i = F ( u , p ) = 0 p ∈ P p ∈ P   11 / 28

KKT Conditions Solution Set Nonlinear optimization problem for f ( x ) = x i Interval Methods  min x f ( x ) Optimization  methods  h ( x ) = 0 New Approach  g ( x ) ≥ 0  Example 1 Example 2 Lagrange function L ( x , λ, µ ) = f ( x ) + λ T h ( x ) − µ T g ( x ) Example 3 Optimality conditions can be solved by the Newton method. Conclusions  ∇ x L = 0   ∇ λ L = 0     µ i ≥ 0  µ i g i ( x ) = 0   h ( x ) = 0     g ( x ) ≥ 0  12 / 28

KKT Conditions - Newton Step Solution Set Interval Methods F ′ ( X )∆ X = − F ( X ) Optimization methods New Approach ∇ 2 x f ( x ) + ∇ 2 � �  x h ( x ) y ∇ x h ( x ) n × m − I n × n  Example 1 n × n ( ∇ x h ( x )) T F ′ ( X ) = Example 2 0 n × m 0 m × n   m × n Example 3 0 n × m Z n × n X m × n Conclusions     ∆ x x  , X = ∆ X = ∆ y y    ∆ z z  ∇ x f ( x ) + ∇ x h T ( x ) y − z  F ( X ) = − h ( x )   XYe − µ k e 13 / 28

Steepest Descent Method Solution Set Interval Methods In order to find maximum/minimum of the function u it is Optimization possible to apply the steepest descent algorithm. methods New Approach 1 Given x 0 , set k = 0. Example 1 2 d k = −∇ f ( x k ). If d k = 0 then stop. Example 2 3 Solve min α f ( x k + α d k ) for the step size α k . If we know Example 3 d T k d k Conclusions second derivative H then α k = k H ( x k ) d k . d T 4 Set x k +1 = x k + α k d k , update k = k + 1. Go to step 1. I. Skalna and A. Pownuk, Global optimization method for computing interval hull solution for parametric linear systems, International Journal of Reliability and Safety, 3, 1/2/3, 235-245, 2009. 14 / 28

New Approach for Finding Guaranteed Bounds Solution Set Interval Methods Optimization methods New Approach Example 1 Example 2 Example 3 Conclusions 15 / 28

New Approach for Finding Guaranteed Bounds Solution Set Interval Methods Optimization methods New Approach Theorem Example 1 Let’s assume that g : P → R is a continuous function, P is a Example 2 Example 3 path-connected, compact subset of R , then Conclusions g ( P ) = { g ( p ) : p ∈ P } = [ g ( p min ) , g ( p max )] = [ x min , x max ] is a closed interval and p min , p max ∈ P, x min = inf { g ( x ) : p ∈ P } , x max = sup { g ( x ) : p ∈ P } . 16 / 28

New Approach for Finding Guaranteed Bounds Solution Set Interval Methods Optimization methods Theorem New Approach Example 1 Let’s assume that g : P → R is a continuous function, P is a Example 2 path-connected, compact subset of R m , we know at least one Example 3 value x 0 = g ( p 0 ) such that p 0 ∈ P, and exists some ε > 0 such Conclusions that x 0 + ∆ x / ∈ g ( P ) for all ∆ x ∈ (0 , ε ] , then x 0 = g ( p max ) = g max is a guaranteed upper bound of the set g ( P ) . 17 / 28

New Approach for Finding Guaranteed Bounds Solution Set Interval Theorem Methods Optimization Let’s assume that g : P → R n is a continuous function that is methods New Approach defined as a unique solution of the equation f ( x , p ) = 0 , P is a Example 1 path-connected, compact subset of R m , we know at least one Example 2 value x 0 = g ( p 0 ) such that p 0 ∈ P, and exists some ε > 0 such Example 3 that x 0 , i + ∆ x i / ∈ g i ( P ) for all ∆ x i ∈ (0 , ε ] , then Conclusions x 0 , i = g i ( p max ) = g i , max is a guaranteed upper bound of the set g i ( P ) = [ g i , min ( P ) , g i , max ( P )] . If the equation f ( x , p ) = 0 has multiple solutions x = g i ( p ) ( i = 1 , ..., s ), then g ( P ) = g 1 ( P ) ∪ g 2 ( P ) ∪ ... ∪ g s ( P ) 18 / 28

Example 1 Solution Set Interval Methods Optimization methods Let’s consider the equation nonlinear equation with uncertain New Approach parameter Example 1 x 2 − 4 p 2 = 0 for p ∈ [1 , 2] . Example 2 Example 3 Presented equation has two solutions x = g 1 ( p ) = 2 p and Conclusions x = g 2 ( p ) = − 2 p . Non-guaranteed solutions are [ x 1 , x 1 ] = g 1 ([1 , 2]) = [2 , 4] and [ x 2 , x 2 ] = g 2 ([1 , 2]) = [ − 4 , − 2]. 19 / 28