Which Method for Solution of the System of Interval Equations Should - PowerPoint PPT Presentation

Which Method for Solution of the System of Interval Equations Should we Choose? A. Pownuk 1 , J. Quezada 1 , I. Skalna 2 , M.V. Rama Rao 3 , A. Belina 4 1 The University of Texas at El Paso, El Paso, Texas, USA 2 AGH University of Science and Technology, Krakow, Poland 3 Vasavi College of Engineering, Hyderabad, India 4 Silesian University of Technology, Gliwice, Poland 21th Joint UTEP/NMSU Workshop on Mathematics, Computer Science, and Computational Sciences 1 / 31

Outline Solution Set 1 Optimization methods 2 Other Methods 3 Interval Methods 4 Comparison 5 Conclusions 6 2 / 31

Solution of PDE Solution Set Parameter dependent Boundary Value Problem Optimization methods Other A ( p ) u = f ( p ) , u ∈ V ( p ) , p ∈ P Methods Interval Methods Exact solution Comparison u = inf p ∈ P u ( p ) , u = sup u ( p ) Conclusions p ∈ P 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 / 31

Mathematical Models in Engineering Solution Set Optimization methods Other Methods Interval Methods Comparison Conclusions High dimension n > 10000. 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 / 31

Two point boundary value problem Solution Set Sample problem Optimization methods � − ( a ( x ) u ′ ( x )) = f ( x ) Other Methods u (0) = 0 , u (1) = 0 Interval Methods and u h ( x ) is finite element approximation given by a weak Comparison formulation Conclusions � 1 � 1 f ( x ) v ( x ) dx , ∀ v ∈ V (0) a ( x ) 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 / 31

The Finite Element Method Solution Set 1 1 � � Optimization a ( x ) u ′ h ( x ) v ′ ( x ) dx = Approximate solution f ( x ) v ( x ) dx . methods 0 0 Other Methods   Interval � 1 � 1 n n � � Methods   v j = 0 a ( x ) ϕ i ( x ) ϕ j ( x ) dxu i − f ( x ) ϕ j ( x ) dx Comparison j =1 i =1 Conclusions 0 0 Final system of equations (for one element) Ku = q where � 1 � 1 K i , j = a ( x ) ϕ i ( x ) ϕ j ( x ) dx , q i = f ( x ) ϕ i ( x ) dx 0 0 Calculations of the local stiffness matrices can be done in parallel. 6 / 31

Global Stiffness Matrix Solution Set Global stiffness matrix Optimization methods Other  Methods � n e n e n n n e � � � � � ∂ϕ e u u a ( x ) ∂ϕ e j ( x ) i ( x ) Interval  U e dxU e i , q u q − Methods j , p ∂ x ∂ x p =1 q =1 e =1 i =1 j =1 Comparison Ω e Conclusions  � n e n e n n e � � � � u u U e f ( x ) 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 Computations of the global stiffness matrix can be done in parallel. 7 / 31

Solution Set Solution Set Optimization methods Nonlinear equation F ( u , p ) = 0 for p ∈ P . Other Methods F : R n × R m → R n Interval Methods Comparison Implicit function u = u ( p ) ⇔ F ( u , p ) = 0 Conclusions u ( P ) = { u : F ( u , p ) = 0 , p ∈ P } Interval solution u i = min { u : F ( u , p ) = 0 , p ∈ P } u i = max { u : F ( u , p ) = 0 , p ∈ P } 8 / 31

Interval Methods Solution Set Optimization methods A. Neumaier, Interval Methods for Systems of Equations Other Methods (Encyclopedia of Mathematics and its Applications, Cambridge Interval University Press, 1991. Methods Comparison Z. Kulpa, A. Pownuk, and I. Skalna, Analysis of linear Conclusions mechanical structures with uncertainties by means of interval methods, Computer Assisted Mechanics and Engineering 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 / 31

Optimization methods Solution Set Optimization methods Other Interval solution Methods Interval Methods u i = min { u ( p ) : p ∈ P } = min { u : F ( u , p ) = 0 , p ∈ P } Comparison Conclusions u i = max { u ( p ) : p ∈ P } = max { u : F ( u , p ) = 0 , p ∈ P }   min u i max u i   u i = F ( u , p ) = 0 , u i = F ( u , p ) = 0   p ∈ P p ∈ P 10 / 31

KKT Conditions Solution Set Nonlinear optimization problem for f ( x ) = x i Optimization methods   min x f ( x ) Other  Methods h ( x ) = 0 Interval   Methods g ( x ) ≥ 0 Comparison Conclusions Lagrange function L ( x , λ, µ ) = f ( x ) + λ T h ( x ) − µ T g ( x ) Optimality conditions can be solved by the Newton method.  ∇ x L = 0     ∇ λ L = 0    µ i ≥ 0 µ i g i ( x ) = 0      h ( x ) = 0   g ( x ) ≥ 0 11 / 31

KKT Conditions - Newton Step Solution Set Optimization methods F ′ ( X )∆ X = − F ( X ) Other Methods Interval � �   Methods ∇ 2 x f ( x ) + ∇ 2 x h ( x ) y ∇ x h ( x ) n × m − I n × n n × n Comparison F ′ ( X ) =  ( ∇ x h ( x )) T  0 n × m 0 m × n Conclusions m × n 0 n × m Z n × n X m × n     ∆ 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 12 / 31

Steepest Descent Method Solution Set Optimization methods In order to find maximum/minimum of the function u it is Other possible to apply the steepest descent algorithm. Methods Interval 1 Given x 0 , set k = 0. Methods 2 d k = −∇ f ( x k ). If d k = 0 then stop. Comparison Conclusions 3 Solve min α f ( x k + α d k ) for the step size α k . If we know d T k d k 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. 13 / 31

The Gradient Solution Set Optimization methods Other Methods After discretization Interval Methods Ku = q Comparison Calculation of the gradient Conclusions ∂ ∂ Kv = q − Ku ∂ p k ∂ p k ∂ where v = ∂ p k u . 14 / 31

Gradient Method and Sensitivity Analysis Solution Set Optimization methods A. Pownuk, Numerical solutions of fuzzy partial differential Other Methods equation and its application in computational mechanics, in: Interval M. Nikravesh, L. Zadeh and V. Korotkikh, (eds.), Fuzzy Partial Methods Differential Equations and Relational Equations: Reservoir Comparison Characterization and Modeling, Physica-Verlag, 308-347, 2004. Conclusions Postprocessing of the interval solution. ε = Cu σ = D ε 15 / 31

Linearization Solution Set Optimization methods Other Methods ∆ f ( x ) = f ( x + ∆ x ) − f ( x ) ≈ f ′ ( x )∆ x Interval Methods Derivative can be calculated numerically. Comparison Conclusions f ′ ( x ) ≈ f ( x + h ) − f ( x ) h The method can be used together with incremental formulation of the Finite Element Method. K ( p )∆ u = ∆ Q ( p ) 16 / 31

Monte Carlo Simulation/Search Method Solution Set Optimization Monte Carlo Method (inner approximation of the solution set) methods Other Methods Interval u ( P ) ≈ Hull ( { u : K ( p ) u = Q ( p ) , p ∈ { random values from P }} ) Methods Comparison Search Method. P ≈ { special points } Conclusions u ( P ) ≈ Hull ( { u : K ( p ) u = Q ( p ) , p ∈ { special points }} ) Vertex Method u ( P ) ≈ Hull ( { u : K ( p ) u = Q ( p ) , p ∈ { set of vertices }} ) 17 / 31

Cauchy Based Monte Carlo Simulation Solution Set Optimization ρ ∆ ( x ) = ∆ 1 methods π · 1 + x 2 / ∆ 2 . Other Methods when ∆ x i ∼ ρ ∆ i ( x ) are indep., then Interval � n � n Methods ∆ y = c i · ∆ x i ∼ ρ ∆ ( x ), with ∆ = | c i | · ∆ i . Comparison i =1 i =1 Conclusions Thus, we simulate ∆ x ( k ) ∼ ρ ∆ i ( x ); then, i ∆ y ( k ) def x 1 − ∆ x ( k ) = � y − f ( � 1 , . . . ) ∼ ρ ∆ ( x ). Maximum Likelihood method can estimate ∆: � N � N 1 + (∆ y ( k ) ) 2 / ∆ 2 = N 1 ρ ∆ (∆ y ( k ) ) → max, so 2 . k =1 k =1 To find ∆ from this equation, we can use, e.g., the bisection 1 ≤ k ≤ N | ∆ y ( k ) | . method for ∆ = 0 and ∆ = max 18 / 31

Theory of perturbations Solution Set Optimization methods J. Skrzypczyk1, A. Belina, FEM ANALYSIS OF UNCERTAIN Other Methods SYSTEMS WITH SMALL GP-FUZZY TRIANGULAR Interval PERTURBATIONS, Proceedings of the 13th International Methods Conference on New Trends in Statics and Dynamics of Comparison Buildings October 15-16, 2015 Bratislava, Slovakia Faculty of Conclusions Civil Engineering STU Bratislava Slovak Society of Mechanics SAS A = A 0 + ε 1 A 1 + ε 2 A 2 + ... J.D. Cole, Perturbation methods in applied mathematics, Bialsdell, 1968. 19 / 31

Interval Boundary Element Method Solution Set Optimization methods Other Methods Interval T. Burczynski, J. Skrzypczyk, Fuzzy aspects of the boundary Methods element method, Engineering Analysis with Boundary Comparison Elements, Vol.19, No.3, pp. 209216, 1997 Conclusions � � � G ∂ u ∂ n − ∂ G cu = ∂ n u dS ∂ Ω 20 / 31