A Posteriori Error Bounds for Two Point Boundary Value Problem with - PowerPoint PPT Presentation

A Posteriori Error Bounds for Two Point Boundary Value Problem with Uncertain Parameters Andrew Pownuk Jazmin Quezada The University of Texas at El Paso Sixteenth New Mexico Analysis Seminar, Department of Mathematical Sciences, New Mexico State University, Las Cruces, May 21, 2017. 1 / 29

Outline Errors in numerical calculations 1 Uncertain Parameters 2 Error estimation 3 Computational method 4 Linearization-Based Algorithm 5 Conclustions 6 2 / 29

Errors in numerical calculations Errors in numerical Boundary value problem. calculations Uncertain Parameters L ( u ) = f , u ∈ V Error estimation u - exact solution, u h - approximate solution. Computational Approximation error � u − u h � = � e � . method Parameter dependent boundary value problem. Linearization- Based Algorithm L ( u , p ) = f , u ∈ V Conclustions u ( p ) - parameter dependent exact solution, u h ( p ) - parameter dependent approximate solution. Maximal approximation error sup � u ( p ) − u h ( p ) � E = sup � e ( p ) � E = � e � E p ∈ P p ∈ P . 3 / 29

Extreme values of the solution Errors in numerical Parameter dependent boundary value problem. calculations Uncertain Parameters L ( u , p ) = f , u ∈ V Error estimation Exact solution Computational method u = inf p ∈ P u ( p ) , u = sup u ( p ) Linearization- Based p ∈ P Algorithm Conclustions 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 )] 4 / 29

Interval parameters (worst case analysis) Errors in numerical calculations Uncertain Parameters Solution of the equation with interval parameters for given x Error can be defined as the following set: estimation Computational method [ u ( x ) , u ( x )] = Linearization- Based = ⋄{ u ( x , p 1 , ..., p m ) : p 1 ∈ [ p 1 , p 1 ] , ..., p m ∈ [ p m , p m ] } Algorithm Conclustions where [ p 1 , p 1 ] , ..., [ p m , p m ] are interval parameters (for example E , A , n etc.) and ⋄ B is the smallest interval that contains the set B . In presented example uncertain parametrs may be E , n , L etc. 5 / 29

Steepest Descent Method Errors in numerical calculations Uncertain Parameters In order to find maximum/minimum of the function u it is Error possible to apply a modified version of the steepest descent estimation algorithm. Computational method 1 Given x 0 , set k = 0. Linearization- Based 2 d k = −∇ f ( x k ). If d k = 0 then stop. Algorithm Conclustions 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. 6 / 29

Two point boundary value problem Errors in numerical Sample problem calculations � − ( a ( x ) u ′ ( x )) = f ( x ) Uncertain Parameters u (0) = 0 , u (1) = 0 Error estimation Computational and u h ( x ) is finite element approximation given by a weak method formulation Linearization- Based Algorithm � 1 � 1 Conclustions 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 7 / 29

Example Errors in numerical Tension-compression problem calculations � − ( E ( x ) A ( x ) u ′ ( x )) ′ = n ( x ) Uncertain Parameters u (0) = 0 , u ( L ) = 0 Error estimation Computational E is a Young modulus and A is an area of cross-section. method u h ( x ) is finite element approximation given by a weak Linearization- Based formulation. Algorithm Conclustions � L � L n ( x ) v ( x ) dx , ∀ v ∈ V (0) E ( x ) 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 8 / 29

The Finite Element Method Errors in Weak formulation numerical calculations 1 1 � � Uncertain Parameters f ( x ) v ( x ) dx , ∀ v ∈ V (0) a ( x ) u ′ h ( x ) v ′ ( x ) dx = h Error estimation 0 0 Computational method Approximate solution Linearization- Based n n � � Algorithm u h = u i ϕ i ( x ) , v = v j ϕ j ( x ) Conclustions i =1 j =1 n � ∂ϕ i ( x ) ∂ u h ∂ x = u i ∂ x i =1 n � ∂ v ∂ϕ j ( x ) ∂ x = v j ∂ x j =1 9 / 29

The Finite Element Method Errors in numerical 1 1 � � calculations Approximate solution a ( x ) u ′ h ( x ) v ′ ( x ) dx = f ( x ) v ( x ) dx . Uncertain 0 0 Parameters   Error � 1 � 1 estimation n n � �   v j = 0 Computational a ( x ) ϕ i ( x ) ϕ j ( x ) dxu i − f ( x ) ϕ j ( x ) dx method j =1 i =1 Linearization- 0 0 Based Algorithm Final system of equations (for one element) Ku = q where Conclustions � 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. 10 / 29

Global Stiffness Matrix Errors in numerical Global stiffness matrix calculations Uncertain  Parameters � n e n e n n n e � � � � � ∂ϕ e u u a ( x ) ∂ϕ e j ( x ) Error i ( x )  U e dxU e estimation i , q u q − j , p ∂ x ∂ x Computational p =1 q =1 e =1 i =1 j =1 method Ω e Linearization-  Based � n e n e n n e � � � � Algorithm u u U e f ( x ) ϕ e i ( x ) ϕ e  v p = 0 j ( x ) dx Conclustions j , p q =1 e =1 i =1 j =1 Ω e Final system of equations Ku = q Computations of the global stiffness matrix can be done in parallel. 11 / 29

The Gradient Errors in numerical calculations Uncertain After discretization Parameters Ku = q Error estimation Calculation of the gradient Computational method Linearization- ∂ ∂ Based Kv = q − Ku Algorithm ∂ p k ∂ p k Conclustions ∂ where v = ∂ p k u . Presented gradient can be used in the optimization process. Derivative with respect to different parameters p k can ba calculated simultaneously by using parallel computing. 12 / 29

FEM approximation Errors in numerical calculations Uncertain The error of the solution can be approximated by the following Parameters inequality Error estimation Computational � u − u h � E ≤ � u − v � E , ∀ v ( x ) ∈ V (0) ⊂ H 1 method 0 h Linearization- this means that the finite element solution u h ∈ V (0) Based is the h Algorithm best approximation of the solution u by the function in V (0) h , Conclustions where 1 � � � 2 dx � u − u h � 2 u ′ ( x ) − u ′ E = a ( x ) h ( x ) 0 13 / 29

FEM approximation Errors in numerical calculations Uncertain (An apriori error estimate). Let u and u h be the solutions of Parameters the Dirichlet problem (BVP) and the finite element problem Error estimation (FEM), respectively. Then there exists an interpolation Computational constant C i , depending only on a ( x ), such that method Linearization- Based � u − u h � E ≤ C i � hu ′′ � a Algorithm Conclustions where 1 � a ( x ) ( u ( x )) 2 dx � u � 2 a = 0 This, however, requires that the exact solution u ( x ) is known. 14 / 29

FEM approximation Errors in numerical calculations (a posteriori error estimate). There is an interpolation constant Uncertain C i depending only on a ( x ) such that the error in finite element Parameters approximation of the Dirichlet boundary value problem (BVP) Error estimation satisfies Computational � method � � 1 � Linearization- 1 � Based � a ( x ) h 2 ( x ) R 2 ( u h ( x )) dx � u − u h � E ≤ C i Algorithm Conclustions 0 where h ( x ) is some weight and R h ( u h ( x )) = f ( x ) + ( a ( x ) u ′ h ( x )) ′ is the residual error and u h is a solution of the Finite Element Method. 15 / 29

Adaptivity Errors in numerical calculations Assume that one seeks an error bound less that a given error Uncertain tolerance TOL: Parameters � e ( x ) � E ≤ TOL Error estimation Computational Then one may use the following steps as a mesh refinement method strategy: Linearization- Based (i) Make an initial partition of the interval. Algorithm Conclustions (ii) Compute the corresponding FEM solution u h ( x ) and residual R ( u h ( x )). (iii) If � e ( x ) � E > TOL refine the mesh in the places for 1 a ( x ) R 2 ( u h ( x )) is large and perform the steps (ii) which and (iii) again. 16 / 29

Adaptivity Errors in numerical calculations Uncertain Parameters Error estimation Computational method Linearization- Based Algorithm Conclustions Figure : Adaptive FEM. 17 / 29