Optimal Preconditioning for the Interval Parametric GaussSeidel - PowerPoint PPT Presentation

Optimal Preconditioning for the Interval Parametric Gauss–Seidel Method Milan Hlad´ ık Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic http://kam.mff.cuni.cz/~hladik/ SCAN, W¨ urzburg, Germany September 21–26, 2014 1 / 12

Interval Linear Equations Interval Linear Equations Ax = b , A ∈ A , b ∈ b , where A := [ A , A ] = [ A c − A ∆ , A c + A ∆ ] , b := [ b , b ] = [ b c − b ∆ , b c + b ∆ ] are an interval matrix and an interval vector, respectively. The Solution Set Σ := { x ∈ R n : ∃ A ∈ A , ∃ b ∈ b : Ax = b } . Problem formulation Find a tight interval vector enclosing Σ . 2 / 12

Preconditioning Preconditioning Let C ∈ R n × n . Preconditioning is a relaxation to A ′ ∈ ( C A ) , b ′ ∈ ( C b ) , A ′ x = b ′ , Preconditioning usually we use C = ( A c ) − 1 , theoretically justified by Neumaier (1984, 1990), optimal preconditioning for the interval Gauss–Seidel method by Kearfott et al. (1990, 1991, 2008). The Interval Gauss–Seidel Method   1 �  , z i :=  ( C b ) i − ( C A ) ij x j x i := x i ∩ z i , ( C A ) ii j � = i for i = 1 , . . . , n . 3 / 12

Interval Parametric Systems Interval Parametric System A ( p ) x = b ( p ) , p ∈ p , where K K � � A k p k , b k p k , A ( p ) = b ( p ) = k =1 k =1 and p k ∈ p k , k = 1 , . . . , K . The Solution Set Σ p := { x ∈ R n : ∃ p ∈ p : A ( p ) x = b ( p ) } . Preconditioning and Relaxation Relaxation to Ax = b , where K K � � ( CA k ) p k , ( Cb k ) p k . A ∈ A := b ∈ b := k =1 k =1 4 / 12

The Parametric Interval Gauss–Seidel Method The Parametric Interval Gauss–Seidel Method � K   K � 1 � ( Cb k ) i p k − � � ( CA k ) ij p k  , z i := x j  �� K � k =1 ( CA k ) ii p k k =1 j � = i k =1 x i := x i ∩ z i , for i = 1 , . . . , n . Optimal Preconditioner Various criteria of optimality: minimize the resulting upper bound, that is, the objective is min z i , maximize the resulting lower bound, that is, the objective is max z i . 5 / 12

The Parametric Interval Gauss–Seidel Method The Parametric Interval Gauss–Seidel Method � K   K � 1 � ( Cb k ) i p k − � � ( CA k ) ij p k  , z i := x j  �� K � k =1 ( CA k ) ii p k k =1 j � = i k =1 x i := x i ∩ z i , for i = 1 , . . . , n . Optimal Preconditioner Various criteria of optimality: minimize the resulting width, that is, the objective is min 2 z ∆ i , minimize the resulting upper bound, that is, the objective is min z i , maximize the resulting lower bound, that is, the objective is max z i . 5 / 12

The Parametric Interval Gauss–Seidel Method The Parametric Interval Gauss–Seidel Method � K   K � 1 � ( Cb k ) i p k − � � ( CA k ) ij p k  , z i := x j  �� K � k =1 ( CA k ) ii p k k =1 j � = i k =1 x i := x i ∩ z i , for i = 1 , . . . , n . Optimal Preconditioner Various criteria of optimality: minimize the resulting width , that is, the objective is min 2 z ∆ i , minimize the resulting upper bound, that is, the objective is min z i , maximize the resulting lower bound, that is, the objective is max z i . 5 / 12

Minimal Width Preconditioner Preliminaries For simplicity assume that 0 ∈ x and 0 ∈ z Denote by c the i th row of C , Normalize c such that the denominator has the form of [1 , r ] for some r ≥ 1. Interval Gauss–Seidel Step Then the operation of the i th step of the Interval Gauss–Seidel iteration is simplified to � K K � � � � ( cb k ) p k − ( cA k z i := ∗ j ) p k x j k =1 j � = i k =1 The objective is min z ∆ i . 6 / 12

Minimal Width Preconditioner � K � K � � � ( cb k ) p k − ( cA k minimize the width of ∗ j ) p k x j . k =1 j � = i k =1 Denote β k := | cb k | , k = 1 , . . . , K , α jk := | cA k ∗ j | , j = 1 , . . . , n , k = 1 , . . . , K , �� K � k =1 ( cA k η j := ∗ j ) p k x j , j � = i , �� K � k =1 ( cA k ψ j := ∗ j ) p k j � = i . x j , 7 / 12

Minimal Width Preconditioner K � 2 p ∆ � min k β k + ( η j − ψ j ) , k =1 j � = i subject to β k := | cb k | , k = 1 , . . . , K , α jk := | cA k ∗ j | , j = 1 , . . . , n , k = 1 , . . . , K , �� K � k =1 ( cA k η j := ∗ j ) p k x j , j � = i , �� K � k =1 ( cA k ψ j := ∗ j ) p k j � = i . x j , 7 / 12

Minimal Width Preconditioner K � 2 p ∆ � min k β k + ( η j − ψ j ) , k =1 j � = i subject to β k ≥ cb k , β k ≥ − cb k , k = 1 , . . . , K , α jk := | cA k ∗ j | , j = 1 , . . . , n , k = 1 , . . . , K , �� K � k =1 ( cA k η j := ∗ j ) p k x j , j � = i , �� K � k =1 ( cA k ψ j := ∗ j ) p k j � = i . x j , 7 / 12

Minimal Width Preconditioner K � 2 p ∆ � min k β k + ( η j − ψ j ) , k =1 j � = i subject to β k ≥ cb k , β k ≥ − cb k , k = 1 , . . . , K , α jk ≥ cA k ∗ j , α jk ≥ − cA k ∗ j , j = 1 , . . . , n , k = 1 , . . . , K , �� K � k =1 ( cA k η j := ∗ j ) p k x j , j � = i , �� K � k =1 ( cA k ψ j := ∗ j ) p k j � = i . x j , 7 / 12

Minimal Width Preconditioner K � 2 p ∆ � min k β k + ( η j − ψ j ) , k =1 j � = i subject to β k ≥ cb k , β k ≥ − cb k , k = 1 , . . . , K , α jk ≥ cA k ∗ j , α jk ≥ − cA k ∗ j , j = 1 , . . . , n , k = 1 , . . . , K , η j ≥ c � K k x j ± � K k =1 A k ∗ j p c k =1 p ∆ k x j α jk , j � = i , η j ≥ c � K k x j + � K k =1 p ∆ k =1 A k ∗ j p c k x j α jk , j � = i , �� K � k =1 ( cA k ψ j := ∗ j ) p k j � = i . x j , 7 / 12

Minimal Width Preconditioner K � 2 p ∆ � min k β k + ( η j − ψ j ) , k =1 j � = i subject to β k ≥ cb k , β k ≥ − cb k , k = 1 , . . . , K , α jk ≥ cA k ∗ j , α jk ≥ − cA k ∗ j , j = 1 , . . . , n , k = 1 , . . . , K , η j ≥ c � K k x j ± � K k =1 A k ∗ j p c k =1 p ∆ k x j α jk , j � = i , η j ≥ c � K k x j + � K k =1 p ∆ k =1 A k ∗ j p c k x j α jk , j � = i , ψ j ≤ c � K k x j ± � K k =1 p ∆ k =1 A k ∗ j p c j � = i , k x j α jk , ψ j ≤ c � K k x j − � K k =1 A k ∗ j p c k =1 p ∆ k x j α jk , j � = i , 7 / 12

Minimal Width Preconditioner K � 2 p ∆ � min k β k + ( η j − ψ j ) , k =1 j � = i subject to β k ≥ cb k , β k ≥ − cb k , k = 1 , . . . , K , α jk ≥ cA k ∗ j , α jk ≥ − cA k ∗ j , j = 1 , . . . , n , k = 1 , . . . , K , η j ≥ c � K k x j ± � K k =1 A k ∗ j p c k =1 p ∆ k x j α jk , j � = i , η j ≥ c � K k x j + � K k =1 p ∆ k =1 A k ∗ j p c k x j α jk , j � = i , ψ j ≤ c � K k x j ± � K k =1 p ∆ k =1 A k ∗ j p c j � = i , k x j α jk , ψ j ≤ c � K k x j − � K k =1 A k ∗ j p c k =1 p ∆ k x j α jk , j � = i , and the condition that the denominator is has the form of [1 , r ] c � K k − � K k =1 A k ∗ i p c k =1 p ∆ k α ik = 1 . 7 / 12

Minimal Width Preconditioner Optimization problem. Optimal preconditioner C found by n linear programming (LP) problems. each LP has Kn + K + 3 n − 2 unknowns c , β k , α jk , η j , and ψ j , and 2 Kn + 4 n − 3 constraints C needn’t be calculated in a verified way. The problem is effectively solved in polynomial time. Practical Implementation Call the standard version using midpoint inverse preconditioner (or any other method), and after that tighten the enclosure by using an optimal preconditioner C . In our examples: one iteration with minimization of the upper bound, and one with maximization of the upper bound. 8 / 12

Example I Example (Popova, 2002) � 1 � � p 3 � p 1 p ∈ p = ([0 , 1] , − [1 , 4] , [0 , 2]) T . A ( p ) = , b ( p ) = , p 1 p 2 p 3 Initial enclosure by the Parametric Interval Gauss–Seidel Method with midpoint inverse preconditioner: direct version: 7 . 66% of the width on average reduced residual form: 0% of the width on average reduced Initial enclosure as the interval hull of the relaxed system: direct version: 50% of the width on average reduced residual form: 12 . 56% of the width on average reduced 9 / 12

Example II Example (Popova and Kr¨ amer, 2008)  30 − 10 − 10 − 10 0   1  − 10 10 + p 1 + p 2 − p 1 0 0 0         A ( p ) = − 10 − p 1 15 + p 1 + p 3 − 5 0 , b ( p ) = 0 ,         − 10 0 − 5 15 + p 4 0 0     0 0 − 5 5 1 0 where p ∈ p = [8 , 12] × [4 , 8] × [8 , 12] × [8 , 12] . Initial enclosure by the Parametric Interval Gauss–Seidel Method with midpoint inverse preconditioner: direct version: 15% of the width on average reduced residual form: 0% of the width on average reduced 10 / 12

Conclusion Summary Optimal preconditioning matrix for the parametric interval Gauss–Seidel iterations. It can be computed effectively by linear programming. Preliminary results show that sometimes can reduce overestimation of the standard enclosures. Directions for Further Research Other types of optimality of preconditioners (S-preconditioners, pivoting preconditioners, etc.) Optimal preconditioners for other methods than the parametric interval Gauss–Seidel one. 11 / 12