SONIC A Solver and Optimizer for Nonlinear Problems based on Interval Computation

Transcription

1 SONIC A Solver and Optimizer for Nonlinear Problems based on Interval Computation Elke Just University of Wuppertal June 14, 2011

2 Contents About SONIC Miranda Borsuk Topological Degree Implementation Branch and bound and partitioning strategies Expanded systems Heuristics SONIC, Elke Just 2/30

3 Basic Facts Main contributors: Thomas Beelitz, Bruno Lang, Elke Just, Paul Willems (Applied Computer Science, University of Wuppertal) Peer Ueberholz (Hochschule Niederrhein, University of Applied Sciences) initial goal: efficient and robust design of dynamic systems SONIC, Elke Just 3/30

4 Basic Facts based on branch-and-bound contains rigorous nonlinear solver and (constrained) optimizer SONIC, Elke Just 4/30

5 Basic Facts based on branch-and-bound contains rigorous nonlinear solver and (constrained) optimizer written in C++ generic interval code provides performance and portability supported interval libraries C-XSC SUN C++ filib++ parallization with OpenMP and MPI SONIC, Elke Just 4/30

6 Key features of SONIC Several preconditioners inverse midpoint preconditioner optimal linear programming preconditioners (width-optimal contraction, mignitude-optimal splitting) SONIC, Elke Just 5/30

7 Key features of SONIC Several preconditioners inverse midpoint preconditioner optimal linear programming preconditioners (width-optimal contraction, mignitude-optimal splitting) Constraint Propagation (CP) on finite unions of real intervals SONIC, Elke Just 5/30

8 Key features of SONIC Several preconditioners inverse midpoint preconditioner optimal linear programming preconditioners (width-optimal contraction, mignitude-optimal splitting) Constraint Propagation (CP) on finite unions of real intervals Taylor refinement order-1 and order-2 optionally integrated into CP SONIC, Elke Just 5/30

9 Key features of SONIC Several preconditioners inverse midpoint preconditioner optimal linear programming preconditioners (width-optimal contraction, mignitude-optimal splitting) Constraint Propagation (CP) on finite unions of real intervals Taylor refinement order-1 and order-2 optionally integrated into CP Hybrid subdivision strategy apply subdivision scheme that seems most promising for current box SONIC, Elke Just 5/30

10 Key features of SONIC Several preconditioners inverse midpoint preconditioner optimal linear programming preconditioners (width-optimal contraction, mignitude-optimal splitting) Constraint Propagation (CP) on finite unions of real intervals Taylor refinement order-1 and order-2 optionally integrated into CP Hybrid subdivision strategy apply subdivision scheme that seems most promising for current box Interval method on a hierarchy of expanded systems SONIC, Elke Just 5/30

11 Verfication of solutions Miranda Borsuk Topological Degree Implementation aim: prove existence (and uniqueness) of solutions for a system f(x) = 0 0 [f(x)] is no verification overestimation different solutions for equations each possible solution box has to be verified separately the following methods need square systems SONIC, Elke Just 6/30

12 Miranda Borsuk Topological Degree Implementation simply apply an interval operator uniqueness provable SONIC, Elke Just 7/30

13 Miranda Borsuk Topological Degree Implementation simply apply an interval operator uniqueness provable diameter too small: epsilon-inflation epsilon inflation SONIC, Elke Just 7/30

14 Miranda Borsuk Topological Degree Implementation Facets For x IR n we define n pairs of opposite facets as: x i,+ := (x 1,..., x i 1, x i, x i+1,..., x n ) T x i, := (x 1,..., x i 1, x i, x i+1,..., x n ) T for 1 i n. Topological boundary n x = x i,s (1) i=1 s {+, } SONIC, Elke Just 8/30

15 Miranda Borsuk Topological Degree Implementation Subfacets To avoid overestimation we use partions of the facets, subfacets x i,+,j and x i,,k. x i,+ x i,+ x i,+,k x i,+,k x i,,l x i,,l x i, (a) unsymmetric x i, (b) symmetric w.r.t. the midpoint of the box SONIC, Elke Just 9/30

16 Theorem of Miranda About SONIC Miranda Borsuk Topological Degree Implementation Generalization of the intermediate value theorem. Theorem Let f : R n R n be a continuous function and x IR n a box. If or f i (x ) 0 f i (x + ) f i (x + ) 0 f i (x ) for all x + x i,+, x x i, and i = 1,..., n holds, f has a zero in x. SONIC, Elke Just 10/30

17 Miranda Borsuk Topological Degree Implementation Implementation nonsingular matrix as preconditioner alternative test for disjoint subfacets and preconditioned function g { sup([gi ](x i,,l )) 0 inf([g i ](x i,+,k )) 0 or { sup([gi ](x i,+,l )) 0 inf([g i ](x i,,k )) 0 for all i = 1,..., n and l resp. k. SONIC, Elke Just 11/30

18 Theorem of Borsuk About SONIC Miranda Borsuk Topological Degree Implementation Theorem Let f : R n R n be a continuous function and x IR n. If f(mid (x) + y) λf(mid (x) y) holds for all y, where mid (x) + y is element of a facet x i,+ and λ > 0, then f has a zero in x. f (mid (x) + y) f (mid (x) y) mid (x) SONIC, Elke Just 12/30

19 Miranda Borsuk Topological Degree Implementation Implementation subfacets to avoid overestimation have to be symmetrical preconditioner nonsingular matrix can be chosen separately for each facet special preconditioner for Borsuk [FL04] alternative test for disjoint subfacets and preconditioned function g n [g j ](x i,+,k ) [g j ](x i,,k (0, ) = ) j=1 for all i = 1,..., n and k and symmetric subfacets SONIC, Elke Just 13/30

20 Topological degree About SONIC Miranda Borsuk Topological Degree Implementation For Ω R n open, bounded f : Ω R n continuous function f(x) 0 for all x Ω the topological degree d(f, Ω, 0) is the number of solutions of f(x) = 0 with x Ω. SONIC, Elke Just 14/30

21 Miranda Borsuk Topological Degree Implementation Test Homotopy h(t, x) = t g(x) + (1 t) (x mid (x)) with t [0, 1], x x and the preconditioned function g(x) = C f (x) Theorem The function f has a zero in x if h has no zero in the 2n boxes [0, 1] x = n i=1 s {+, } [0, 1] x i,s. See: [FHL07] SONIC, Elke Just 15/30

22 Miranda Borsuk Topological Degree Implementation A sufficient condition ([BFLW09]) Theorem Let ˇx int x and let t [0, 1], y x. For j 1,..., n let [g j (y)] be an enclosure of the range of the preconditioned function g j over y. Then, if we have n j=1 ( [g j (y)] 1 1 ) = (y ˇx) j t the homotopy h has no zero in t x. less costly for chosen testset SONIC, Elke Just 16/30

23 Miranda Borsuk Topological Degree Implementation Modifications other homotopies tried t subdivided equidistantly (good choice for us: 4 parts) or by same heuristic as used for x epsilon inflation bigger box more successful SONIC, Elke Just 17/30

24 Further implemented About SONIC Miranda Borsuk Topological Degree Implementation verification separated from finding solutions hierarchy of verification tests (using epsilon inflation) verification for non-square systems (using QR) SONIC, Elke Just 18/30

25 Branch and bound and partitioning strategies Expanded systems Heuristics Branch and bound x SONIC, Elke Just 19/30

26 Branch and bound and partitioning strategies Expanded systems Heuristics Numerical results 7erSys. Brent7 Eco9 Trig. min original MinNew= MinNew= MinNew= MinNew= MinNew= parts parts+MinNew= Table: box numbers for different settings SONIC, Elke Just 20/30

27 Branch and bound and partitioning strategies Expanded systems Heuristics Numerical results 7erSys. Brent7 Eco9 Trig. min original MinNew= MinNew= MinNew= MinNew= MinNew= parts parts+MinNew= Table: time for different settings (in s) SONIC, Elke Just 21/30

28 Branch and bound and partitioning strategies Expanded systems Heuristics Expanded systems (Split level) decomposition of arithmetic expressions using intermediate variables results in another termnet SONIC, Elke Just 22/30

29 Branch and bound and partitioning strategies Expanded systems Heuristics Expanded systems - An example System: f 1 (x 1, x 2, x 3 ) = x 2 1 ex 2, f 2 (x 1, x 2, x 3 ) = x x 2 x 3 f 1 = 0 f 2 = 0 f 1 = 0 f 2 = x 4 x 5 x 6 sqr exp sqr exp x 1 x 2 x 3 x 1 x 2 x 3 (a) original (b) expanded (full split) SONIC, Elke Just 23/30

30 Branch and bound and partitioning strategies Expanded systems Heuristics Expanded systems can be used for CP, extended method default in SONIC original CST (common subterms) full split more splits [Wil04] more effective = more costly SONIC, Elke Just 24/30

31 Branch and bound and partitioning strategies Expanded systems Heuristics Old Heuristic Use original, CST, full split in this order restart with original when contracted by certain percentage stop when box is small enough or a certain number of tests have been applied SONIC, Elke Just 25/30

32 Branch and bound and partitioning strategies Expanded systems Heuristics New heuristic also original, CST, full split no restarts splits used depending on their success success= time * ratio of contraction averaged success over previous runs use all split levels in the beginning stored in box for each level SONIC, Elke Just 26/30

33 Branch and bound and partitioning strategies Expanded systems Heuristics restart hierarchy when contracted by a certain rate managment of splits by success Original CP, Taylor Original CP, Taylor CST CP, Taylor CST CP, Taylor Full split CP, Taylor Full split CP, Taylor box small enough box small enough maximum number of newton tests is reached Figure: old heuristic Figure: new heuristic SONIC, Elke Just 27/30

34 Branch and bound and partitioning strategies Expanded systems Heuristics Comparison (selection): old and new heuristic old boxes new boxes old time (in s) new time (in s) Arithmetic mean of ratios of boxes: 1.12 (over 16 problems) Arithmetic mean of ratios of times: 0.58 SONIC, Elke Just 28/30

35 new test with different bisection methods using Taylor models for better enclosures acceleration of SONIC for more automatic proofs of spherical designs GUI SONIC, Elke Just 29/30

36 Contact For further questions or giving suggestions even after the workshop write to SONIC, Elke Just 30/30

37 Thomas Beelitz, Andreas Frommer, Bruno Lang, and Paul Willems. A framework for existence tests based on the topological degree and homotopy. Numer. Math., 111(4): , A. Frommer, F. Hoxha, and B. Lang. Proving the existence of zeros using the topological degree and interval arithmetic. J. Comput. Appl. Math., 199(2): , Andreas Frommer and Bruno Lang. On Preconditioners for the Borsuk Existence Test. PAMM, 4(1): , Paul Willems. Symbolisch-numerische Techniken zur verifizierten Lösung nichtlinearer Gleichungssysteme, Mai Diplomarbeit, RWTH Aachen. SONIC, Elke Just 30/30