On Preconditioners for the Borsuk Existence Test

Frommer, Andreas; Lang, Bruno

On Preconditioners for the Borsuk Existence Test

Bruno Lang

2004, PAMM

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Existence or fixed point theorems, combined with interval analytic methods, provide a means to computationally prove the existence of a zero of a nonlinear system in a given interval vector. One such test is based on Borsuk's existence theorem. We discuss preconditioning techniques that are aimed at improving the effectiveness of this test. *

PAMM · Proc. Appl. Math. Mech. 4, 638–639 (2004) / DOI 10.1002/pamm.200410300 On Preconditioners for the Borsuk Existence Test Andreas Frommer∗1 and Bruno Lang∗∗1 1 Bergische Universität Wuppertal, Fachbereich Mathematik und Naturwissenschaften, Scientific Computing, D-42097 Wuppertal, Germany Existence or fixed point theorems, combined with interval analytic methods, provide a means to computationally prove the existence of a zero of a nonlinear system in a given interval vector. One such test is based on Borsuk’s existence theorem. We discuss preconditioning techniques that are aimed at improving the effectiveness of this test. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 The existence theorems of Miranda and Borsuk Given an interval vector (a “box”) [x] = [x1 ] × · · · × [xn ] = [x1 , x1 ] × · · · × [xn , xn ] ⊂ IRn , let ¾ [x]i,− = [x1 ] × · · · × [xi−1 ] × {xi } × [xi+1 ] × · · · × [xn ], i = 1, . . . , n, [x]i,+ = [x1 ] × · · · × [xi−1 ] × {xi } × [xi+1 ] × · · · × [xn ], denote the pairs of opposite facets of [x]. Miranda’s theorem [3] generalizes the intermediate value theorem to higher dimensions. It states that if the components fi of a continuous function f : [x] → IRn take opposite (constant) sign on opposite facets of a box [x], that is, fi (x− ) ≤ 0 ≤ fi (x+ ) for all x− ∈ [x]i,− , x+ ∈ [x]i,+ , i = 1, . . . , n, (1) then f has a zero in [x]. Borsuk’s theorem [1] guarantees the existence of a zero provided that the function does not point in the same direction at any pair of opposite points on the boundary of the box. More precisely, let x̌ be the midpoint of the box. If f (x̌ + y) 6= λf (x̌ − y) for all y such that x̌ + y is contained in some facet [x]i,+ and all λ > 0 (2) then f has a zero in [x]. (Note that x̌ + y ∈ [x]i,+ implies x̌ − y ∈ [x]i,− , the opposite facet.) 2 Computational existence tests and preconditioning In practice, the conditions (1) and (2) are not checked for the function f under consideration, but for a preconditioned function g = A · f , where A is some n-by-n matrix. If A is nonsingular then (1) or (2) holding for g also implies a zero of f in [x]. Furthermore, the conditions (1) and (2) are replaced with weaker range-based conditions that can be checked automatically with the use of interval arithmetic. To this end, let [gj ]i,± denote an enclosure for gj ’s range over the facet [x]i,± . Such an enclosure can be obtained, e.g., by plain interval evaluation of the function f over [x]i,± , followed by a multiplication with the j-th row of A. More sophisticated ways for computing the [gj ]i,± involve derivatives or slopes of the function f ; see the discussion in [2]. Given the enclosures [gj ]i,± , the condition (1) is certainly fulfilled for g if sup [gi ]i,− ≤ 0 ≤ inf [gi ]i,+ for each i = 1, . . . , n. (3) Analogously, considering the ratios of g’s components at opposite points, we see that (2) holds for g if n \ [gj ]i,+ ∩ (0, ∞) = ∅ [g ]i,− j=1 j for each i = 1, . . . , n. (4) Extended interval arithmetic [4] comes into play whenever an interval in the denominator contains zero. Other range-based (sufficient or even equivalent) criteria for Borsuk’s theorem can be formulated; see [2]. Overestimation of the ranges may cause the computational Miranda test (3) or the Borsuk test (4) to fail even if the prerequisites of the respective existence theorem are fulfilled. In order to reduce such overestimation we may partition each S facet [x]i,± into a set of subfacets, [x]i,± = k [x]i,±,k , such that the subfacets are disjoint except for their relative boundaries. For the Miranda test, opposite facets [x]i,± may be partitioned independently from each other; cf. theSleft picture in Fig. 1, and the enclosures [gi ]i,± in (3) are simply replaced by the unions of the enclosures over the subfacets, k [gi ]i,±,k . ∗ ∗∗ e-mail: frommer@math.uni-wuppertal.de e-mail: lang@math.uni-wuppertal.de © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Section 21 639 [x]i,− [x]i,− ✛ [x]i,+,k ✛ [x]i,+,k f+ [x]i,+ Fig. 1 [x]i,−,k ✲ [x]i,+ f− Partitioning of opposite facets for the Miranda test (left picture) and for the Borsuk test (right picture). For the Borsuk test, the partitionings of opposite facets must be symmetric w.r.t. the midpoint of the box; cf. the right picture in Fig. 1. Here, the condition (4) is replaced with its subfacet analogon, n \ [gj ]i,+,k ∩ (0, ∞) = ∅ [g ]i,−,k j=1 j for each i = 1, . . . , n and all k. (5) The effectiveness of the tests is determined to a large degree by the choice of the preconditioner, A. For the Miranda test, the same preconditioner must be used on all (sub)facets of the box. Popular choices are a) A = In (i.e., no preconditioning) and b) the midpoint-inverse preconditioner, i.e., A ≈ Jˇ−1 , where [J] denotes an enclosure of the Jacobian J = ∂f /∂x over the whole box [x], and Jˇ is the midpoint of [J]. These two preconditioners can also be used for the Borsuk test, but there we have considerably more freedom. In fact, a different preconditioner A = Ay might be chosen for every offset y occuring in (2). We do not completely exploit this additional flexibility but use a constant preconditioner Ai,k on each pair [x]i,±,k of opposite subfacets. The preconditioner is chosen such that the function g = Ai,k · f should point in almost opposite directions on [x]i,+,k and [x]i,−,k . To achieve this goal, we consider the function values at the midpoints of the subfacets, f + = f (mid [x]i,+,k ) and f − = f (mid [x]i,−,k ); cf. Fig. 1. We choose Ai,k such that g+ = Ai,k · f + = (+100, 1, 0, . . . , 0)T and g− = Ai,k · f − = (−100, 1, 0, . . . , 0)T , i.e., g+ ≈ −g− . (In general we cannot achieve g+ = −g− with a nonsingular matrix Ai,k .) To obtain Ai,k , we first compute the two n-by-2 QR decompositions (f + , f − ) = Qf · Rf and (g+ , g− ) = Qg · Rg , and then set Ai,k = (g+ , g− , Qg (:, 3 : n)) · (f + , f − , Qf (:, 3 : n))−1 . If this matrix is very ill-conditioned or even singular then either f + and f − already point in almost opposite directions, which means that no preconditioning is necessary, or they point in the same direction, so that preconditioning will not help anyway. Therefore we set Ai,k = In in this case. 3 Numerical results and conclusions Borsuk test, identity preconditioner Borsuk test, midpoint-inverse preconditioner Borsuk test, new preconditioner Miranda test, identity preconditioner Miranda test, midpoint-inverse preconditioner Max. diam To assess the effectiveness of the different existence tests and preconditioners we consider the two-dimensional function √ √ f (u, v) = (4 − 2(u − 1)2 , (2 − (u + 1)2 ) · (2 − (v − 1)2 ))T , which has a unique zero x∗ = (1 − 2, 1 − 2)T in the box [x] = [−1, 1] × [−1, 1]; cf. [2]. The data given in Fig.2 indicate that, while the Borsuk test with identity or midpoint-inverse preconditioning is already more powerful than the Miranda test, its effectiveness is further enhanced with our new subfacet-based preconditioner. Indeed, if we allow a moderate number of subfacets per facet (16, say) then with the new preconditioner we can verify the existence of the zero in a box of double size, as compared to the identity or midpoint-inverse preconditioner. 4 3.5 3 2.5 2 1.5 1 0.5 1 2 4 8 16 32 64 128 256 512 1024 Max. subfacets per facet Fig. 2 Diameter of the largest box of the form [x]α = x∗ + α · ([−1, 1]2 − x∗ ), such that the zero can be verified with the Miranda or Borsuk test using the identity, midpoint-inverse, or our new preconditioner. Acknowledgements The second author gratefully acknowledges support by VolkswagenStiftung within the project “Konstruktive Methoden der Nichtlinearen Dynamik zum Entwurf verfahrenstechnischer Prozesse”, Geschäftszeichen I/79 288. References [1] [2] [3] [4] K. Borsuk, Drei Sätze über die n-dimensionale Sphäre, Fund. Math. 20, 177–190 (1933). A. Frommer and B. Lang, Existence tests for solutions of nonlinear equations using Borsuk’s theorem, submitted for publication. C. Miranda, Un’ osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. (2) 3, 5–7 (1940). W. Walster, The extended real interval system, http://www.mcsc.mu.edu/∼globsol/walster-papers.html (1998). © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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