Global Optimization for Algebraic Geometry Computing Runge Kutta Methods

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1 Global Optimization for Algebraic Geometry Computing Runge Kutta Methos Ivan Martino 1 an Giuseppe Nicosia 2 1 Department of Mathematics, Stockholm University, Sween martino@math.su.se 2 Department of Mathematics an Computer Science, University of Catania, Italy nicosia@mi.unict.it Abstract. This research work presents a new evolutionary optimization algorithm, EVO-RUNGE-KUTTA in theoretical mathematics with applications in scientific computing. We illustrate the application of EVO-RUNGE-KUTTA, a two-phase optimization algorithm, to a problem of pure algebra, the stuy of the parameterization of an algebraic variety, an open problem in algebra. Results show the esign an optimization of particular algebraic varieties, the Runge- Kutta methos of orer. The mapping between algebraic geometry an evolutionary optimization is irect, an we expect that many open problems in pure algebra will be moelle as constraine global optimization problems. 1 Introuction In science an engineering, problems involving orinary ifferential euations (ODEs) can be always reformulate a set of N couple first-orer ifferential euations for the functions y i, i =1, 2,...,N, having the general form yi(t) t = f i (t, y 1,...,y N ),i = 1, 2,...,N. A problem involving ODEs is completely specifie by its euations an by bounary conitions of the given problem. Bounary conitions are algebraic conitions, they ivie into two classes, initial value problems an two-point bounary value problems. In this work we will consier the initial value problem, where all the y i are given at some starting value x 0, an it is esire to fin the y i s at some final point x f. In general, it is the nature of the bounary conitions that etermines which numerical methos to use. For instance, the basic iea of the Euler s metho is to rewrite the y s an x s of the previouseuationas finite steps δy an δx, an multiply the euations by δx. This prouces algebraic formulas for the change in the functions when the inepenent variable x is increase by on stepsize δx; for very small stepsize a goo approximation of the ifferential euation is achieve. The Runge-Kutta metho is a practical numerical metho for solving initial value problems for ODEs [1]. Runge-Kutta methos propagate a numerical solution over an N +1-imensional interval by combining the information from several Euler-style steps (each involving one evaluation of the right-han f s), an then using the information obtaine to match a Taylor series expansion up to some higher orer. Runge-Kutta is usually the fastest metho when evaluating f i is cheap an the accuracy reuirement is not ultra-stringent. In this research work, we want to esign a methoology that allow us to fin new Runge-Kutta methos of orer with minimal approximation error; such a uestion Y. Hamai an M. Schoenauer (Es.): LION 6, LNCS 7219, pp , c Springer-Verlag Berlin Heielberg 2012

2 450 I. Martino an G. Nicosia can be tackle as a constraine optimization problem. At first, we efine the problem from a geometrical point of view, using the theory of labelle trees by Hairer, Norsett an Wanner [2], an then the stuy of the parameterization of the algebraic variety RKs = {s-level Runge-Kutta methos of orer } suggests the use, with goo results, of a new class of evolutionary optimization [3]. Algebraic geometry provies justification for why it is important to use evolutionary optimization algorithm to esign effective new Runge-Kutta methos uner several constrains. 2 The Algebraic Variety RK s Let Ω R n+1 be an open set an f : Ω R n a function such that the following Cauchy problem makes sense: y = f(t, y) uner y(t 0 )=y 0. We can always reuce that to the autonomous system { y = f(y) y(t 0 )=y 0 (1) where we abuse of the notation f, but the meaning is clear. The Runge-Kutta methos are a class of schemes to approximate the exact solution of (1). The structure of s- level Implicit Runge-Kutta metho (RK metho) is k i = f(y 0 + h s j=1 a i,jk j )with i =1,...,san h is the step size. The final numerical solution y 1 R n of the problem (1) is given by y 1 = y 0 + h s i=1 w ik i. A RK metho is calle explicit if a i,j =0if i j. The approach will use all the parameters of the Butcher Tableau [4]. All c i, w i an a i,j are in R an characterize a given metho with respect another one. To unerstan the relation between the parameters of the Butcher Tableau an the orer of the approximate solution we nee to express the local truncation error, σ 1 = y(t 0 + h) y 1, with respect to h an then we have to force that the coefficients of h k must be zero for k =0, 1,...,p.Then the Runge Kutta metho has orer p+1. Forcing the coefficients of h k to be zero prouces the so calle orer conition euations.itis well known a combinatorial interpretation of the orer conitions for a Runge Kutta metho, involving roote labelle trees an elementary ifferentials. This connection is carefully constructe in [2], an we refer you there for all the etails. We enote T p the set of the roote labelle trees of orer p. The reaer shoul simply know that it possible state a bijection between the structure of the p-elementary ifferentials an each p-orer conitions. The following Theorem explains also how to use this information to state the orer conition euations. Theorem 1. If the Runge-Kutta metho is of orer p an if f is (p +1)- times continuously ifferentiable, we have y J (y 0 + h) y J 1 = hp+1 (p +1)! t T p+1 α(t)e(t)f J (t)(y 0 )+ϑ(h p+2 ) where e(t) =1 γ(t) s j=1 w jφ j (t) is calle the error coefficient of the tree t. Using this result it is possible to compute symbolically the system of euations [5]; an it is easy to see that the number of euations blow up like the factorial: this fact plays a key role in the choice of the evolutionary algorithm for the solution of the

3 Global Optimization for Algebraic Geometry Computing Runge Kutta Methos 451 problem. Moreover the integer number Φ j (t) an γ(t) are also use to construct the local truncation error (Theorem 1). We skip their combinatorial efinition but we remark that Φ j (t) epens by {a i,j } an using e(t) epens by the {a i,j } an {w i }. We note that F J (t)(y 0 ) is the J-component of the elementary ifferential of f corresponing to the tree t evaluate at the point y 0. Now we call RK s the set of all s-level Runge-Kutta methos an RKs RK s the set of all s-level Runge-Kutta methos with accuracy orer ; moreover the subset of explicit s-level Runge Kutta will be enote as ERK s an similarly we efine also ERKs. The coefficients that control the methos are all in the Butcher Tableau. For this reason, a priori, the RK methos have s + s 2 free coefficients. Moreover, it is simple to prove that the orer conitions in Theorem 1 are polynomials, so we can consier the affine algebraic variety Vs in the affine real space As(s+1) (R), minimally efine by the following polynomials in s(s +1)variables: s j=1 w j Φ j (t) = 1 γ(t), t T 1 T 2 T ; (2) this set of polynomials has a particular algebraic structure, in fact it is an ieal of the ring of polynomials in s(s +1)variables. We will enote it with Is. Thus, we can rewrite the Theorem of the Local Error as: Theorem 2. Let x =((a i,j ) 1 j i s, (w 1,w 2,...,w s )), thenx RKs x V s. Similarly the algebraic variety EVs in A s(s 1) 2 (R) minimally efine by the same polynomial euation an by {a i,j =0 i j} is the variety of the explicit Runge Kutta metho of s levels; EVs is also a subvariety of Vs. We remark that claiming Vs an EVs being an algebraic variety has some unerlying effect. One of the main ifference concerns the topology: it is use the Zariski topology an in contrast to the stanar topology, the Zariski topology is not Hausorff (one can not separate two points with ifferent open sets). We are going to show the harness of stuying the imension an the parameterization of the varieties Vs an EVs. From now on, avoiing repetitions of EVs, every result that we state for Vs is true also for EVs, with the obvious opportune changes. Again, to simplify the reaing, we put V = Vs an I = Is (the ieal of the variety V ). The goal of the present research work is to esign particular features of the RK methos; hence, if we want to use symbolical or numerical methos, we nee to parameterize V : in fact, we nee to control the set RKs using free parameters {p 1,...,p m } where m is at least the imension of V. More explicitly given a connecte component ecomposition V = i I U i we nee to know the function θ s,i : Ri U i This problem translate in the algebraic geometry θ s,i : Ai (R) U i is the problem of local parameterization of an real algebraic variety; we remark, that, even if the meaning an the appearance of the two θ s,i are exactly the same, the algebraic one carries all the ifferences in the topologies an in the map. If we narrow own our investigation, for a moment, an consier only curves (one imension algebraic varieties) in the affine plane [6,7], the uestion is the following: is the curve rational? We know that non-rational curve exists. We can suppose that not all varieties X are birational euivalent (generalization of rational concept) to A (R). The theory of Gröebner basis [8] or the Newton Polygon [9] coul be applie to tackle this

4 452 I. Martino an G. Nicosia problem, but, with a lot of generators, computational time blows up. The problem is harly structure. It is extremely ifficult to compute the connecte component ecomposition an their local imension i s [10]: there are some methos in computational algebra where the complexity of computation epens on number of generators m, the number of variables s(s +1)an the egree of the polynomial as m o(1) o(s(s+1)) [11,12]; so if s an increase (so increase) the computational time blow up. For the same reason a symbolic approach of the problem is not feasible [5]. Hence excluing some particular cases, fining a global solution for the parameterization of an algebraic variety is an open problem [13]. Now we want to state clearly the results in the most general conition. Let X R m be the set of real solutions of a system of n polynomial euations f i =0. Let f be a positive real values function efine over X, an consier the optimization problem consisting of fining x X such that the value f(x) is minimal in f(x) R +. Then for bigger value of m an n it is an open problem to fin the connecte components {U i } of X, their local imensions i an their local parameterization {θ i : R i U i }. For this reason, we suggest the use of evolutionary algorithms to search a goo solution of the corresponing optimization problem. Of course, any kin of optimization over the varieties V s an EV s are of this type; thus we are going to show how this optimization shoul be. 3 The Approach an the Results In this section, we efine the optimization problem that we want to solve. Even if a Runge-Kutta metho of orer has many features that we want to control (for instance, the convergence region for implicit RK, S a ), the aim of this research work is to obtain new explicit or implicit Runge-Kutta methos of maximal orer that minimizes local errors of orer +1. We enote, x =((a i,j ) 1 j,i s, (w 1,w 2,...,w s )) R s(s+1) where {a i,j } an {w i } are the coefficients of Butcher Tableau of an implicit, an respectively explicit, Runge-Kutta metho. Using these results, x is a feasible solution, i.e., it is in Vs (or EVs ) if it respects the constrains in 2. The fitness function is so efine fitness(x) = +1 i=0 t T i α(t) e(t). Moreover, we want that x minimizes the local errors. The impossibility of analytically computing the functions θ s,i an the ifficulty of trying to numerically optimize local error gives us the motivation to use evolutionary algorithms to face this ifficult global optimization problem [3]. The problem has been ivie an tackle in two parts: I) to fin solution for orer conition, an II) to optimize the RK metho provie by the solution of the system. Let us fix the number s of level an let us iscuss the implicit RK methos case; what shall follow hols for the explicit case too. Since a priori we o not know the local imension of the varieties Vs 2, we nee to fix =2an exploring if the variety Vs has points; if it is, we can also prouce a suitable set of feasible solutions. Thus, we will consier the next orer until we will not fin any solutions, i.e. for the maximal orer the variety Vs +1 =. Without to moify the esigne evolutionary algorithm, it is possible to explore the solution space computing the non-ominate solutions (the Pareto optimal solutions) of the given problem. The evolutionary algorithm for a fixe level s an orer has the structure shown in Appenix A. We want to prouce new Runge-Kutta methos of high orer approximation, but to verify our theory an methoology we have teste the

5 Global Optimization for Algebraic Geometry Computing Runge Kutta Methos 453 evolutionary algorithm with a 3-level explicit Runge-Kutta methos of orer 3 an with 4-level an 5-level explicit Runge-Kutta methos of orer 4: we have fin respectively 146, 364 an 932 new explicit Runge-Kutta methos. Moreover we have prouce a remarkable set of feasible solution that can be use for ifferent optimization problem over the algebraic varieties Vs an EVs. We show the results in the following table. Table 1. Feasible solutions in the explicit Runge-Kutta methos Orer/Level Conclusion The esigne an implemente evolutionary algorithm, EVO-RUNGE-KUTTA, optimizes the Butcher Tableaux an implicit or explicit Runge Kutta methos in orer to fin the maximal orer of accuracy an to minimize theirs local errors in the next orer. The results presente in this article suggest that further work in this research fiel will avance the esigning of Runge-Kutta methos, in particular, an the use of the evolutionary algorithm for any kin of optimization over an algebraic variety. To our knowlege this is the first time that algebraic geometry is use to state correctly that evolutionary algorithms have to be use to face a particular optimization problem. Again we think this is the first time that algebraic geometry an evolutionary algorithms are use to tackle a numerical analysis problem. Further refinement of our evolutionary optimization algorithm will surely improve the solutions of these important numerical analysis problems. Acknowlegements. Ivan Martino want to thank his avisor Professor Torsten Ekeahl that has recently passe away. It has been a privilege to learn mathematics from him. His genius an his generosity will always inspire me. A The Algorithm: Evo-Runge-Kutta EVO-RUNGE-KUTTA() 1. t=0; 2. inizializepopulation(pop (t) ); /* ranom generation of RKs*/ 3 initialize(newsolution); /* new array of feasible solutions*/ 4. evaluationpopulation(pop (t) ); /* evaluation of RK systems */ 5. while ((t<i max )&&(meanerror< accuracy)&&(besterror< accuracy)) o { 6. Copy (pop (t), popmut(t),p c); 7. mutationoperator(popmut (t), p m,σ); 8 isfeasible(popmut (t), olsolutions);

6 454 I. Martino an G. Nicosia 9. evaluationpopulation(popmut (t) ); 10. pop (t+1) =Selection(pop (t), popmut(t), r s); 11. computestatistics(pop (t+1) ); 12 savesolutions(newsolution); 13. t=t+1; 14. } In the following table we show the parameters use. Parameter Value orer of the RK-methos 2 level of the RK-methos 3 orer of the optimization of the error 3 = population size 10 3 I max = Max iterations of the first & secon cycle p m = mutation probability of weight vector 0.5 part of element of Butcher Tableau that oes not change uring mutation 0.3 r s = part of population selecte for elitism in the first & secon cycle 0.3 p c = selection probability in the first & secon cycle 0.5 σ = variance of Gaussian perturbation 0.1 References 1. Kaw, A., Kalu, E.E.: Numerical Methos with Applications. Autarkaw (2010) 2. Hairer, E., Norsett, S.P., Wanner, G.: Solving Orinary Differential Euation I: Nonstiff Problems, 3r en. Springer (January 2010) 3. Golberg, D.: Design of Innovation. Kluwar (2002) 4. Butcher, J.C.: Numerical methos for orinary ifferential euations. John Wiley an Sons (2008) 5. Famelis, T., Papakostas, S.N., Tsitouras, C.: Symbolic erivation of runge kutta orer conitions. J. Symbolic Comput. 37, (2004) 6. Fulton, W.: Algebraic Curves. Aison Wesley Publishing Company (December 1974) 7. Kollar, J.: Rational Curves on Algebraic Varieties. Springer (1996) 8. Cox, D., Little, J., O Shea, D.: Ieals, Varieties, an Algorithms, An Introuction to Computational Algebraic Geometry an Commutative Algebra, 3r en. Springer (2007) 9. D Anrea, C., Sombra, M.: The newton polygon of a rational plane curve. arxiv: Castro, D., Giusti, M., Heintz, J., Matera, G., Paro, L.M.: The harness of polynomial euation solving. Foun. Comput. Math. 3(4), (2003) 11. Giusti, M., Hägele, K., Lecerf, G., Marchan, J., Salvy, B.: The projective Noether Maple package: computing the imension of a projective variety. J. Symbolic Comput. 30(3), (2000) 12. Giusti, M., Lecerf, G., Salvy, B.: A Gröbner free alternative for polynomial system solving. J. Complexity 17(1), (2001) 13. Greene, R.E., Yau, S.T. (es.): Open Problems in Geometry. Proceeings of Symposia in Pure Mathematics, vol. 54 (1993)