F. Augustin, P. Rentrop Technische Universität München, Centre for Mathematical Sciences

Transcription

1 NUMERICS OF THE VAN-DER-POL EQUATION WITH RANDOM PARAMETER F. Augustin, P. Rentro Technische Universität München, Centre for Mathematical Sciences Abstract. In this article, the roblem of long-term integration in the Wiener-Hermite calculus for ordinary differential equations with random arameters is discussed. The Wiener-Hermite calculus is known to lose accuracy with increasing time, even if it rovides good results for short times. This is demonstrated for the Van-der-Pol equation with a random arameter. To reduce this roblem, the adated stochastic sectral method (assm), which is based on the work of P. Vos [8], is resented. Alying assm to the Van-der-Pol equation reduces the error considerably and allows an accurate long-term integration. 1 Introduction This aer deals with the method of stochastic sectral exansion of random variables with finite variance. The stochastic sectral exansion is based on the work of N. Wiener [11] and was considered in the framework of generalized Fourier series by R. Ghanem and P. S. Sanos, see [2]. Karniadakis, LeMaıtre and co-workers develoed the method [14, 6, 5], which is still an active field of research [1]. Problems arising in industry ossess large dimensions and are time consuming to comute, even in the deterministic case. Therefore in the case of uncertain data, they are often not treatable with standard techniques like Monte Carlo simulation. The method of stochastic sectral exansion, or one of its derivates, often rovides a good means to handle such roblems. We choose the Van-der-Pol equation as benchmark roblem, following P. E. Zadunaisky, who stated "I have a theory that whenever you want to get in trouble with a method, look for the Van-der-Pol equation.", see [3]. Although its simlicity in alication, the calculus of stochastic sectral exansion does not rovide a general urose method. For examle the costs increase exonentially with the number of random arameters in the underlying model. Furthermore, no mechanisms for error estimation and control are available. U to now, the standard way to check accuracy is to comare the results with a reference solution comuted by Monte-Carlo simulation. This is not satisfactory in ractical use. To reduce this roblem, we develoed the adated stochastic sectral method (assm). By adating the density while comuting the solution, we are able to reduce the error considerably in contrast to the standard stochastic sectral decomosition. The work is organized as follows. At first the Van-der-Pol equation is derived from modeling the electrical circuit shown in figure 1. Subsequently, the alication of the stochastic sectral decomosition method to the randomness deendent Van-der-Pol equation is discussed. After that, the algorithm of assm is elaborated and numerical results of its alication to the random Van-der-Pol equation are resented. We close this work by stating conclusions and objectives for future work. 2 Derivation of the Van-der-Pol equation Preliminarily we derive the Van-der-Pol equation (VDP) from the modeling of the electrical circuit shown in figure 1. It consists of an inductance L, a caacitor C, a tunnel diode and the grounding U 0 as well as an oerational voltage U o. We are interested in the voltage at the nodes N 1 and N 2. Modeling electrical circuits is based on the two fundamental rules of Kichhoff, which can be stated as follows: 1. the sum of all currents in each node equals zero, 2. the sum of all artial voltages in a loo equals zero. 1

2 Furthermore, we need the model equations for the basic comonents of the electrical circuit [4] U L = Lİ L, I C = CU C, U R = RI R, I D = f(u D ). (1) (2) (3) (4) Here, U and I denote the resective voltages and currents for L the inductance, C the caacitor, R the resistance and D the tunnel diode. The function f describes the characteristics for the tunnel diode. We use the notation := d/dt. R N 1 L N 2 U o C tunnel diode U 0 = 0 Figure 1: Circuit with an inductance L, a caacitor C, a resistance R, a tunnel diode and the grounding U 0 as well as an oerational voltage U o. Additionally two nodes N 1 and N 2 are lotted. Having settled the basic hysical laws, we develo the mathematical model for the circuit in figure 1. We denote the voltage at node N i by U i, i = 1,2. The current I 1 o between N 1 and the oerational voltage, the current I 2 C 0 from node N 2 to the ground via the caacitor and the current I 2 D 0 from node N 2 to the ground via the tunnel diode are given by I 1 o = (U 1 U o )/R, I 2 C 0 = C( U 2 U 0 ), I 2 D 0 = f(u 2 U 0 ), (5) which are direct alications of (2)-(4). Using Kirchhoff s rules and equations (5) we get 0 = 1 R (U 1 U o )+I 1 2, (6) 0 = I 2 1 +C ( U 2 U 0 ) + f(u2 U 0 ). (7) The currents I 1 2 and I 2 1 are defined by the same systematics as indicated above, denoting the currents from node N 1 to N 2 and reverse. Differentiation of (6) with resect to time yields 0 = 1 R ( U 1 U o ) + 1 L (U 1 U 2 ), (8) 0 = 1 R (U 1 U o )+C U 2 + f(u 2 ), (9) 2

3 distribution orthogonal olynomial suort continuous Gaussian Hermite (, ) Gamma Laguerre [0, ) Beta Jacobi [a, b] Uniform Legendre [a, b] discrete Poisson Charlier {0, 1, 2,...} Binomial Krawtchouk {0, 1,..., n} Negative binomial Meixner {0, 1, 2,...} Hyergeometric Hahn {0, 1,..., n} Table 1: Distributions and the corresonding orthogonal olynomials (n N and a, b R). where we already introduced (1) for İ 1 2, exloited the fact that I 2 1 = I 1 2 and set U 0 0. The combination of equations (8) and (9) results in ( RC 0 = CÜ + L + d ) du f(u) U + 1 L (R f(u)+u U o), (10) omitting the subscrit of U 2. We fit a olynomial of degree 3 to the characteristic curve of the diode f(u) = a 1 U + a 2 U 2 + a 3 U 3, so that the well known nonlinear Van-der-Pol oscillator results from equation (10) 0 = ÿ(t)+ ( y(t) 2 1 ) ẏ(t)+y(t). (11) The arameter reresents a combination of the olynomial coefficients a i, i = 1,2,3, and denotes the uncertainty in the following. 3 Stochastic sectral exansion and stochastic domain decomosition In this section we introduce the stochastic sectral exansion, also known as Wiener-Hermite exansion, which is an exansion of random variables with finite variance into a series of Hermite olynomial functionals. It is based on the ioneering work [11] of N. Wiener in In [2] R. Ghanem and P. D. Sanos ut it into the framework of generalized Fourier series, in order to work more easily. Further advancements were introduced in [14] by D. Xiu and G. E. Karniadakis. They roosed the use of arbitrary basis functionals of L 2 (P) with resect to an aroriate choice of weight function ρ. Assume X(ξ) L 2 (P) is a random variable and ρ is the density function, which we assume to exist, of the robability measure P. Further we denote the resective orthogonal olynomial functionals by {Ψ i }. If they belong to the Askey scheme of orthogonal olynomial functionals, the generalized Wiener-Hermite exansion X(ξ) = q i Ψ i (ξ) is called Wiener-Askey exansion of X. It is a remarkable fact, that the density functions for all common distributions corresond to weight functions of orthogonal olynomial functionals in the Askey scheme, see table 1. But not only orthogonal olynomial functionals from the Askey scheme are suitable for the stochastic sectral exansion. Every comlete basis of L 2 (P) can be chosen. In [6, 5] O. LeMaître et. al. used Wavelets as basis functionals. In this article, we concentrate on orthogonal olynomial functionals, which are orthogonal with resect to arbitrary weight functions. 3

4 A further advancement of the Wiener-Hermite exansion is the stochastic domain decomosition. That means, that the underlying robability sace is divided into elements. This method was roosed by G. E. Karniadakis et. al. in [9], where it is called Multi Elements generalized Polynomial Chaos (MEgPC). For the reason, that it will be sulementary used in the adative Stochastic Sectral Method, we give a short introduction on stochastic domain decomosition. For simlicity we restrict to the one dimensional case. Let y(ξ) : S I be a random variable on the robability sace (S,F(S),P) onto an interval I R and assume a decomosition of S into n disjunct intervals {S k } n is given. We denote the density of y by ρ : I R + 0. The decomosition of S into disjunct intervals will be rescribed by assm as we will see in section 5.2. The stochastic sectral exansion of y restricted to S k, denoted by y k, can be comuted as follows: at first we comute the conditional density ρ k of y k, k = 1,...,n, by ρ k (ξ) = ρ(ξ) P(S k ). Secondly, orthogonal olynomial functionals {Ψ i,k } with resect to ρ k are constructed on S k, using a stable Gram-Schmidt orthogonalization rocedure. This rocedure results in a number of n stochastic sectral exansions y k (ξ) = q i,k Ψ i,k (ξ), for k {1,..., n}. Using these exansions, we are able to comute the moments of y. For examle the exected value is comuted by E(y) = S y(ξ)ρ(ξ)dξ = n Analogously, the variance can be comuted by V(y) = S P(S k ) y(ξ) 2 ρ(ξ)dξ E(y) 2 = S k y k (ξ)ρ k (ξ)dξ = n P(S k ) n P(S k )q 0,k. q 2 i,k E(y)2, where we used Parseval s equality, see f.e. [10]. The variance will be aroximated by V(y) n P(S k ) m q 2 i,k E(y)2, and m N is called the order of the stochastic sectral exansion. In all the calculations above we assumed the olynomial functionals to be normalized to have the norm equal to 1. 4 The stochastic Van-der-Pol equation We now return to the Van-Der-Pol equation (11) and assume the arameter to be a random variable. We choose the range measure to be P 0 := U [1.9,2.1]. Its density is given by { ρ (0) 5 if [1.9,2.1] () = 0 else. The solution becomes a stochastic rocess y(t, ) and the Van-der-Pol equation becomes a random differential equation 0 = ÿ(t, )+ ( y(t, ) 2 1 ) ẏ(t, )+y(t, ). (12) We assume the initial conditions to be rescribed deterministically by y(0, ) 0.1 and ẏ(0, ) 0.0. The density of y(t i, ) for a fixed time t i will be denoted by ρ (i). There are two ossible ways to aly the Wiener-Askey calculus to the random Van-der-Pol equation. The first one works as follows: we insert the truncated exansion y(t, ) m 4 q i (t)ψ i ()

5 into (12) 0 = m q i (t)ψ i ()+ ( m 2 q i (t)ψ i ()) 1 m q i (t)ψ i ()+ m q i (t)ψ i (), and roject onto the finite dimensional subsace F := san{ψ 0,...,Ψ m } L 2 (P 0 ), so that we arrive at an m-dimensional system of deterministic ordinary differential equations in the unknown coefficient functions {q i } m. These fully couled differential equations can be solved by using suitable integrators. The second way of alying the Wiener-Askey calculus is called the non-intrusive method, or stochastic collocation, [12]. Here we use the roerties of the Fourier series and comute the coefficient functions directly by integration: q i (t) = y(t, )Ψ i ()ρ()d, (13) S for i = 1,..., m. Although the following method of adated stochastic sectral exansions can be alied to both methods of alication, we will restrict to the non-intrusive method. We now discuss the roblem arising from the long-term integration of random differential equations. Therefore we consider the Wiener-Askey exansion y(t, ) = q i (t)ψ i (), and comute the coefficient functions {q i } using formula (13). The olynomial functionals {Ψ i} are chosen, so that they are orthogonal with resect to the weight function ρ (0), which is the density of the random arameter. This choice is reasonable, if we consider the maing { L2 (Ω,F,P) L T(t, ) : 2 (Ω,F,P) y(t, ) being well aroximated by a linear maing for early time t > 0. In this case, the distribution of y(t, ) is aroximately a linear transformation of P 0 and the Wiener-Askey exansion of y(t, ) is suosed to converge very fast. If t becomes large, the influence of the in general non-linear character of T becomes rominent and P 0 becomes a bad aroximation of the distribution of y(t, ). Hence, convergence of the Wiener-Askey exansion becomes very slow for increasing time t. 5 The adative Stochastic Sectral Method In this section we introduce the assm. At first, a resonse surface in the stochastic domain of the solution of the random Van-der-Pol equation is constructed in order to comute the integrals (13). Knowing the coefficients {q i } m of the stochastic sectral exansion, we comute the density of y(t j, ) at certain time stes t j, j N. For the selection of the time stes, we consider the magnitude of the third coefficient of the truncated stochastic sectral exansion. The density will be udated, if q 2 (t j ) exceeds a rescribed tolerance TOL. By adating the density we are able to reduce the aroximation error, which is introduced by the truncation of the stochastic sectral exansion. 5.1 Comutation of the resonse surface in random sace The resonse surface is constructed by comuting solutions of the deterministic roblem for fixed arameter values i, i = 1,..., n. Having these data oints we reresent the resonse surface by B-slines of degree 3. This aroach is analogous to the stochastic collocation method, see [13, 12]. The resonse surfaces at time stes t = 0.5 and t = 3.5 are shown in figure 2. The differentiation of the resonse surface in the reresentation of B-slines can be readily obtained, which will be imortant in section 5.2, where the adation of the density is discussed. The resonse surface, once comuted, serves as initial data for the comutation of the resonse surface at the next time ste. That means, that the number of deterministic evaluations at time t j+1 is indeendent of the number of deterministic evaluations at time t j, so that exensive comutations of additional solutions are not needed. 5

6 Figure 2: Resonse surfaces of the solution rocess: y(0.5, ) (left) and y(3.5, ) (right) for the arameter [1.9, 2.1]. The squares reresent the deterministic solutions for arameters {1.900, 1.933, 1.967, 2.0, 2.033, 2.067, 2.100}. 5.2 Comutation of the density Having comuted the resonse surface we can readily determine the density of the random variable y(t j, ). This method is based on the density transformation formula I R + 0 ρ ( j) : ŷ ρ (0) (r) d d y(t 1 j, ), r R =r with I R being the range of y and the sum going over all re-images r R of y(t j, ) = ŷ, see [7]. This formula was first used in the context of stochastic sectral exansions by P. Vos in [8]. In figure 3 (left) we show the resonse surface at time t = 5.3. The dotted line is the tangent at oint (1.985, 1.991). By formula (14) we comute the density of the solution, which is lotted in figure 3 (right). The determination of the re-images of ŷ [1.965, 2.009] is done by Newton s method. (14) Figure 3: Resonse surface of y(5.3, ) (solid line) with resective tangent at oint (1.985, 1.991) (left) and weight function of the distribution of the random variable y(t, 5.3) (right). The need of the determination of the re-images of the range of y(t, ) is a major drawback of the adation of the density. In figure 4 (left) the resonse surface y(3.5, ) is shown. At this time ste the resonse surface has a local minimum and the re-image comutation in a neighborhood E of the local extrema, at = 1.965, is an ill-osed roblem. To handle this roblem, we eliminate intervals {E i } q i=1 in I, where the derivative of y with resect to the random arameter, falls below a rescribed threshold. Figure 4 (right) shows the resonse surface y(3.5, ) for [1.961,1.970]. The area below the dotted line will be excluded in the comutation of the density. We obviously lose information by the elimination of {E i } q i=1. 6

7 We will address this roblem in the section 6. For values in the remaining set Î := n I k := I \ q E i i=1 we determine the re-images using Newton s method, so that we are able to comute the conditional density with resect to Î. Note, that the intervals {I k } n, together with the intervals {E i} q i=1 define a artition of I. The elimination of the intervals E i from I is equivalent to set their masses to 0. Thereby the total mass of Î, m T := P(Î) = n P(I k ), is unequal to one. We correct this discreancy by choosing the masses P(I k )/m T as artial masses of the sets I k, k = 1,...,n Figure 4: Cut-off of the area of the resonse surface of y(3.5, ) (solid line) where re-image comutation is an ill-osed roblem (left) and magnified to the interval [1.961, 1.970] (right). 6 Numerical results We return to the Wiener-Askey calculus for solving random differential equations and aly it to the random Van-der-Pol equation (12), choosing m = 2 for the length of the truncated exansion. We comute the exected value and variance by the formulas from section 3. The results are shown in figure 5. In E(y(t, )) V(y(t, )) t t Figure 5: Exected value (left) and variance (right) of y(t, ) in the time interval [0,60] comuted by Wiener- Askey calculus with 3 terms in the truncated exansion. order to estimate accuracy, we comute a reference solution by Monte-Carlo simulation, using 200, 000 samles. The absolute error of the variance is lotted in figure 6 (left). As we discussed in section 4, the variance aroximated by Wiener-Askey calculus loses accuracy with increasing time. Adding more terms to the truncated series exansion does not reduce the roblem. To illustrate this 7

8 situation we comuted the solution by Wiener-Askey calculus with 6 terms in the truncated series exansion. Although the magnitudes of the coefficients decrease with increasing olynomial order, the gradient of the descend becomes larger with increasing time. Thus adding more terms to the truncated Wiener-Askey series is not an aroriate way to comute an accurate aroximation of the solution rocess. The gradient of the descent of the series coefficients q i has to be drawn into consideration as well. Instead of increasing the length of the truncated Wiener-Askey exansion, we aly assm to reduce the error. The choice of initializing values are ste size h = 0.1, U [1.9,2.1] for the distribution of, y(0, ) 0.1, ẏ(0, ) 0.0 and TOL = 10 5 (see aendix A). The associated lots of the exected value and the variance are omitted, because they differ only slightly from figure 5. The error of the variance, comuted by assm comared to the Monte-Carlo simulation, is shown in figure 6 (right). It is reduced considerably in comarison to the error of the Wiener-Askey calculus, c.f. figure 6 (left). In figure 7 the arithmetic mean of the absolute values of the third coefficients of the truncated stochastic sectral exansions are lotted in the interval [0, 60]. Although these values are suosed to be below TOL = 10 5, they sometimes exceed this tolerance. For examle, the value at time t = 16.9 is much larger than TOL. This haens, because of the exclusion of the intervals E i of the range of the solution y(t, ). If Î I, the density of y(t, ) is only aroximated, so that the absolute value of the third coefficient of the truncated stochastic sectral exansion has to cature the offset between the aroximated density and the exact density. Future work will deal with how to find a good aroximation of the exact density. 7 Conclusions and Objectives In this article we resented the method of adative stochastic sectral exansion. We have shown, that assm is an adequate method to reduce the error arising in the long-term integration of the random Vander-Pol equation. The standard way to cature the error, increasing the length of the truncated series exansion, is often not feasible due to the increasing comutational effort and further more does not guarantee a good aroximation. assm is based on the adation of the stochastic sectral exansion to the density of the solution rocess y(t, ). Doing so, the comutational time is about one day for the roblem in section 6. This is, comared to the Wiener-Askey method, which needs only an hour, a considerable drawback. But nevertheless using Monte-Carlo simulation took several weeks to aroximate the solution with 200, 000 samles. The secification of comutational time are for a Sun Fire V40z machine with 4x Dual Core AMD Oteron(tm) Processor 875 (2,2GHz). We see, that assm is a very romising method for high accuracy comutation of solutions of random differential equations. The most challenging roblem of this method is, that the determination of the exact density may lead to ill-osed roblems. In those cases, an alternative density has to be chosen. We roose the aroximation by elimination of the sets E i as described in section 5.2. Of course, this is not the only ossible choice. In order to introduce an error, which is not worse than in the Wiener-Askey t t Figure 6: Left: absolute error of variance comuted by Wiener-Askey calculus in comarison to the Monte- Carlo reference solution using 200, 000 samles. Right: the resective absolute error of assm. 8

9 t Figure 7: Arithmetic mean of the absolute values of the third coefficients of the truncated stochastic sectral exansions in the assm. calculus we can chose ρ ( j) = ρ (0) whenever Î I. To find an aroriate choice of aroximation in order to minimize the truncation error is u to future work. 8 Acknowledgement We are very much indebted to Dr. U. Wever, Dr. M. Paffrath from Siemens Cororate Design under Prof. Dr. A. Gilg for their suort. 9 References [1] F. Augustin, A. Gilg, M. Paffrath, P. Rentro, and U. Wever. Polynomial chaos for the aroximation of uncertainties: Chances and limits. Euroean Journal of Alied Mathematics, 19(2): , Aril [2] R. G. Ghanem and P. D. Sanos. Stochastic finite elements: A sectral aroach. Sringer-Verlag New York, [3] E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equations 1. Sringer-Verlag Berlin Heidelberg, [4] W. Kamowski, P. Rentro, and W. Schmidt. Classification and numerical solutions of electric circuits. Surveys on Mathematics for Industry, 2:23 65, [5] O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem. Uncertainty roagation using wiener-haar exansions. Journal of Comutational Physics, 197(1):28 57, June [6] O. P. Le Maître, H. N. Njam, R. G. Ghanem, and O. M. Knio. Multi-resolution analysis of wienertye uncertainty roagation schemes. Journal of Comutational Physics, 197(2): , July [7] A. Paoulis and S. U. Pillai. Probability, Random Variables and Stochastic Processes. McGraw- Hill, New York, 4 (international edition) edition, [8] P. Vos. Time-deendent olynomial chaos. Master s thesis, Delft University of Technology, Faculty of Aerosace Engineering, November [9] X. Wan and G. E. Karniadakis. An adative multi-element generalized olynomial chaos method for stochastic differential equations. Journal of comutational hysics, 209: , [10] D. Werner. Funktionalanalysis, volume 5. Sringer-Verlag Berlin Heidelberg, [11] N. Wiener. The homogeneous chaos. American Journal of Mathematics, 60(4): , October [12] D. Xiu. Efficient collocation aroach for arametric uncertainty analysis. Communications in Comutational Physics, 2(2): , Aril [13] D. Xiu. Fast numerical methods for stochastic comutations. Communications in Comutational Physics, 5: , [14] D. Xiu and G. E. Karniadakis. The wiener-askey olynomial chaos for stochastic differential equations. SIAM J. Sci. Comut., 24(2): ,

10 A The algorithm of assm We introduce the algorithm of the adated Stochastic Sectral Method (assm). It is based on a stochastic sectral exansion with adated density: ÁÆÁÌÁ ÄÁ Ë Ì Ò Ø Ð Ø Ñ Ø ¼ Ò Ø Ô Þ Ë Ì Ò Ø Ñ Ì Ë Ì Ò ØÝ ¼ Ó Ô Ë Ì Ò Ø Ð Ú ÐÙ Ý ¼ Ô Ë Ì ØÓÐ Ö Ò ÌÇÄ Æ ÏÀÁÄ Ø Ì Ø Ø ÈÈÊÇ ÁÅ Ì Ý Ø Ô Ý ¹ ÔÐ Ò ÇÅÈÍÌ Ø Ö Ó ÒØ ÛºÖºØº Ø ¹ Á Õ ¾ ÌÇÄ ÇÅÈÍÌ Ò ØÝ Ø Ó Ý Ø Ô Ê Í ÙÔÔÓÖØ Ó Ò ØÝ ÇÆËÌÊÍ Ì ÓÖØ Ó ÓÒ Ð ÔÓÐÝÒÓÑ Ð ÛºÖºØº Ø ÄË Ø Ø ¹ Æ ÇÅÈÍÌ ½ Ø Ò ¾Ò Ó ÒØ ÛºÖºØº Ø ÇÅÈÍÌ ÜÔ Ø Ú ÐÙ Ø ÇÅÈÍÌ Ú Ö Ò Î Ø Æ 10