TIME SAMPLING OF DIFFUSION SYSTEMS USING SEMIGROUP DECOMPOSITION METHODS

Transcription

1 TIME SAMPLING OF DIFFUSION SYSTEMS USING SEMIGROUP DECOMPOSITION METHODS J. M. Lemos J. M. F. Moura INESC-ID/IST, R. Alves Reol 9, -29 Lisboa, Portugal, Carnegie Mellon University, Pittsburgh PA , U. S. A., Abstract: The paper consiers the time sampling of the a priori probability istribution function of the state of a iffusion process by semigroup ecomposition methos. We approximate the semigroup generate by the evolution equation associate to the iffusion by Trotter s formula. We benchmark the approximation against an exact metho, viz. the Wei- Norman ecomposition). We illustrate both methos through the iscretization of the Fokker- Planck equation associate with the PLL estimation error probability istribution function. Keywors: Sampling, iffusions, Fokker-Planck equation, semigroups, Trotter s formula, Wei-Norman metho, istribute parameter.. INTRODUCTION Diffusion moels (Jazwinski, 97; Lipster an Shiryayev, 984) fin many important applications in iversifie fiels of science an technology. In Biology examples range from cell (e.g. ion currents) (Smith, 22) to population moels (population ispersal) (Britton, 23). Inustrial processes with large spatial imensions provie also examples. In the class of transport/reaction-iffusion processes (Grinro, 996), one or more variables with a space epenency are escribe by a iffusion moel or a egenerate iffusion moel as in the case of istribute collector solar fiels (Silva et al., 23). The classical problem of course is heat iffusion in a bar heate at a central point, whose temperature satisfies the well known heat equation. Diffusion moels in ynamic congestion control protocols for communication networks allow to aress traffic variability (Mukherjee et al., 99). Analysis of tracking properties of aaptive algorithms for time varying systems, both for signal processing (Benveniste, 987) an control (Coito et al., 995) relies on iffusion moels. As a final example, if- Part of the work of J. M. Lemos has been one uner the project AMBIDISC, contract POSI/SRI/36328/2. fusion moels are a basic starting point for signal processing an communication problems. The phase lock loop (PLL) estimation error verifies a iffusion moel (Viterbi, 963) an will hereafter be taken as a case stuy. The time evolution of the state of a iffusion process satisfies a stochastic orinary ifferential equation, an its probability ensity function (p..f.) is the solution of the Fokker-Planck partial ifferential equation (Jazwinski, 97). Computer base esign an implementation of estimation an control algorithms require iscrete time moels that aequately represent the system ynamics. When a probability istribution (or other space functions such as temperature) is use as a state, this becomes the problem of time sampling an infinite imensional system. The sampling problem consists of, given the state istribution at time t k, compute (eventually approximate) the state istribution at the next sampling time t k+. Once the iscrete time moel relating both space functions is available, for the sake of illustration, it may be applie to a control problem by selecting the value of the manipulate value to apply to the plant such that the state istribution attains some prescribe value. This, together with a time varying sampling

2 interval, is the path followe in (Silva, 23; Silva et al., 23a) to esign preictive controllers for istribute collector solar systems. In relation to estimation problems, the Fokker-Planck equation correspons to the preiction step in optimal nonlinear filtering problems where the message is continuous an the observations are iscrete. As such, aressing the iscretization of the Fokker-Planck equation shes light on the relation between continuous an iscrete optimal nonlinear filters. To solve the sampling problem for infinite imension systems, we resort here to semigroup ecomposition methos. For the class of problems consiere, a semigroup is (McBrie, 987) a family of boune (continuous) linear operators, inexe by time, which escribe the time response of ynamic systems evolving from an initial conition state. Semigroups may be ecompose by expressing them in simpler semigroups that can be exactly compute. This paper consiers two ecomposition methos. The Wei-Norman metho (Wei et al., 964; Ocone, 98) is an exact metho (which may not always be applie), relying on a change of time scale, given by the solution of the so calle Wei-Norman equations. The secon is an approximate metho known as Trotter s formula (Trotter, 959). Actually, Trotter s formula can be obtaine as a first orer Taylor s expansion of the solution of the Wei-Norman equations, whenever these represent the unerlying semigroup. The Wei-Norman metho was originally evelope for finite imensional operators an applie to express the solution of orinary ifferential equations as a prouct of exponentials (Wei, 964). In Numerical Analysis, this set of ieas was the subject of recent interest within the context of geometric integration an iscretization methos for ifferential equations on manifols, base on the use of Lie group methos (Celleoni et al., 2). The metho was extene to infinite imensional operators an applie to the iscretization of optimal non-linear filters (Ocone, 98; Brockett, 98; Cohen e Lara, 997). Trotter s formula has been use in Quantum physics to solve Schröeinger s equation (Weinholtz, 982) an is sometimes reiscovere to solve engineering problems in a somewhat empirical way without any reference to Trotter (Dresnack, 968; Camacho et al., 988). In (Silva, 23), a controller for a istribute collector solar fiel is esigne on the basis of a iscrete time moel obtaine by the application of Trotter s formula. In (Neihart et al., 998) error bouns for Trottter s formula approximation are erive (see also the references therein). The contribution of this paper consists in showing how semigroup ecomposition methos may be applie for sampling the Fokker-Planck equation associate to iffusion processes an in relating the time propagation in this way an by sampling the original iffusion moel. The paper is organize as follows: After stating the paper s purpose an placing its contribution in perspective in this introuction, basic concepts on semigroups incluing the Wei-Norman metho an Trotter s formula are reviewe in section 2. The application to iffusion systems is presente in section 3, where the Fokker-Planck equation is sample using Trotter s formula. In the case of linear iffusions, the Fokker-Planck equation is solve by the Wei-Norman approach an this is explore in orer to she light on the type of approximation yiele by Trotter s formula. Furthermore, it is shown that Trotter s formula yiels a probabilistic interpretation of the iscretization of the Fokker-Planck equation. Conclusions are rawn in section SEMIGROUP CONCEPTS This section presents basic concepts an results concerning semigroups that will be use subsequently. Consier the Banach space X of continuous functions equippe with the supremum norm. Definition (McBrie, 987) A semigroup of operators of class C is a family of operators T t efine in X an inexe by the parameter t R (time) such that: () T t is efine t ; (2) T t satisfyies the semigroup conition: s, t R T t+s = T t T s () (3) T t satisfyies the continuity conition lim T tx = x x X t (4) T t is boune t : c R : x X T x c x Definition 2 (McBrie, 987) The infinitesimal generator of the semigroup T t is the operator efine by A = lim t t (T t I) where I is the ientity operator. Remark The set B(X) of boune linear operators in a Banach space X is itself a Banach space with respect to the norm inuce by the norm efine in X: { } T t = Tt x sup : x X \ {} x Uner this norm, efinition 2 states that the semigroup T t satisfies the following ifferential equation, known as an evolution equation: t T t = AT t = T t A (2) with the initial conition T = I. The solution T t of (2) is referre to as the integral operator corresponing

3 to A or, by abuse of notation, as the exponential of A. 2. Trotter s formula Consier the situation in which A is the sum of two operators A an A 2. Let T t an T 2 t be the corresponing operators (semigroups), i. e. assume that t T t i = A i Tt i i =, 2 (3) while T t satisfies t T t = (A + A 2 ) T t (4) In general, it it is not true that T t results from the composition of Tt an Tt 2. However, this is approximately true for small t, meaning that T t can be approximate by the iterate composition of T an T 2 over small intervals of time. This is state in the following theorem: Theorem (Trotter, 959) Let T t an T 2 t satisfy the norm conition: ω R : t > T i t e ω it i =, 2 an that D (A + A 2 ) = D (A ) D (A 2 ) is ense in X, where D (A) enotes the omain of A. Then, (the closure of) A + A 2 generates a semigroup of class C iff (the closure of) R (λi A A 2 ) is ense in X for some λ > ω + ω 2, where R(A) enotes the range of A. If A + A 2 (or its closure) generates a semigroup of class C, this is given by T t = lim ( T T 2 ) t/ (5) where t/ represents the greatest integer that oes not excee t/. Expression (5) is commonly known as Trotter s formula. The approximation emboie by Trotter s formula may be extene to a finite sum of operators. Remark 2 Trotter s formula is one of a family characterize by the equality of the first erivative at t = of the semigroup generate with its approximation. Another approximation formula is the semi-sum formula (Weinholtz, 98), given by [ ] n T n 2 (T 2 + T2 ) 2 (6) 2.2 The Wei-Norman metho Since, in (4), T t is in general not given by the composition of Tt an Tt 2, one can ask whether there is a change of time variable such that the composition results in an exact equality. This is the approach of the Wei-Norman metho (Wei, 964; Ocone, 98). The Baker-Campbell-Hausorf (BCH) formula plays here a key role. Let L be the Lie Algebra generate by the linear operators A,..., A m. This is the vector space generate by all the operators, together with the operators generate, in all possible combinations, by the Lie prouct efine by [A i, A j ] = A i A j A j A i A i, A j L (7) Theorem 2 (Ocone, 98; Lara, 997) Assume that L is of finite imension an enote by e Ait the integral operator corresponing to the infinitesimal generator A i. The following equality, known as the BCH formula, hols for A, B L: e A Be A = B + [A, B] + [A, [A, B]] + 2! Remark 3 + [A, [A, [A, B]]] +... (8) 3! Remark that for commuting operators, i.e. when AB = BA, the formula reuces to e A Be A = B (9) In orer to explain the Wei-Norman metho, consier the evolution equation (4) in the situation in which A an A 2 form a basis of the Lie algebra that they generate, with the commuting rule where γ R is a known parameter. An example is provie by the PDE [A, A 2 ] = γa 2 () p(x, t) = u(t) p(x, t) + R(t) () t x which provies a simple moel for a istribute collector solar fiel (Silva, 23). Here p(x, t) is the temperature of a moving flui (oil) at position x an time t, u is the oil flow an R is a continuous function moelling incient sun raiation. Defining the infinitesimal generators A p(x, t) = u(t) p(x, t) (2) x A 2 p(x, t) = R(t) (3) equation () is associate to the evolution equation (4), where T t is the integral operator transferring the temperature spatial istribution at time t = to the temperature spatial istribution at time t >. An abuse of notation is mae because the operators epen on t. Assume that the exact solution of (4) can be written as T t = e Ag(t) e A2g2(t) (4) an look for conitions satisfie by the real value functions g an g 2. To fin these, ifferentiate (4) an use (3) an the special case (9) of the BCH formula to get t T t = ġ e A g A e A 2g 2 + ġ 2 e A 2g 2 A 2 =

4 = [ ġ e A g A e A g + +ġ 2 e A g e A 2g 2 A 2 e A 2 g 2 e A g ] e A g e A 2g 2 = = [ ġ A + ġ 2 e A g A 2 e A g ] Tt (5) From (8) an (), it follows that e Ag A 2 e Ag = e γg(t) A 2 (6) an hence, combining (5, 6) an equating to (4)) A + A 2 = ġ A + ġ 2 (t)e γg (t) A 2 (7) Since A an A 2 form a basis, the corresponing coefficients shoul be equal an the Wei-Norman equations for the case at han follow: ġ (t) = (8) ġ 2 (t) = e γg (t) Since T = I, the initial conitions must be (9) g () = g 2 () = (2) This ODE initial problem has the solution g (t) = t (2) g 2 (t) = ( e γt ) (22) γ Hence, the Wei-Norman ecomposition of the semigroup that represents the exact solution of (4) is T t = e A t e A2( γ (e γt ) ) (23) The integral operators corresponing to A an A 2 are exp(ta )p(x, ) = p(x t exp(ta 2 )p(x, ) = p(x, ) + u(σ)σ, ) (24) t R(σ)σ (25) Their composition accoring to (23) yiels the solution of () as p(x, t) = p(x t u(σ)σ, ) + t R(σ)σ (26) THis is the formula use in (Silva, 23) as a basis for control esign. Although (26) can be obtaine using the stanar Laplace s metho, it is erive here as a simple application of the Wei-Norman metho. In general, the Wei-Norman equations always have a solution, although it may not be global. It shoul also be remarke (Brockett, 98) that the functions g i only epen on the constants γ that efine the commuting rules of the infinitesimal operators A i an not on the actual form of the operators. Thus, by solving the Wei-Norman equations, a whole family of linear evolution equations is solve. 2.3 Error bouns Whenever the Lie algebra generate by the partial operators is finite, the Wei-Norman metho can be use to establish error bouns on approximate semigroup ecomposition methos such as Trotter s formula. Consier again the example of section 2.2, in which the Wei-Norman ecomposition is given by (4), while Trotter s formula approximation, enote Tt T is given by T T t = e A t e A 2t (27) The norm of the error over a small interval is given by T T T = e A g ( ) e A 2g 2 ( ) e A e A 2 (28) where enotes the operator norm efine in Remark. Expan g ( ) an g 2 ( ) in Taylor series aroun = an assume that (as for boune operators) the following series expansions converge e Ai = I + A i + 2! A2 i (29) By expaning the error operator in (28), it is possible to obtain the following boun: T T T 2 [A, A 2 ] 2 + O( 3 ) (3) Remark 4 Trotter s formula equates terms up to first orer an, if the operators commute, it is exact. This is a consequence of the fact that the first orer expansion of the solution of the Wei-Norman equations is g i ( ) (3) Actually, Trotter s formula correspons to the highest orer Taylor s approximation of the solution of the Wei-Norman equations that oes not imply the computation of the Lie brackets of the infinitesimal operators. For error bouns for unboune operators see (Neihart, 998). 3. THE FOKKER-PLANK EQUATION The semigroup ecomposition methos reviewe in the previous section are now applie to sample the PDE that propagates in time the p.. f. of the state of a iffusion moel. 3. Diffusion moels Consier the iffusion process with state x given by the solution of the stochastic ifferential equation x t = f(x t )t + σw t (32) where σ is a constant parameter, the initial conition x() = x is a ranom variable with p..f. p x an w t is a stanar Wiener process. For t > the probability ensity function p(x, t) of the state x of the iffusion process satisfies the Fokker-Planck equation (Jazwinski, 97), given by p t = f x(x)p f(x) p x + σ2 2 p 2 x 2 (33)

5 with the initial conition an the bounary conitions p(x, ) = p x (x) (34) p(±, t) =, t > (35) Two particular cases of interest to be consiere hereafter are the linear case, in which f(x) = ax (36) an the PLL error ynamics (Viterbi, 963), in which f(x) = ax K P LL sin(x) (37) Here, a, an K P LL are constant parameters. 3.2 Applying Trotter s formula To apply Trotter s formula to the Fokker-Planck equation (33), efine the infinitesimal generators L, L 2, an L 3 by L p(x, t) = f x (x) p(x, t) (38) L 2 p(x, t) = f(x) p(x, t) (39) x L 3 p(x, t) = p(x, t) (4) 2 x2 The Fokker-Planck equation (33) is written t p(x, t) = (L + L 2 + L 3 ) p(x, t) (4) σ2 2 The following proposition summarizes a number of facts neee for applying Trotter s formula. The proof is reaily one using stanar techniques. Proposition A) The semigroups T i t generate by the infinitesimal generators L i, i =, 2, 3, are given by T p(x, t) = p(x, t) exp ( f x (x) ) (42) T p(x, 2 t) = p(α (α(x) + ), t) (43) with x ξ α(x) = f(ξ) (44) α enoting the inverse of α, an T 3 p(x, t) = p(x, t) G(x, ) (45) where stans for convolution an G is a Gaussian kernel given by ) G(x, ) = exp ( x2 (2πσ 2 /2 ) 2σ 2 (46) B)By using a Taylor series evelopment, the following approximations are seen to hol for small : T p(x, t) p(x, t) (47) + f x (x) an T 2 p(x, t) p(x f(x), t) (48) C)The operators T i, i =, 2, 3 satisfy the conitions of Definition as well as the norm conition of Theorem. Continuous state equation Propagate the a priori pf of the state (continuous time) Fokker-Planck equation Discretize in time (first orer) Trotter's formula Discrete state equation Propagate the a priori pf of the state (iscrete time) Operators for iscrete time pf propagation Fig.. Probabilistic interpretation of sampling the Fokker-Planck equation using Trotter s formula. Remark 5 Instea of using the expressions (42-45) or the approximations (47, 48), the semigroups T i, i =, 2, 3 may be approximate by applying finite ifferences to each of the equations p t = L ip (49) The semi-sum formula (6) reuces then to the explicit metho for the iscretization of the PDE an Trotter s formula to the scheme propose heuristically in (Bella et al., 968). Proposition 2 Given the operators efine in Proposition, Trotter s formula yiels the following approximation of the semigroup generate by the Fokker-Planck equation (33): p(x, t + ) T T 2 T 3 p(x, t) (5) 3.3 Probabilistic interpretation The following proposition states a probabilistic interpretation of the application of Trotter s formula to the Fokker-Planck equation. Sample the iffusion moel (32) to obtain the stochastic ifference equation: x k+ = x k + f(x k ) + σ(w k+ w k ) (5) where x k := x(k ), w k := w(k ) an R is the iscretization step. As the solution of (5) converges to the solution of (32) in mean square (Jazwinski, 97). Proposition 3 For small the iagram of fig. is vali, i. e., the operators which propagate in iscrete time the p..f. of the state of the iscrete moel are the same as the ones obtaine by applying Trotter s formula to the Fokker- Planck equation. Proof of proposition 3 Since the ranom variable ζ = θ(x k ) = x k + f(x k ) (52)

6 is inepenent of the increment w k+ w k of the Wiener process it follows that p x(k+) = p ζ N (, σ 2 ) (53) where N enotes a Gaussian kernel an p z enotes the p.. f. of the ranom variable z. Furthermore From (52) p ζ (ζ) = p x(k) θ(x k) x(k)=θ (ζ) (54) x k θ(x k) = + f x (x k ) x k (55) an the action of the operator T is recovere (compare with (47)). Furthermore, again from (52) When is small, x k ζ an x k = ζ f(x k ) (56) x k ζ f(ζ) (57) i.e. by (48) the change of variable in (54) is seen to be approximately the same as the one efine by the operator T 2. In conclusion, computing the p..f. of x k+ from the p..f. of x k involves the operations efine by Trotter s formula approximate up to first orer in. Remark 6 For (54) to be vali, the function θ(x k ) must be monotonous, i.e., its total erivative must be either strictly positive or strictly negative in all its omain. This is true in the linear case (36). For the PLL error ynamics (37) θ x k = + a K P LL cos x k (58) Since K P LL > (Viterbi, 963) an a, the monotonicity conitions are then either K P LL a > (59) for positive erivative or, for negative erivative or K P LL a > an K P LL > a (6) K P LL a < an K P LL < a (6) 3.4 The linear case Although in the linear case the Fokker-Planck equation may be solve exactly by stanar methos, it is interesting to apply the Wei-Norman metho an compare the exact ecomposition thereby obtaine with the approximation resulting from Trotter s formula. Assuming therefore that (36) hols, the infinitesimal operators L i, i =, 2, 3 efine in (38-4) are a basis for a Lie Algebra with the commuting rules [L, L 2 ] = (62) [L, L 3 ] = (63) [L 2, L 3 ] = 2aL3 (64) Let T t be the integral operator expressing the solution of the Fokker-Planck equation. Accoring to the Wei- Normal metho, assume a ecomposition of the form T t = e Lg(t) e L2g2(t) e L3g3(t) (65) By following the proceure escribe in section 2.2, the functions g i (t), i =, 2, 3 are seen to satisfy the set of ODE s with zero initial conitions: Therefore g (t) = t (66) g 2 (t) = t (67) g 3 (t) = e 2ag 2(t) t (68) (69) e 2at T t = exp(l t) exp(l 2 t) exp(l 3 t) (7) 2at Observe now that the integral operator efine by U t = exp(l 3 g 3 (t)) (7) is to be interprete as the solution of the evolution equation t U t = L 3 ġ 3 (t)u t (72) Therefore, the integral operator e exp (L 2at ) 3 t 2at is to be obtaine by solving the PDE σ2 2 p(x, t) = e 2at p(x, t) (73) t 2 x2 with the initial conition p(x, ) = p (x) < x < + (74) This problem can be solve by applying the Fourier transform in x to both sies of (73). The resulting integral operator is a convolution of the initial conition with the Gaussian kernel Q(x, t) = exp [ ] x 2 σ 2 a ( e 2at ) [ π σ 2 a ( e 2at ) ] /2 (75) Assume that the initial conition is p(x, ) = δ(x) where δ is the Dirac impulse function. Eq. (7) expresses therefore the solution of the Fokker-Planck equation in the linear case by the following operations:.a convolution with the Gaussian kernel (75). Since the initial conition is an impulse, the Gaussian kernel is left unchange.

7 .6 Variance Exact Trotter s approximation t Fig. 2. Time evolution of the variance of the solution of the Fokker-Planck equation for the linear case. 2.A replacement of x by xe at (corresponing to operator T 2 ). 3.Multiplication by e at. The resulting solution is the Gaussian function { } p(x, t) = 2πσ 2 x (t) exp x2 2σx(t) 2 (76) with the variance σx(t) 2 = σ2 ( e 2at ) (77) 2a Remark 7 The approximation yiele by Trotter s formula is obtaine by replacing g 3 = ( e 2at )/2a in (7) by its first orer Taylor series evelopment an hence must be only a local approximation. This may be evience by observing that the approximation given by Trotter s formula is a Gaussian as in (76) but with the variance replace by σ 2 T (t) = σ 2 te 2at (78) Fig. 2 compares the exact variance σ 2 x(t) with σ 2 T (t). As it is apparent they are close only for small values of t. Hence, Trotter s formula must be use iteratively, over a small time interval each time. The reason for this is apparent from (49). Trotter s formula implicitly assumes that the r.h.s. of the thir equation in (66) is, neglecting the influence of operator T 2. In general, Trotter s formula neglects the interaction between the operators along the intervals uring which it is applie. 3.5 Numerical example The PLL error ynamics In the cases in which the Lie algebra associate to the infinitesimal generators is not of finite imension, the Wei-Norman ecomposition cannot be compute an one has to resort to numerical methos. Trotter s formula is then a possibility, being particularly useful if a control application of the sample moel is planne. p(x) x 2 Time [PLL time constant] Fig. 3. Time evolution of the solution of the Fokker- Planck equation for the PLL, as compute by Trotter s formula. The numerical implementation of the sampling algorithm efine by (5) raises a number of issues concerne with space iscretization. The first ifficulty relies on the computation of T 2 p(x, t) since this requires p(x f(x), t), whose value is usually not available in the computer memory (because it oes not exactly correspon to one of the points of the spatial gri). Although a time varying sampling interval can be use such as to match the space an time gris (Silva, 23), interpolation is neee if the time increment is require to be constant. Linear or cubic interpolation may be use. Cubic interpolation provies a better approximation but has the rawback of not ensuring the positivity of the solution. Furthermore, the use of interpolation inuces an unesirable effect referre in (Dresnack, 968) as pseuo-iffusion an which amounts to not preserving space impulse functions, but instea spreaing them over ifferent gri points. Another important issue are the stability conitions to be respecte when approximating the ifferent operators with a iscrete gri. For T 3 (convolution with a Gaussian kernel) this amounts to 2 h 2 < C (79) where C is a constant an h is the spatial step (length of the interval between two gri points). This means that, for a given spatial resolution, the time step cannot be lower than a certain interval. Fig. 2 shows the time evolution of the solution of the Fokker-Planck equation for the PLL, as compute by Trotter s formula when a = an K P LL =. In this case, the error will iffuse along the real line, spreaing over wier an wier intervals. In orer to reuce the computational buren, the space gri is graually increase by aing more an more intervals of length 2π.

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