The infinite dimensional parameter estimation for stochastic heat diffusion equations is considered using the method of sieves. The consistency property is also studied for the long run data

Aihara, ShinIchi

Bagchi, Arunabha

English

University of Twente Research Information

Statistics & Probability Letters 8 (1989) 279-287 North-Holland August 1989 INFINITE DIMENSIONAL PARAMETER IDENTIFICATION FOR STOCHASTIC PARABOLIC SYSTEMS ShinIchi AIHARA Departmeni of Electrical Engineering, Science University of Tokyo, I-3, Kagurazaka, Shinjuku-ku, Tokyo, 162, Japan Arunabha BAGCHI Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlandr Received April 1988 Revised October 1988 Abstract: The infinite dimensional parameter estimation for stochastic heat diffusion equations is considered using the method of sieves. The consistency property is also studied for the long run data. Ah4S 1980 Subject Classifications: 93C29, 93E12. Keywords: parameter estimation, inference for stochastic process, stochastic heat diffusion equation. 1. Introduction The identification of infinite dimensional parameters for stochastic distributed parameter systems becomes important in a wide variety of physical phenomena: stochastic control (Balakrishnan, 1975) and large flexible structures (Curtain and Kotelenez, 1987). In contrast with the identification problem for a finite dimensional unknown parameter vector, it may be impossible to construct an algorithm for generating directly an estimate for an infinite dimensional parameter vector. One possible way to construct a practical estimator is to project the infinite dimensional parameter vector to a finite dimensional subspace. For consistency, one must then check whether the finite dimensional estimate converges to the true one as the number of samples and also the dimension of the projected subspace increase without bound. In this paper, by using the method of sieves, originally proposed by Grenander (1981), an identification algorithm for identifying the unknown diffusion coefficient of a heat equation is proposed, based on a certain log likelihood type cost function. Nguyen en Pham (1982) used this technique for studying the estimation of time-dependent parameters for lumped systems based on independent observations, and recently Bag&i (1985) succeeded in applying this method to the identification problem for a hereditary system with distributed delay. Here we apply the method proposed in Bag&i (1985) to identify spatially dependent parameters for stochastic parabolic systems. 2. Stochastic parabolic systems We set G c R”, an open bounded domain with a regular boundary r, T =]O, tr[, and we work with the following Hilbert spaces: 0167-7152/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland) 279 Volume 8, Number 3 STATISTICS & PROBABILITY LETTERS August 1989 L2(G) is the space of square integrable functions on G with the norm 1 . 1 and inner product (. , -). (We also use 1 .I as the norm in R”.) H”(G) is the mth order Sobolev Space, i.e. the space of functions + such that @EL’(G), D*+EL*(G) VLY, JaJ <rn (a = (a,, . . . , a,), a, natural numbers, 1 a I = a1 + . . . +a, and D%$ = (a,,+ ... +an/axpl . . . axzn)+). H;(G)= {+I+EH~(G), D”$=Oand r, J(Y\ <m-l}. H-“(G) is the dual of Horn(G). Define V= H,(G) c H = L2(G) c V’ = H-‘(G) where we assume that the injection of V + H is compact (Adams, 1975, p. 144). Define the linear operator A(a”) E L(V, V’) such that (2.1) where (Cl) O<a<a’(x),<fi VxEG and (C2) .’ E C2(G). From (C2), the operator A(a’) with range restricted to H defines an unbounded operator which is still denoted by A(a”) with the domain D = H,’ n H*. Let w be a H-valued Brownian motion process with the incremental covariance Q E L,(H; H) where L,( H; H) denotes the space of trace class operators. Consider the following stochastic diffusion equations: (2.2) Theorem 1 (Bensoussan, 1971, p. 189; Paradoux, 1975, p. 48). Under (Cl), (C2) and (C3) u. E L2(Q; H), there exists a unique solution u(t) of (2.2) such that u E L2(9; C(T; H) n L2(T; V)). 0 (2.3) Theorem 2 (Mizohata, 1973, Theorem 5.5, Chapter 5; Tanabe, 1979, Theorem 1.4, Chapter 3). Under (Cl) and (C2), -A generates strongly continuous, contraction semigroup T, such that d7;/dt = -AT, T, =I, 17;@12~Ce-at1+)2 forsomeC>O, a>O, +EH. •I Define the auto-correlation operator A(t, s) = E{ u(t) @ u(s)}, where 41@4*=41(4,, *I. (2.4) 280 Volume 8, Number 3 STATISTICS & PROBABILITY LETTERS August 1989 For C#I E H and noting that Q = Cz,q,e, @ e, (e,; orthonomal basis in H), we have A(t, s) = T,E{u,@ u,}r,* + jtAST,_.Qr,Z,dr. 0 (2.5) We are interested in the stationary system state and, therefore, we choose the initial condition uO to be j ’ T_, dw(r). (2.6) -m The auto-correlation A (r, s) then becomes A(r - s) =A(t, s) = j?_,Qq’, dr. --m (2.7) Theorem 3. J 0 mlA(r) I&v;v) dr< 00 (2.8) where L,( - ; .) denotes the space of Hilbert-Schmidt operators. IJ This result can be easily established using the property of T,. For more details, refer to Aihara and Sunahara (1989). 3. Parameter estimation procedure Noting that the incremental covariance Q of the Brownian motion process w is a trace class operator, it is not clear how a likelihood function may be defined for the infinite dimensional evolution equation (2.2). Hence, instead of introducing the log-likelihood function as the cost, we consider L(a) = -j”i (a(x)%, $(-A)-’ du(r)) 0 r=l I I -; &4(t) a 4")~ , , a,(&(-A)-']*a(x)y] dt where A = 2 ~‘(+‘k3x,‘, i=l and from Lions (1969, p. 195), (-A)-’ is defined by (-A)-’ = E (l/h,)+, @ +k, k=l (3.1) (3.2a) (3.2b) 281 Volume 8, Number 3 STATISTICS & PROBABILITY LETTERS August 1989 where { #, } is a special orthonormal basis in H, which satisfies -W,c = h4,, $k E v, (3.2~) and then (3.2d) Suppose that { $, } is an orthonormal basis in H with values in H’. We approximate the unknown parameter a(x) by Using this approximation, the cost functional L(a) defined by (3.1) can be expressed by ~(a”) = -Ltr,$l( u’$, &(-A)-' du(t)) I I I au(t) akr, &( &(-A)-j*a’?$-)j dt. (3.4) The estimate of a0 (true value) is then determined by maximizing (3.4) with respect to {a,, u2,. . . , uk} and the estimate becomes Q= ; G,Mx> (3.5) i=l where B,(t,) satisfies, by defining a^(tr) = { Bl(tf), a^,(t,), . ., a^,(tf)}‘, p”a^( tf) = p) and where the matrix F(‘f) and vector gCtO have components (3.6) i:‘)=;j,tf * au(t) 5 (O,$$ &-(&(-A)-') +Jax, p,m=l P and au(t) +j ax -, &(-A)-‘du(t) P P (3.7) (3.8) respectively. Remark. ( -A)-’ can be numerically realized by using the finite difference and/or finite element methods. We do not get into this ques+% in the present paper. 4. Consistency of the estimate In order to check the consistency property of the estimate (3.5), the basic equation (3.6) is rewritten as 282 Volume 8, Number 3 STATISTICS & PROBABILITY LETTERS August 1989 follows. First, from (3.8) we have + au(t) a - -( $(-A)-lj*aa(x)%$-$) dt J ax, ’ ax, m (p au(t) -, $(-A)-’ dw(t) J ax, P i x a”(x) - ; a;+&> y dt i q=l i r m + au(t) -, &(-A)-' dw(t) , J ax, P i where ai = (a’, +q). Hence, setting A:“) as the 2nd and 3rd terms of the right hand side of (4.1), (3.6) becomes F(“)( ik( tf) - a”(k)) = h(“) where aO(k) = {a?, u; )...) a;}‘, and /z(“) denotes the jth component of I?(“). J Lemma 1. 3C, > 0 (independent of i undj) such that qp-E(f,:“‘) I*) G c,/rt. Proof. From (3.7), it is easy to show that where TJct>= it au(t) p,m=l +r ax - --( &(-A)-lj*$j~) P 9 aa M m (4.1) (4.2) (4.3) (4.4) (4.5) (4.7) 283 Volume 8, Number 3 STATISTICS & PROBABILITY LE’ITERS Noting that E{ X,,(t)T.j(s)} = E +,F, $( &(-A)-‘)*$$ P P m m + a+) a p,m=l - -( L(_d)‘il~j!!p)] 1 ax, 9 ax, ax, m and using the Gaussian property of the state U, we get E{ X;,(t)X,j(s)} p,m=l q,r=l (&( +-‘)* m where [ ., .] denotes the Hilbert-Schmidt inner product. From (3.2d), we have (from Green’s formula) (from Lions, 1969, p. 192) ~C1512 (from Lions and Magenes, 1972, p. 85, a/ax; EZ( H; H-l)) August 1989 (4.9) (4.10) 284 Volume 8, Number 3 STATISTICS & PROBABILITY LETTERS and from the fact that & is an orthonomal basis in H, Icp;12=l, we have August 1989 (4.11) E{ Xi,(~>xjj(s>} <Const. IA(r, s) I&(v:~). (4.12) Now, we get Lemma 2. i Const. (1 m 1, -03 IAh) I~2~vv~ d7 1 G Const./t,. •i n a = = [i p,m=l L(-A)Y’)’ 3 +I% (+j&AtOJ)*] a*.~. q ax, ax, (4.13) (4.14) The proof of this lemma can be easily derived from Theorem 3 and Gaussian property of u. For more details, refer to Aihara and Sunahara (1989) and Wong (1971, p. 69). Lemma 3. 3C, > 0 (independent of j and tr), qh:“q2} G c,(vt, + I I,(k) I ‘> where Z,(k) is given by (4.17) below and 5 II,(k)12-,0* ask-+oo. j=l Proof. We set (4.15) (4.16) P m (-A)-) ( a”(x) - 5 a&(x) q=l 285 Volume 8, Number 3 STATISTICS & PROBABILITY LETTERS August 1989 It is easy to show that from Lemma 2, G Const. IA(O) ILz,~y~;v~ (4.18) au(t) $7 , T&(-A)~’ dw(t) +311,(k) 12. (4.19) P P By using the same procedure to derive the results of Lemma 1, we can show that there exists a C > 0 such that the 1st term of the right hand side of (4.19) < C/t,. (4.20) Furthermore, it follows that au(t) +,‘I P , +-(-A)-’ dw(t) ~_ au(t) --.-, -&(-A)-‘( &(-A]-‘Q)*+~~) dt) J ax, P m m < (C/if) IA(O) I L,(v’;v). q Theorem 4. If k + 00, with k/t, + 0, as t, + 00, then (4.21) (4.22) in probability. 286 Volume 8, Number 3 STATISTICS & PROBABILITY LEITERS Proof. From Lemma 3, it follows that E(IF”“(ri;‘-a”(k))12) = ; “(lh)‘q2} j=l i k gConst. -+ tf Using Lemma 2, we can derive (4.22) from (4.23). q ask, tf-+w. August 1989 (4.23; 5. Conclusion We studied the identification problem for parabolic systems with spatially dependent parameter. Our state was assumed to be completely observed. The problem becomes more complicated for the case of partial observation. For the case of finite dimensional parameter vectors, this problem has been analyzed in Bag&i and Borkar (1984). The extension to infinite dimensional parameter vectors is not obvious at present. References Adams, R.A. (1975) Sob&u Spaces (Academic Press, New York). Aihara, S.I. and Y. Sunahara (1988), Identification of an infinite dimensional parameter for stochastic diffusion equations, SIAM J. Control Optim. 26, 1062-1075. Bagchi, A. (1985), Identification for a hereditary system with distributed delay, System Control Lett. 5, 339-345. Bag&i, A. and V. Borkar (1984). Parameter identification on infinite dimensional linear systems, Stochastics 12, 201-213. Balakrishnan, (1975), Identification and stochastic control of a class of distributed parameter systems with boundary noise, in: Lecture Notes in Econom. Math. Systems, Vol. IO7 (Springer, New York). Bensoussan, A. (1971), Filtrage Optimal des System Lineaires (Dunod, Paris). Curtain, R.F. and P. Kotelenez (1987). Stochastic model for uncertain flexible systems, Automatica 23, 657-661. Grenander, U. (1981) Abstract Inference (Wiley, New York). Lions, J.L. (1969), Quelques Methodes de Resolution des Prob- lemes aux Limites Non Lineaires (Dunod, Paris). Lions, J.L. and E. Magenes (1972) Non-Homogeneous Boundary Value Problems and Applications, Vol. I (Springer, Berlin). Mizohata, S. (1973) The Theory of Partial Differential Equa- tions (Cambridge Univ. Press, Cambridge). Nguyen, H. and T.D. Pham (1982), Identification of nonsta- tionary model by the method of sieves, SIAM J. Control Optim. 20 603-611. Paradoux, E. (1975) Equations aux derivees partielles stochas- tiques non lineaires monotones, Thesis, Universite Paris XI (Paris). Tanabe, H. (1979), Equations of Euolutions (Pitman, London). Wong, E. (1971) Stochastic Processes in Information and Dy- namical Systems (McGraw-Hill, New York). 287 

Infinite dimensional parameter identification for stochastic parabolic systems

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Abstract: The infinite dimensional parameter estimation for stochastic heat diffusion equations is considered using the method of sieves. The consistency property is also studied for the long run data

Shinichi Aihara

Arunabha Bagchi

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INFINITE DIMENSIONAL PARAMETER IDENTIFICATION FOR STOCHASTIC PARABOLIC SYSTEMS

Abstract Inference

Equations aux derivees partielles stochastiques non lineaires monotones, Thesis, Universite

Identification and stochastic control of a class of distributed parameter systems with boundary noise, in:

Identification for a hereditary system with distributed delay,

Identification of an infinite dimensional parameter for stochastic diffusion equations,

Infinite dimensional parameter identification for stochastic parabolic systems

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