Abstract Inverse problems for elliptic and parabolic partial dierential equations are considered. It is assumed that a solution of the equation is obs

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1 Statistical approach to inverse boundary problems for partial dierential equations Golubev G. and Khasminskii R. eierstra{institut fur Angewandte Analysis and Stochastik Mohrenstae 39 D{7, Berlin, Deutchland ayne State University Detroit, MI 48, USA This work was partially supported NSF grant DMS 9645 and ONR grant N Mathematics Subject Classications. 6G5. Key words and phrases. Partial dierential equation, minimax estimator, boundary problem.

2 Abstract Inverse problems for elliptic and parabolic partial dierential equations are considered. It is assumed that a solution of the equation is observed in white Gaussian noise with a small spectral density. The goal is to recover smooth but unknown boundary or initial conditions based on the noisy data. It is shown that the second order minimax estimators are linear as the spectral density of the noise goes to zero. Introduction Ill-posed inverse problems for ordinary and partial dierential equations are very popular in modern mathematics. e refer to the books Hadamard (93), Ivanov & Vasin & Tanana (978), Tihonov & Arsenin (978), avrentiev & Romanov & Shishatskii (98). Usually these problems are associated with recovering of an unknown dierential operator or unknown boundary conditions based on the observations of a solution of the equation in some domain. There are two important questions, which arise naturally in this context. In the rst place one wants to prove the uniqueness of a solution of the inverse boundary problem if it is known that the restored boundary functions belong to a certain functional class. If the solution represents a noisy data, one has to show next that the error in the recovered functions tends to zero when the noise in the data tends to zero too, attes & ions (967). Statistical approach to inverse problems is based on the assumption that solutions are observed in a white noise with a small spectral density. The goal is to nd estimators which are at least consistent. The main advantage of the statistical approach lies in the fact that it helps to compare dierent approaches to inverse problems. Thus we have a mathematical basis for choosing solutions, which are optimal in dierent senses. Sudakov & Khaln (964) and Khaln (978) were among the rst who propose to use the statistical approach to ill-posed problems. In Chow & Khasminskii (997a) and Chow & Ibragimov & Khasminskii (997b) this approach was used for estimation of a source based on solutions of ordinary and partial dierential equations. In the present paper we consider from statistical point ofviewtwo very popular inverse problems i) recovering of initial conditions for parabolic equations based on the observations in a xed time strip, ii) recovering of boundary conditions for elliptic equations based on the observations in an internal domain. Many scientists have paid attention on these problems. For instance, the major part of the monograph attes & ions (967) is devoted to the problem i). From statistical point of view the authors have proposed very interesting estimates and demonstrated that they are consistent. e show in this paper how to construct estimates with optimal rates of convergence when the spectral density of the noise goes to zero ( ). Moreover

3 we indicate these rates up to constants. Here we follow the well-known paper Pinsker (98). Unfortunately these rates are very low and there exist many estimators having the optimal rates. In order to discriminate between the estimators in this situation we nd the second order term in the expansion of the minimax risk and propose second order minimax estimators. In the problem of estimation initial conditions for parabolic equations we propose also exponentially minimax estimators.. Estimation of initial conditions for parabolic equations To simplify technical details we consider the problem in the simplest setting. et u(t x) bea solution of the heat u in the domain D = f x gft>g with the periodic boundary conditions and with the initial condition u(t ) = @u (t ) = (t ) u( x)=(x) (3) Identifying points x = and x =, one could say that the equation () is considered on the surface of the cylinder Cft>g, where C is the circle of the unit length. It is also assumed that (x) x Chas Sobolev smoothness, so where j = ;j and (x) = j=; j=; j exp(ijx) i= p ; (4) j j j jjj (5) The set of all function satisfying to (4) and (5) constitutes the Sobolev class (). Next we consider two models of observations. Model a. Assume that a solution of the equation () is observed at t = T in a white Gaussian noise with the spectral density. Thus we observe the generalized random process Y (x) x [ ], which has the form Y (x) =u(t x)n(x) (6) where n(x) is a white Gaussian noise, t.e. a generalized Gaussian process with the covariance function En(x)n(y) = (x ; y) here () is the Dirac -function. It means that for any ' () (C) we can observe the family of random variables Y = Z C u(t x)' (x)dx (7)

4 where are Gaussian random variables with zero mean and the covariance function E = Thus, our problem is to nd the minimax risk r ( ()) = inf ~ Z C ' (x)' (x)dx sup () E ZC ~(x Y ) ; (x) dx where the inf is taken over all estimators based on observations (6). e want also to nd a minimax estimator i.e. the estimator such that r ( ()) = sup E ( ZC (x Y ) ; (x)) dx () The above problem can be easily reduced to the following one. et u(t x) be a solution of (){(3). Since this function is periodic, it can be expanded into the Fourier series u(t x) = j=; u j (t)exp(ijx) (8) Using the Fourier method (cf. Petrovskii (95), s. 38) one obtains u j (t) = j exp(;4 j t) (9) where j are the Fourier coecients associated with the function (x) (see (4)). Since the Fourier basis is a complete orthonormal system in (C), the problem of estimation (x) is equivalent to estimation of the Fourier coecients j j = based on the observations Y j = j exp(;4 j T ) j () where j are independent N ( ). Or equivalently, we have to estimate j based on the data Z j = j j j () where j =exp(4 j T )and j belong to the following set = 8 < j j=; 9 = j j j jjj () Note the minimax risk admits the following representation in the terms of the Fourier coecients r ( ()) = inf ~ sup E j=; ~j ; j (3) Model b. It is assumed that the solution of () is observed into the strip T t T h inatwo-dimensional white Gaussian noise Y (t x) =u(t x)n(t x) t [T T h] x C 3

5 where n(t x) is a generalized two-dimensional Gaussian eld with the covariance function En(t x )n(t x )=(t ; t x ; x ) By (8) and (9) we get the equivalent model (see also (7)) where ~Z j = j ~ j j (4) ~ j = Z T h T exp(;8 j t)dt ;= = p jjj exp(4 j T ) ; exp(;8 j h) ;= Based on the observation from (4) we have to nd the minimax estimator and to calculate its risk (cf. (3)). Thus we see that the only dierence between models a and b is in the denition of j and ~ j.. Estimation of boundary conditions for elliptic equations Consider the simplest case of the Dirichlet problem for the aplace equation on the circle of radius u = (5) u(cos ' sin ') =f(') It is known (see e.g. Petrovskii (95), s. 9) that the solution of the above problem can be rewritten in the polar coordinates as u(r ')= p p k= where k are the Fourier coecients of f('), so that f(') = p p k= r k ( k cos(k') ;k sin(k')) (6) ( k cos(k') ;k sin(k')) et us assume that a solution of (5) is observed on the circle C of the radius < in a white Gaussian noise. Thus we observe a generalized Gaussian eld Y (r ')=ru(r ') p rn(r ') r ' [ ] (7) where n(r ') is a white Gaussian noise i.e. a generalized Gaussian eld with the covariance function En(r ' )n(r ' )=(r ; r ' ; ' ) 4

6 Substituting (6) in (7), multiplying it by cos(k') sin(k') andintegrating by parts over ', we arrive to the equivalent family of the observations (7) k (r) = k r jkj= n k (r) k = (8) where r and n k (r) are one-dimensional independent (for dierent k) white Gaussian noises. Thus the problem of estimation k based on (8) is equivalent to estimation of k based on the observations Y k = Z r jkj= k (r)dr Z r jkj dr Z ;= = k r dr jkj k where k are independent N ( ). Finally we arrive to the following equivalent problem to estimate k based on the data where k are i.i.d. N ( ) and Y k = k k k k = k = q jkj ;jkj; e see that each problem under consideration is equivalent to the problem of estimation unknown parameters k based on the observations (), while the prior information is provided by (). The type of the considered problem is reected only in the denition of k. For parabolic equations k grow like exp(ck ). In the elliptic case we have a growth of the order exp(ck). This dierence plays an essential role only when we prove that second order minimax estimators are linear for the both models. inear estimation. An upper bound for the minimax risk Consider the following linear estimator ^ j = h j Y j of the parameters j based on the observations (). It is assumed j are subjected to restrictions (). The problem of calculation of the linear minimax risk r ( ())=inf h j sup E has the well-known solution, Pinsker (98). j=; (h j Y j ; j ) 5

7 Proposition r ( ()) = j=; j j=; where is a root of the equation j jjj 4 j ; ; j 3 5 # (9) = () The estimator j = ; j # Y j () is the minimax linear estimator. Proof follows directly from the fact that the functional [h ] = j=; ( ; h j ) j j=; j h j = E j=; (h j Y j ; j ) has a saddle point on l (; ) (see e.g. Pinsker (98)). The components of this saddle point are given by h j = ; j () where [x] = max( x). j = j 4 j # ;. A lower bound for the minimax risk 3 5 (3) To show that the linear estimator dened by (), () is a rst order minimax estimator we will use the following fact. Proposition et ( j ) are dened by (3). Then r ( ()) (j ) ; 4 ( j ) 4 j ; (4) j6= j6= Proof. et j be independent random variables taking values j ; j with the probabilities =. It is clear that r ( ()) j6= inf EEf( ~ j ; j ) j j g (5) ~ j 6

8 It is also evident that the inf in the right-hand side of (5) is attained when ~ j is the Bayesian estimator Y ~ j = j j j th j where Then we have th x = exp(x) ; exp(;x) exp(x) exp(;x) EEf( ~ j ; j ) j j g = Ef( ~ j ; j ) j j = j g (6) 8 < = (j ) E exp Y jj j 8 < ; 9 = ; j = j 9 = ; j = j (j ) E ; exp Y jj j 9 = (j ) 4 ; 3 exp 4( j ) 4 exp ( j ) j j ( j ) 5 ; 3 exp 4( j ) j Since 5 ; 3 exp(x) is a convex function, one easily obtains that for x [ log(5=3)]. Note also that if 5 ; 3exp(x) ; x= log(5=3) (7) 4( j ) j > log(5=3) the right-hand side in (6) is negative. Then (7) and (6) yield EEf( ~ j ; j ) j j g( j ) ; 4( j ) j thus, by virtue of (5), proving the required inequality (4)..3 Asymptotic behaviour of the minimax risk (8) At the rst glance it seems that the lower bound given by Proposition is not very good. This observation is true in the case when j grow not very rapidly. But in our models we have at least an exponential growth of j, and this fact plays an essential role in the proving of the following theorem. 7

9 Theorem et then j j j j j j > j (9) r ( ()) = r ( ()) O (3) where the bandwidth is dened by (). Proof. et us simplify the lower bound for the minimax risk (4). First of all note that j j6=( ) = j ; j # j (3) j6= = j ; j # j ; ; j6= j ; j ; j ; j # j6= j6= In order to estimate from above the last term in the right-hand side of the above equation we use (9) and the inequality ; ( ; x) max( )x, which is valid for x. Then we have j6= j ; j # b c = k= b c;k # 4 ; b c;k ( ) ; b c; k= 3 5 ;(k;) k = C b c; ; (3) On the other hand it is evident that r ( ()) = j ; j j=; b c; ; b c; A O b c; ; (33) Therefore from (3) and (3) we see that j6= ( j ) r ( ()) O (34) Thus to complete the proof it remains to consider the last term in the right-hand side of (4). e have ; (j ) 4 j ; = j ; j # j (35) j6= j6= 8

10 = <jjj= j ; j # <jjj= j j6= j j ; j jjj>= # j ; j # j From (9) we arrive at <jjj= Hence from (3), (33) and (35) it follows that j ( ; ) ; b c; b c ; (j ) 4 j ; O j6= ; r ( ()) This inequality together with (34) completes the proof of the theorem. Our next goal is to specify the asymptotic behaviour of r ( ()) as. Proposition 3 et (9) is fullled, then r ( ()) = where is a root of equation ( ) O (36) ; b c = C (37) here bxc means integer part of x and C is an arbitrary positive constant not depending on. Proof. A simple algebra easily reveals (see (9), ()) that r ( ()) = ; () j j j=; j j=; where is dened by (). At the same time by virtue of (3) and (33) ; j # = O ; r ( ()) # (38) Hence we get by (38) r ( ()) = () O (39) 9

11 Thus it remains to obtain a lower and an upper bound for dened by (). By () = j ; j j jjj j=; # = () j j ; j j b c = () b c;j(b c;j) j= b c;j ; A By Taylor expansion and (9) we get from the above equation that for some constants C C do not depending on () j j ; j j () j j ; j j # # C ; b c C ; b c; Therefore is located between two roots and of the equations ; b c C = ; b c; C = (4) Taking the logarithm in (4) one concludes that j ; j < C. This inequality together with (39) completes the proof. The following theorem easily follows from Proposition 3 and Theorem. Theorem et (9) is fullled and is a root of equation (37). Then r ( ()) = ( ) O et us look how this theorem works in the case of elliptic equation. Remind that in this case k =(jkj )e ;(jkj) log Taking the logarithm of (37) we get where Hence r ( ()) = ; log log(= ) = ~ O() (4) ~ = ; log log ; log log 4 ; log log(= ) log log(= ) log(= ) O log ;;

12 Thus Theorem gives us the expansion of the minimax risk up to the second order term. The linear estimator ~ j = 4 ; j ~ 3 5 Y j (4) with ~ given by (4), is the second order minimax estimator. Remark. The estimator (4) is robust with respect to the constant in the denition of the class () if the true value of is unknown, but it is known that < < then we can use the new bandwidth ~ = ; log log ~ ; log log ~ with an arbitrary ~ [ ], and the estimator (4) is again a second order minimax estimator. In the problem of estimation of the initial condition for the parabolic equation j have the following form (c.f. ()) j = exp(8 j T ) Therefore taking the logarithm of (37), we get = 8 T log ( ; ) ; log log O() (43) Hence, by Theorem r ( ()) = T O log ;;= log(= ) Thus the estimator from (4) with the bandwidth from (43) is a rst order minimax estimator. It is not very dicult to check that the projection estimator ( Zj jjj ^ ^ j = jjj > ^ with ^ = 8 T log ; 8 T log log where >is also the rst order minimax estimator. Remark. It is easy to see that the estimator (4) with the bandwidth dened by (43) and the projection estimator ^ j are also robust w.r.t. in the sense of Remark. Unfortunately Theorem gives us only the rst term in the expansion of the minimax risk. This situation is quite dierent from the elliptic case, where this theorem provides us with the second order term. Therefore, we consider in the next section the parabolic case in more detail.

13 3 Estimation of initial conditions for parabolic equations In this section we assume that j = ;j and log j ; log j = C j o(j) j (44) where C > is some constant. Dene the integer as follows = arg min j j (j) ; ; (45) Denote for brevity by bold letters two dimensional vectors, that is x = (x x ) T, and kxk = x x. The asymptotic behaviour of the minimax risk is given by the following theorem. Proposition 4 et be given by (45) and condition (44) be fullled. Then, as r ( n ()) = inf sup Ek (Y) ; yk (46) ( ) kyk ( ;kyk )) O (exp(;c )) where Y = y s N( E) s = ( ) p here C > is a certain constant and E is the identity matrix. Proof. et us obtain rst a lower bound. Assume that only =( ; ) T and = ( ;;) T are unknown. The others j are assumed to be. Then the observations have the form Z = (47) Z = Based on these observations we have to estimate and provided that k k ( ) k k (( )) (48) Introduce the new variables y = ( ) ;= y = (( )) ;= Then equation (48) can be rewritten as kyk ky k

14 and the observations (47) are equivalent to the following ones Y = y s (49) Y = y s ; ( ) ; where and are two-dimensional independent N ( E). Thus assuming that y = zq ;kyk, where z z are independent random variables taking values ;= p = p with the probabilities =, we have r ( ()) inf sup kyk ky k inf ( ( ) E k (Y Y ) ; yk (5) ) (( )) E k (Y Y ) ; y k sup kyk ( ( ) EE fw ( (Y Y ) ; y) jzg (( )) EE ( (Y Y ) ; z q )) ;kyk z where the new loss function w(x) = minfkxk 4g, and the observations Y Y are given by Y = y s (5) Y = z q ;kyk s ; ( ) ; To simplify the right-hand side of (5) rst of all note that by (8) and (44) ( q EE (Y Y) ) ; z ;kyk z ( ;kyk ) Hence from (5) we have r ( ()) inf sup kyk ; 4( ;kyk ) ;kyk s ( ) O(exp(;C )) ( ( ) EEfw ( (Y Y ) ; y) jzg (5) (( )) ;kyk ) O(exp(;C )) To simplify more the right-hand side of the above equation consider the auxiliary observations Y = s ; ( ) ; (53) It is easy to see that -distance between the corresponding densities of the observations Y Y and Y Y dened by (5) and (53), is suciently small. Indeed 3

15 by Cauchy-Schwartz inequality one obtains Z Z py (x Y x ) ; p (x Y Y x ) dx dx Z Z = py (x )p (x Y ) ; p Y (x )p (x Y ) dx dx Z = py (x ) ; p (x Y ) dx = Z p = Y Z (x ) ; p = p = Y Y (x ) ; p = (x ) Y p = Y (x ) dx = (x )p = Y (x ) dx Hence for any estimator () = p ; exp ; ( ;kyk ) 8s ( ) = exp(;c ) EEfw ( (Y Y ) ; y) j zg = EEfw ( (Y Y ) ; y) j zg O (exp(;c )) Since the inf in (5), which is taken over all mesurable functions (), can be replaced by the inf taken over k k, then using the above equation and (5) we arrive at the following lower bound r ( ()) inf sup ( ) kyk O (exp(;c )) (E k (Y) ; yk ( ) ;kyk ) ( ) In order to prove that this lower bound is attainable consider the following estimator 8 >< Y k jkj < ^ k = k(y k Y ;k ) jkj = > jkj > where k( ) is an arbitrary function R R. Note that by (44) and (45) Therefore ; 5( ) =5( ) ;e (o())c ; j= Denote for brevity = f j r ( (l)) = inf sup = 8 < j ( o()) ; =e ;(o())c P j infsup jjj< jjj= j (j) g. Then we get 8 < jjj= E( j(y j Y ;j ) ; j ) inf sup ( ) kyk j E( j(y j Y ;j ) ; j ) jjj> jjj= jjj= ) (Ek (Y) ; yk ( ;kyk ) ( ) 4 j 9 = O e ;C (54) 9 = O e ;C

16 Together with (54) this inequality completes the proof of the theorem. At the rst glance it seems that Proposition 4, which species the minimax risk expansion up to the exponential term, has very restrictive practical meaning since we have to solve the suciently complicated variational problem to calculate the vector-valued function ( ), at which the inf in (46) is attained. Fortunately, it can be done very easily and second order minimax estimators can be found explicitly. Moreover we will see that there exists a so-called exponentially ecient estimator, whose risk approximates the minimax risk up to terms of the order exp(;c ). Theorem 3 Assume that (44) is fullled and is given by (45), then r ( ()) = ( ) exp(;c ) and the following linear estimator ^ k = is exponentially ecient. 8 >< > Y s ; A (55) jkj < ; Y k jkj = jkj > Proof. According to Proposition 4 it suces to solve the following problem. Assume that we are given the noisy data Y = y s where is N ( E). Based on Y we have to estimate the unknown vector y. More precisely, our goal is to calculate the minimax risk inf sup kyk ( y) up to terms of the order O(exp(;C )), where ( y)=ek (Y) ; yk ( ;kyk ) To getan upper bound consider the linear estimator ^y = hy with h =; Then elementary algebra reveals that inf sup ( y) sup kyk kyk ( ( ; h) kyk h s ( ;kyk )) (56) = s ; 5

17 To get a lower bound assume that y is the Gaussian two-dimensional vector N ( E). Then we have inf sup ( y) inf kyk Pfkyk g E ( y)fkyk g (57) = inf k k Pfkyk g E ( y)fkyk g 5Pfkyk > g inf E ( y) ; Pfkyk g Since y is Gaussian the inf E ( y) is attained when et us chose = s ; (Y) = Noting that s < we have on the one hand Pfkyk > g C exp s y ; C exp ; s On the other hand noting that under such choice of and we obtain that (cf. (56)) inf E ( y) = h = s This inequality together with (56), (57) reveals that ; exp(;c ) s ; inf sup ( y) = s ; exp(;c ) kyk The proof of the theorem follows now from Proposition 4. 6

18 References [] Hadamard, J. (93). e probleme de Cauchy et les equations aux derivees partielles lineares hyperboliques. Paris. Herman et Cie. [] Chow, P.-. Khasminskii, R. (997). Statistical approach to dynamical inverse problems. Comm. Math. Phys. To appear. [3] Chow, P.-. Ibragimov, I. Khasminskii, R. (997). Statistical approach to some inverse problems for linear PDE. Submitted to PTRF [4] Ivanov, B. K. Vasin, B. V. Tanana, V. P. (978). Theory of linear ill-posed problems and its applications. Moscow, Nauka. [5] avrentiev, M. M. Romanov, V. G. and Shishtatskii, S. P. (98). Ill-posed problems of mathematical physics and analysis. Moscow, Nauka. [6] Petrovskii, N. G. (95). ectures on partial dierential equations. Moscow, Gostekhizdat. [7] Khaln,. A. (978). The statistical approach to the solution of ill-posed problems of geophysics. Studies in statistical estimation theory. Zap. Nauchn. Sem. eningrad. Otd. Inst. Steklov. (OMI) 79, 67{8. [8] attes, R. and ions, J.. (967). Methode de quasi-reversibilite et applications. Travoux et Recherche Mathematiques. 5. Dunod, Paris. [9] Pinsker, M. S. (98). Optimal ltering of square integrable signals in Gaussian white noise. Problems of Information Transmission, {33. [] Sullivan, F. O'. (996). A statistical perspective on ill-posed inverse problems. Statist. Science. 5{57. [] Sudakov, V. N. Khaln,. A. (964). Statistical approach to ill-posed problems in mathematical physics. DAN USSR {96. [] Tihonov, A. N. Arsenin, V. Ja. (986). Methods of solutions of ill-posed problems. Moscow, Nauka. [3] alsh, J. B. (984). An introduction to stochastic PDE. ect. Notes in Math. 8, Springer-Varlag, 65{439. 7

Mathematical Institute, University of Utrecht. The problem of estimating the mean of an observed Gaussian innite-dimensional vector

On Minimax Filtering over Ellipsoids Eduard N. Belitser and Boris Y. Levit Mathematical Institute, University of Utrecht Budapestlaan 6, 3584 CD Utrecht, The Netherlands The problem of estimating the mean