Estimation of time dependent parameters in general parabolic evolution systems

PARAMETERS IN GENERAL

PARABOLIC EVOLUTION SYSTEMS

A. S. Ackleh

B. G. Fitzpatrick y January 17, 1995

Abstract

In this paper we develop a convergence theory for estimating time dependent parameters in a general class of parabolic system. We apply this theory to several problems of interest, such as estimating transport and biodegredation parameters in a contaminant transport model and the stiness coecient in an Euler-Bernoulli beam equation.

Center for Research in Scientic Computation and Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205. Research supported in part by AFOSR grant F49620-93-0198.

1. Introduction.

Many important scientic problems require the estimation of pa-rameters in distributed or partial dierential equations. These inverse or parameter identication problems arise in several contexts, including bioremediation of contami-nated groundwater [12], in population biology problems [1], and in physical models for exible sturctures [2, 3, 7, 8]. The inverse problems then consist of estimating these parameters, using data obtained from experimental observations. The goal of this paper is to present a general convergence and stability theory for approximation methods for the treatment of time dependent parameter identication problems involving distributed parameter systems.

General theory for parameter estimation in an abstract setting can be found in [6]. In that work (and its many references) one nds that key components in inverse problem analyses are continuity of the system state with respect to the parameter, compactness of parameter spaces, and convergence of numerical approximations that is uniform with respect to the parameters and consistent with the topology of the observation space. For general autonomous linear parabolic problems, the paper [4] contains the relevant analysis. The sesquilinear form approach contained therein provides a unied way to handle a wide variety of problems, with conditions that can be veried in a straight-forward manner. In the present paper we seek results which extend the framework of [4] to nonautonomous parabolic problems in order to allow general coverage of many problems, together with veriable conditions on the sesqulinear form that determines the dynamics.

Estimation of time dependent parameters is crucial in certain applications. For example, the growth of an individual in a population model depends on the available resource, which is a function of time. The stiness of a beam in a exible structure

may decrease over a long period of time, due to aging. Fluctuations in water tables due to precipitation cause changes in groundwater velocity elds which in some cases must be determined from tracer movements. To analyze these situations, it is essential to generalize the existing theory to include time dependent parameter problems.

Our theory is based on the weak version of the system in terms of sesquilinear forms used in [4]. The theory depends on the following properties of the time and parameter (q 2Q) dependent sesquilinear form (t;q)(;) describing the system: continuity with

respect to the parameter, uniform boundedness (both in time and the parameter), and uniform coercivity in time and the parameter.

The paper is organized as follows. In Section 2, we present a theoretical framework for the approximation. Applications of this theory to a contaminated groundwater model and the Euler-Bernoulli beam equation is discussed in Section 3. Concluding remarks and further studies are the topics of Section 4.

2. Approximation theory for identication problems.

LetH be a Hilbert space

with inner product h;i and corresponding norm jj. Let V be a Hilbert space that

is densely and continuously imbedded in H, with norm kk and imbedding constant K : for each 2V; we have jj Kkk: We use these spaces to form a Gelfand triple

structure V ,!H =H

,!V

: We consider the following abstract dierential equation

on H

8 > < > :

_

u(t;q) =A(t;q)u(t;q)+F(t;u(t;q);q) u(0;q) =u

0( q)

(2:1)

with parameter q belonging to a compact separable metric space (Q;d). The operator Ais assumed to be determined by a time and parameter dependent sesquilinear form on V; i.e, (;)(;) : [0;1)QV V !C, where (t;q)(;) is sesquilinear for each

t2[0;1) and q2Q. Concerning , we make the following assumptions

(

0

)

The function(;q)(; )is measurable on [0;1);for xed; 2V andq 2Q:

(

1

)

There exists K 0

> 0 3 j(t;q)(; )j K 0

kk k k 8; 2 V; q 2 Q

uniformly int on each interval [0;T].

(

2

)

There exists c 0

> 0; 0

2R 3 (t;q)(;) + 0

jj 2

c 0

kk 2

; 82V; q 2 Q

uniformly int on each interval [0;T].

(

3

)

For q;q^2Q;t 0; and all; 2V, we have that

j(t;q)(; )?(t;q^)(; )jd(q;q^)kkk k:

Under these assumptions there exists a family of uniquely determined linear op-erators A(t;q) : dom(A(t;q)) ! H, with dense domains, satisfying (t;q)(; ) = h?A(t;q); i; 82 dom A(t;q); 2V.

For the nonlinear term, we shall need the following.

(F1)

The function F : [0;1)H Q ! H is continuous. Moreover, F is locally

Lipschitz continuous in H ; uniformly for q2Q and for t2[0;T], for eachT >0:

The weak form of (2.1) is then given by

8 > < > :

hu ;_ i+(t;q)(u;) = hF(t;q;u(t;q));i 82V u(0) = :

(2:2)

As is the case for linear nonautonomous problems (see [15]), we seek solutions of (2.2) in the Hilbert space

W(0;T) = ff 2L 2((0

;T);V): _f 2L 2((0

;T);V )

g;

whose norm is given by

kfk 2 W =

Z T 0

kf(t)k 2+

kf_(t)k 2 V

dt:

For the weak form we will show the following existence and uniqueness result.

Theorem 2.1

Suppose that the hypotheses 0, 1, 2, and (

F1

) hold. Then there existT >0 such that the problem (2:2) has a unique solution u2W(0;T).

Proof.

To begin the proof, we rst assume that 0 = 0

: Once we have the result

established in this case, we will extend it to the general setting. We consider the following sequence of dierential equations.

8 > < > :

hu_ 0

;i+(t;q)(u 0

;) = 0 82V u

0(0) = ;

(2:3)

and for n1; 8 > < > :

hu_ n

;i+(t;q)(u n

;) =hF(t;q;u n?1(

t;q));i 82V u

n(0

;q) = :

(2:4)

We note that for each xed n and each T > 0; there exist a unique solution u

n

2 W(0;T) (see [15]): these systems are linear. Our rst step is to show that u n

is Cauchy in L 2((0

;T);V) for some T 2 (0;1): To do this we rst demonstrate the

uniform boundedness of the solutions u

n of (2.3) and (2.4).

First, set b= jj; let B = maxf2;2bg and consider the bounded set U =fu :juj Bg H : Fix T

0

>0 and choose M so that jF(t;0)j M; for all t 2 [0;T 0]

: Hence for

any u2U we have

jF(t;u)jjF(t;0)j+jF(t;u)?F(t;0)jM +L B

B

where L

B is the local Lipschitz constant for the set

U. Now choose T > 0 such that T T

0and (( M+L

B B)

2 T+b

2) e

T

B

2 ( since

h(t) = ((M+L B

B) 2

t+b 2)

e

tis increasing

function of t for t 0 and h(0) = b 2

<B

2 and lim t!1

h(t) = 1; such a choice of T is

possible). As in the arguments of [15], we see thatju 0

jB:Suppose now thatju n

j B.

We will use an induction argument to show that ju n+1

jB:

Toward this end we have that for any t2[0;T] hu_

n+1 ;u

n+1

i+(t;q)(u n+1

;u n+1) =

hF(t;u n)

;u n+1

i:

Integrating from 0 to t and using coercivity (recall we are assuming

0 = 0), we get Z t 0 1 2 d dt ju n+1 j 2 dt+ Z t 0 c 0 ku n+1 k 2 dt Z t 0

jF(t;u n)

jju n+1

jdt:

This implies that

Z t 0 1 2 d dt ju n+1 j 2 dt Z t 0

jF(t;u n)

jju n+1

jdt:

Using the induction assumption and the inequality 2jabja 2+

b

2 we get ju

n+1( t)j

2

Z t 0

jF(t;u n) j 2 dt+ Z t 0 ju n+1 j 2

dt+jj 2

From Gronwall's inequality we then obtain that

ju n+1

j 2

((M +L B

B) 2

t+b 2) e t B 2 ;

for t 2 [0;T]; as desired. This step is crucial in order to employ the local Lipschitz

property of F :

Next we will show that the sequence u n satises Z t 0 ku n ?u n?1 k 2

dtM^ L 2(n?1) K 4(n?1) c 2(n?1) 0 t N n! where ^ M = TM 2 K 2 c 2 0 Z T 0 ku 1 ?u 0 k 2 dt;

where M is as above and K is the imbedding constant. To obtain this estimate, we

subtract the dierential equation for u

0 from that of u

1 to see that Z

t 0

hu_ 1

?u_ 0 ;u 1 ?u 0 idt+ Z t 0

(t;q)(u 1 ?u 0 ;u 1 ?u 0) dt = Z t 0

hF(t;u 0) ;u 1 ?u 0 idt:

Using coercivity we see that

Z t 0 1 2 d dt ju 1 ?u 0 j 2 dt+ Z t 0 c 0 ku 1 ?u 0 k 2 dt Z t 0

jF(t;u 0) jju 1 ?u 0 jdt:

Multiplying the above inequality by 2 and using the inequality 2jabja 2+

1 b

2together

with the Sobolev imbeding constant K, we see that

ju 1(

t)?u 0(

t)j 2+ Z t 0 2c 0 ku 1 ?u 0 k 2 dt Z t 0 K 2

jF(t;u 0) j 2 dt+ Z t 0 ku 1 ?u 0 k 2 dt:

Choosing =c

0 we have Z t 0 2c 0 ku 1 ?u 0 k 2 dt Z t 0 K 2 c 0

jF(t;u 0) j 2 dt+ Z t 0 c 0 ku 1 ?u 0 k 2 dt; so that Z t 0 ku 1 ?u 0 k 2 dt M 2 K 2 c 2 0 Z T 0

dt= ^M;

as desired. For the induction argument, we suppose that

Z t 0 ku j ?u j?1 k 2

for 1 j n: Subtracting the dierential equation for u

n from that of u

n+1 and using

a similar argument to that above, we see that

Z t 0 2c 0 ku n+1 ?u n k 2 dt Z t 0 K 2 L B ku n ?u n?1 kku n+1 ?u n kdt Z t 0 K 4 L 2 B ku n ?u n?1 k 2 dt+ Z t 0 ku n+1 ?u n k 2 dt:

Again, we choose = c 0

: The induction hypothesis then produces the desired result.

Now standard arguments of ODE's will show that u

n is Cauchy in L

2((0

;T);V): Since

for each N; u

n satises the equation 8 > < > : _ u n(

t;q) =A(t;q)u n(

t;q)+F(t;u n?1(

t;q);q) u

n(0

;q) =u 0(

q);

we have (from the continuity of the operator A(t) from V to V

and continuity of F

in H) that _u

n is Cauchy in L

2((0 ;T);V

). Hence the above observations imply that u

n

is Cauchy in W(0;T). By completeness of W(0;T) we have a limitu 2 W(0;T); and

straightforward arguments show that u is a solution of (2.2). In order to relax the assumption that

0 = 0

; we consider the usual transformation u(t) =z(t)exp(

0

t):It is easily veried that u satises the dierential equation (2.2) if

and only if z satises the equation 8

> < > :

hz;_ i+(t;q)(u;) =hG(t;q;z(t;q));i 82V z(0) =

in which(t;q)(; ) = (t;q)(; )+ 0

h; i;and G(t;q;z) =e ?

0 t

F(t;q;e

0 t

z):The

assumptions in the above special case hold for the z problem, which then produces a

unique solution to the problem of interest.

Our main goal in this paper is a convergence theory for least squares based parameter estimation. Toward that end, we next consider an approximation method based on

a sequence of Hilbert spaces H N

; N = 1;2;:::, with orthogonal projections P N : H !H

N. The following assumption about these approximations will be needed for our

convergence results.

(A1)

The subspacesH

N are subsets of

V, and8v2V;we have thatkP N

v?vk!0.

This assumption is satised by many nite elementand spectral schemes (see [6, 10, 14]). The Galerkin approach to approximation involvesrestricting(t;q)toH

N H

N, yielding

bounded linear operators A N(

t;q) satisfying (t;q)(

N ;

N)= ?hA

N( t;q)

N ;

N i:

Theorem 2.3.

Suppose that (0)?(3), (

F1

), and

(A1)

hold, and that q N

! q in Q. Then we have that u

N( t;q

N)

!u(t;q), in H, uniformly on [0,T].

Proof.

We rst denez N =

u N

?P N

u:It is sucient to show thatz N(

t)!0;uniformly

in t:Now, 82H N

;we have that hz_

N ;i

V

;V = hu_

N

?u_ + (d=dt)(u?P N

u);i V

;V

= hu_ N

?u;_ i V

;V + (

d=dt)hu?P N

u;i V

;V

= hu_ N

?u;_ i V

;V + (

d=dt)hu?P N

u;i H

= hu_ N

?u;_ i V

;V

;

Using this fact we have that 1 2 d dt jz N j 2 =

hz_ N

;z N

i=hu_ N

?u;_ z N

i

= ?(t;q N)(

u N

;z N) +

(t;q)(u;z N)

+hP N

F(t;q N

;u N)

?F(t;q;u);z N

i:

Adding and subtracting a few terms, we see that 1 2 d dt jz N j 2 =

?(t;q N)(

z N

;z N) +

(t;q N)(

u?P N

u;z N)

+(t;q)(u;z N)

?(t;q N)(

u;z N)

+hP N

F(t;q N

;u N)

?P N

F(t;q N

;u);z N

i

+hP N

F(t;q N

;u)?F(t;q;u);z N

i:

Using the coercivity, boundedness, and continuity of ;we see that

1 2 d dt jz N j 2+ c 0 kz N k 2 ? 0 jz N j 2 K 0

ku?P N

ukkz N

k+d(q N

;q)kukkz N k +L B( jz N j 2+

ju?P N

ujjz N

j)

+jP N

F(t;q N

;u)?F(t;q;u)jjz N

j:

Simplifying the above and using standard inequalities (such as 2ab (a 2

=) +b 2 ) we obtain 1 2 d dt kz N k 2 C 1 jz N j 2+ C 2 d(q

N

;q)kuk 2+

C 3

ku?P N uk 2+ C 4 jP N

F(t;q N

;u)?F(t;q;u)j 2

:

An application of Gronwall's inequality and the dominated convergence theorem will provide the nal result.

We have thus obtained, based on the assumptions given above, that u N(

t;q N)

! u(t;q) in H ; when q

N

! q in Q: To put this result into the context of least squares

estimation, we consider a continuous mapC:H !Z ;where Z is a normed linear space.

Given z 2 Z ; one determines an appropriate parameter value for the system by

mini-mizing

J(q) =kCu(q)?zk 2

:

The continuous dependence results above indicate that a minimizer exists within the compact set Q:

In order to compute minimizers,we must makesome approximations. Approximation

u

N of the state variable

u, as discussed above, lead to a cost functional J

N(

q) = kCu N(

q)?zk 2 Z

to be minimized. The convergence results of the previous sections guarantee that if

q N

!q then J N(

q N)

!J(q);which will give us (as in [6]) subsequential convergence of

minimizers.

Below we discuss two example problems, illustrating the application of this general theory.

3. Examples.

The two examples we discuss in this section indicate not only the variety of problems which may be examined with our methods but also the straightforward manner in which the assumptions above may be veried.

1) Remediation of contaminated groundwater

In this example we consider the following model

(x)c t(

t;x) + r(v(t;x)c(t;x))

= r(D(v(t;x))rc(t;x))?(t) c k(t) +c

; (3

:1)

for t > 0; x 2 ; with initial condition c(0;x) = c 0(

x): In this model c(t;x) is the

contaminant concentration at timet, position x; v=v(t;x) is the groundwater velocity

vector, D = D(v(t;x)) is the dispersion matrix, (x) is a parameter that is related to

[9, 13] for a full description of this model. The dispersion matrixD takes the form

D =d 0

I+D 1(

v); (3:2)

whered 0

I models molecular diusion andD 1(

v) models the hydrodynamic dispersion of

the solute that is due to the pore system of the medium and collisions of solute particles with soil particles. Empirical evidence and statistical considerations (see [9]) suggest the following model for D

1:

(D 1(

v)) ij =

a T

jvj ij+ (

a L

?a T)

v i

v j jvj

: (3:3)

Herea

T is called the \transverse dispersivity," and a

Lis called the \longitudinal

disper-sivity." See [9] for a detailed exposition of dispersion.

Plugging the above terms in equation (3.1), we arrive at a model equation of the form

(x)c t+

X i

@ @x

i

(v i

c) = X

i;j @ @x

i

a T

jvj ij+ (

a L

?a T)

v i

v j jvj

! @c @x j

!

?f(t;c);

with

f(t;c) = maxf0;(t) c k(t) +c

g;

and D(v) given in (3.3). The form of f is a mathematical device to guarantee

(F1)

:

in the presence of a suciently weak parabolic maximum principle (we do not know of such a result), this form is unnecessary. The unknown parameters a

L ;a

T, and

; are

to be estimated, as is the velocity eld v. It is of particular importance to handle time

dependent kinetic rate parameters in the biodegradation terms due to the experimen-tally observed behavior of bacteria to learn to degrade compounds more eciently and

to degrade preferentially contaminants in the presence of multiple compounds.

We will now relate the abstract formulation developed in the preceding section to this problem. We begin with the Hilbert spaces H = L

2() and

V = H

1(), which

are the usual Lebesgue and Sobolev spaces. We assume(x) = 1 for simplicity (similar

analysis will hold for >0), and dene the sesquilinear form by (t;q)(; ) =

Z

(D(v)rr ?vr )dx;

which, when integrated by parts (using the no-ux boundary conditions), yields the convection and dispersion terms. To satisfy the requirements of the previous section, we will need to make some restrictions on the parameter q. We set

e

= (0;T) and e

Q= (L 1(

e

)))3 (L

1( e

))2

(C[0;T]) 2

Here L

1 denotes the usual space of essentially bounded measurable functions, and C[0;T] is the space of continuous functions on [0;T] (which is a Banach space when

equipped with the sup norm).

V =f(v 1

;v 2

;v 3)

2(W 1 1(

e

))3 : kv

i k

W 1 1

K

1

;i= 1;2;3g; A=f(a

L ;a

T) 2(W

1 1(

e

))2 : 0 a

T a

L ka

L k

W 1 1

K

2 ; ka

T k

W 1 1

K

3 g F =f(;)2(C

1[0 ;T])

2 :

0; 0;kk 1

K

4 kk

1

K

5 g:

Where the space C 1[0

;T] is the usual space of functions with continuous rst

deriva-tive on [0;T] with the norm kfk

1 = sup t2[0;T]

jf(t)j+jf_(t)j

Using Arzela-Ascoli's and Sobolev embeddings, one can easily show thatQ=VAF

is compact in e

Q. We remark that this choice of Q is not the optimal choice and that

there are many other choices for the set Q; the analysis, however, is similar for dierent

choices.

The assumptions (0)?(2) can be easily veried using standard PDE arguments

(see, e.g., [4, 15]), and the fact that d 0

> 0 is known and xed. For verication of 3,

we let; 2V, and suppose thatq N

!q in Q. We put D N v N ij = d 0 ij + a N T jv N j ij + a N L ?a N T v N i v N j jv N j :

Then we have that

j(t;q)(; )?(t;q N)(

; )j

Z

jD(v)rr ?D N(

v N)

rr jdx

+Z

jv N

?vjjjjr jdx kD(v)?D

N( v N) k 1 kk H 1k k

H 1

+kv?v N

k 1

kk H

1k k H

1

=d(q N

;q)kk H

1k k H

1 !0:

Since v N

!v in [L 1( e )]3 ; a N L !a L ; a N T !a T in L 1( e ).

The hypothesis

(F1)

is easily veried due to the simplicityof the Monod nonlinearity. In the transport model above, data for the inverse problem is typically of the one of the following forms:

^ C d ij Z j

c(t i

;x)dx;

which denotes the dissolved contaminant level (mass as opposed to concentration) at timet

i in the region j;

~ C t ij Z j

(x)c(t i

;x)dx;

which denotes the total contaminant level (adsorbed and dissolved) at time t

i in the

region j

: Note that the operator u 7! R

A

u(x)dx is a bounded linear operator from H =L

2 to the real line. Thus the least squares cost functional J(a

L ;a

T

;;) = X

i;j jC^

d ij

? Z

j

c(t i

;x;v;a L

;a T

;;)dxj 2

;

has a minimizer, and when the state is approximated, the minimizers of the approximate cost functional converge (at least subsequentially) to minimizers of J:

2) Euler-Bernoulli beam equation with Kelvin-Voigt damping

We consider the following equation:

@ 2

y @t

2 + @

2 @x

2( EI

@ 2

y @x

2 + c

D I

@ 3

y @x

2 @t

) =f(t;x) 0<x<l ;t>0 (3:4)

y(t;0) = @y @x

(t;0) = 0

EI @

2 y @x

2(

t;l) +c D

I @

3 y @x

2 @t

(t;l) = 0; @

@x

(EI @

2 y @x

2(

t;l) +c D

I @

3 y @x

2 @t

(t;l)) = 0 y(0;x) =y

0( x);

@y @t

(0;x) =y 1(

x)

This equation describes the transverse vibrations of a cantilevered Euler-Bernoulli beam with Kelvin-Voigt internal damping. The state variableu(t;x) is the displacement

along the beam at timetat positionx. The parameterEI(t;x) is the stiness coecient,

and c D

I(t;x) is the "Kelvin-Voigt" coecient which reects the assumption that the

bending moment depends not only on the strain, but also on the strain rate as well. The function f represents external distributed forces applied to the beam. For further

details on this model, see [3].

The equation above may be written as a rst-order system in the usual way: Dene

w(t;x) = [y(t;x); @y

@t

(t;x)]

T and ^

F = [0;f(t;x)]

T. Then denoting @ @x

2 by D

2, we see that

(3.4) is equivalent to

w t=

A(t;q)w(t;x)+ ^F(t;x); (3:5)

where (formally)

A= "

0 I

?D 2(

EID 2)

D 2(

c D

ID 2)

#

with the initial condition w(0) =w 0 = (

y 0

;y 1)

:

To write equation (3.5) in a weak formulation, we dene the following spaces

H 2 L(0

;l) =fu2H 2(0

;l)ju(0) =u

x(0) = 0 g H =H

2 L(0

;l)L 2(0

;l) V =H

2 L(0

;l)H 2 L(0

;l):

The inner product on the L 2(0

;l) will be denoted by (;) and for the space H 2 L(0

;l)

we use the inner product

hh; ii= ( xx

; xx)

and the associated norm jk kj (which is equivalent to the usual H 2 L(0

;l) norm by

Poincare's inequality). Finally, the inner products for the spaces H, V will be taken

to be the usual product space inner products, and will be denoted as in the abstract formulationh;i and jj, and h;i

V and

kk respectively. Then as in [4] the weak form

of the beam can be written in terms of the following sesquilinear form: with w= (u;v)

and X = (; ) elements of V, dene

(t;q)(w ;X) =?hhv;ii+ 1(

t;q)(u; )+ 2(

v; )

Where

1(

t;q)(u; ) = (EIu xx

; xx),

2(

t;q)(v; ) = (c D

Iv xx

; xx).

Then with w= (y;y_), the weak form of equation (3.4) can be written

hw_(t);Xi+(t;q)(w(t);X) =hF^(t);Xi; forX 2V

In terms of the abstract formulation developed in the previous section we will set

q= (EI;c D

I), = (0;l)(0;T), e Q=L

1() L

1()

and,

Q = f(EI;c D

I)2(C

0;1())2 : c

0

EI(t;x)c 1

; c

2 c

D

I(t;x)c 3

jEIj C

0;1 c 4

; jc D

Ij C

0;1 c 5

g

Here the spaceC

0;1() is the space of Lipschitz continuous functions with norm kfk

C

0;1 = sup

jf(x)j+ sup x;y 2x6=y

f(x)?f(y) x?y

:

Using same arguments as in former example one can establish the compactness of the set Q in

e

Q . Standard arguments and the fact that EI(t;x) c 0 and

c D

I(t;x) c 2

easily verify the assumptions (0)?(2). For verication of (3) we letw= (u;v);X =

(; )2V, and suppose that q N

!q inQ. Then we have that j(t;q)(w ;X)?(t;q

N)( w ;X)j

j

1(

t;q)(u; )? 1(

t;q N)(

u; )j+j 2(

t;q)(v; )? 1(

t;q N)(

v; )j

kEI?EI N

k 1

Z l 0

ju xx

j 2

dx Z

l 0

j xx

j 2

dx 1

2

+

kc D

I?c D

I N

k 1

Z l 0

jv xx

j 2

dx 1

2 Z

l 0

j xx

j 2

dx 1

2 d(q

N

;q)kw kkXk

To illustrate the computational methods analyzed herein, we estimated the stiness parameter EI as a function of time from computationally generated data. Our

FOR-TRAN program uses cubic B-splines to approximate the solution of the Euler Bernoulli dierential equation, and we used linear splines to estimate the parameter. In the com-putations given below, we use a beam of length 1, with = 1; = :01;c

D

I =:01: We

used 15 cubic B-splines for the computations. For the generated data, we used for EI

the function

EI(t) = 10 +

2

1 + expf15(t?:5)g

which is constant with respect to the spatial variable. The initial displacement and velocity are taken to be 0, and the forcing function is given by

f(t;x) = 100sin(5t)

1

:01

[:495;:505]( x);

which approximates a function in the spatial variable.

For data, we sampled the displacement u(t i

;x = 1) at 200 uniformly spaced time

points t

i in the time interval [0

;1]; as generated with the above model. In order to

examine the behavior of theleast squares identication procedure, we used as data the actual model generated signal, as well as the signal modied by Gaussian noise: z

i = u(t

i

;1)(1+ i)

;was used for data, with

i a random sample from a zero mean Gaussian

random number generator. We used =:01 and =:1 for standard deviations for the

noise.

Our identication algorithm was given the known values of ;;c D

I and f, and was

used to estimateEI(t) using 5 and again using 11 linear splines. In order to implement

the above mentioned compactness constraints, we used a penalized (or regularized) least squares cost of the form

J(EI) = 200 X i=1

jz i

?u(t i

;1;EI)j 2+

Z

1 0

jEI_ (t)j 2

dt

with two dierent choices of, depending on the noise level.

In Figure 1, we see the results of minimizing J

with 5 (dashed) and with 11

(dot-dash) linear splines. The true EI function is the solid line. We used the constant

function EI = 12 as our initial guess in the optimization, which was carried out using

the package lmdif1fromnetlib. The regularization parameter used was 10 ?4

:

0 0.2 0.4 0.6 0.8 1

9 9.5 10 10.5 11 11.5 12 12.5 13

Time

EI

True and Estimated EI

Figure 1.

EI estimates using 5 and 11 splines, no noise in the data.

0 0.2 0.4 0.6 0.8 1 9

9.5 10 10.5 11 11.5 12 12.5 13

Time

EI

Estimates of EI from noisy data

Figure 2.

EI estimates using 11 splines, for =:01;:1.

In Figure 2, we used 11 splines to estimateEI when the data was corrupted by the

above described noise. The solid line is again the trueEI; the dashed line is the estimate

with = :01; and the dot-dash line, =:1: The regularization parameter values used

for the two estimation runs were = 10

?4 and

= 10 ?2

; respectively.

4. Concluding Remarks.

In this paper we have developed a unied theoretical and computational approach for estimating time dependent parameters in abstract parabolic systems. These results are readily applied in problems ranging from insect dispersal to contaminant transport in groundwater to vibration in exible structures (with strong damping). Future research in this area will focus on (temporally) discontinuous

ter estimation as well as ecient implementation in two and three dimensional transport problems.

Acknowledgment-

The authors would like to thank Professor H.T. Banks for helpful discussions regarding the convergence of parameter estimation proof.

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