SIAM J. MATH. ANAL.Vol. 18, No. 4, July 1987

(C) 1987 Society for Industrial and Applied Mathematics004

SOME APPROXIMATION FORMULA FOR STOCHASTIC EIGENVALUES*

DAVID C. BARNES"

Abstract. We consider some random eigenvalue problems of the form L(-) AM(. ), where L(. andM(. may be ordinary or partial differential operators which depend on a (perhaps multi-dimensional)random variable to. We generalize some formulas due to Boyce Probabilistic Methods in Applied Mathematics,Academic Press, New York, 1968, pp. 1-73] which give estimates for the eigenvalues of the form A(to)=;t*/ g(,o)+ 0(11,o I1). Here A (to) is an eigenvalue corresponding to the random variable to while A* is aneigenvalue corresponding to some approximating deterministic problem. The term K(to) will bealthough it may not be linear in to. These formulas may be used with very general boundary conditions,including those which contain random coefficients and those which may be nonlinear and nonhomogeneous.The boundary conditions also may contain the eigenvalue parameter A and they need not be self-adjoint.We will first give the theory for ordinary equations, then generalize to partial differential equations, first ina deterministic domain, then in a random domain.

Key words, stochastic eigenvalues, approximation

1. Introduction. Consider the eigenvalue problem

(1) L(y) Agy, L(y) -(fy’)’- qy, 0 < x < I.

We assume the coefficients f, g and q depend on x as well as a random variableto (tol, to2, , toN), taken from a sample space 1) fl x f2 x. x fN where, foreach x [0, l], probability measures/zi(" are defined on fi. The to may or may notbe independent. When appropriate boundary conditions are given, and with some mildrestrictions on the coefficients, the problem will have eigenvalues A (to) which will thenbe random variables defined on f. In this work, we will be concerned with findingapproximations of the form A (to) A * + K (to) + O(11,o 112). Here, A* is an eigenvalueof an approximating deterministic problem

(2) L*(y*) A*g*y*, L*(y*) -(f’y*’)’- q’y*.

We will first develop the theory for (1), then generalize it to problems of the formL(y) AM(y), where L(. and M(. are (possibly partial) differential operators havingrandom coefficients. Finally, we will give a modest treatment of the simple equationV2u + Au 0 with u 0 on 0, where is a random domain. Throughout we will usethe notation A(.)= (.)-(.)*, and *-ed quantities will all be deterministic; so, forexample, AL L- L*, AA A A * and so on. We denote the mean ofa random variableby (.) and also use the notations

II,oll=-max {I,o,,o1}, (u, v)- u(x)v(x) dx.l,J

2. The second order equation. Methods used by D. C. Barnes [3] can be used toprove the following theorem.

THEORErVI 1. Let y, A and y*, A* be eigenpairsfor (1), (2). Define a random variableJ(to) by

(3) j(to) [A,gy,2+ qy,_f(y,,2)] dx.

* Received by the editors March 28, 1985; accepted for publication April 7, 1986.f Mathematics Department, Washington State University, Pullman, Washington 99164-2930.

933

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934 DAVID C. BARNES

Define also a boundary term BT and a random variable K (o)) by,Ix=l(4) BT=f*y*’y-fy’y*-f*y*’y x--o, K(0)) A +BT-J(0)).

Suppose that y* is normalized so that

(5) g,y,2 dx 1.

Then

(6) A(0)) K(0))+ 02 A*+ BT-J(0))+ 02where the term 02 is defined by

(7) 02= [AAAyg*y*+A*AgAyy*+AqAyy*-AfAy’y*’] dx.

In the context of [3], 0) was simply a real parameter and (6) was used to developsome variational properties ofthe eigenvalues A. However, (6) shows that the functionalK (0)) is tangent to A (0)) at 0) 0 and thus will provide a good approximation formulawhen 110)11 is small. Now we need to make O2 small by selecting the base problem (2).One good way to do this is to take

(8) f* (f), g* (g), q* (q).

As an example, we will take the simple boundary conditions, y(0)= y(l)= 0, and use(6) to approximate (A). We see that

(9) (A(0))) A*- [A*(g)y*2+(q)y*2-(f)(y*’2)] dx+02.

Now multiply (2) by y* and integrate. This shows that the integral term in (9) vanishes,so (A(0)))=A*+(O2). This estimate has been given by Boyce [4]. Generally, in thecase of homogeneous, linear, self-adjoint, and deterministic boundary conditions, andin the choice of base problem (8), the result (9) is well known.

It is, however, not at all necessary that the boundary conditions be homogeneous,linear, self-adjoint or deterministic. Equation (6) holds in any case. The boundaryconditions might even involve the eigenvalue parameter A.

There are only two conditions which are necessary in order to use the approxima-tion (6) effectively. First of all BT will, in general, involve the values of y at x -0 andat x I. So we must be able to use the boundary conditions to eliminate, or at leastapproximate, the terms in BT which involve y. The second requirement is that theeigenvalue problem must be well posed when considered as a function of 0). That is,the eigenvalues, the eigenfunctions and their derivatives must depend continuously on0). This will insure that the error term O2 is small. Other than these requirements, theboundary conditions are quite arbitrary.

Suppose we are given constants a, b and random variables 0)i taken from samplespaces (Ii, i) with 0) (0)1,0)2, (’03) E ’1X ’2 X ’3" Consider the boundary conditions

(10) y(O) + (a + 0)2)f(0, 0)1)y’(0) O, (b + 0)3)y(1) +f(l, 0))y’(1) O.

We select the base problem by letting 0)= O, so that

(11)f*(x) f(x, 0), g*(x) g(x, 0), q*(x) q(x, 0),

y*(O) + af*(O)y*’(O) O, by*( l) +f*( l)y*’( l) O.

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APPROXIMATION FORMULA 935

Using (10) and (11) in (4), it follows that

BT (by*+ oayy*)lx=- (af*y*’ off*y*’y’)l=o.

We now use the approximations y- y*+ O(to) and y’= y*’/ O(to) to eliminate theterms involving y and obtain the first order approximation

BT y.E(b + toa)lx=- (a o)f*y*’l=o+ O.Since we have eliminated the terms in BT which involve y, we may substitute thisformula in (6) to obtain first order approximation to A (to) which can be easily computed.

As a second example, consider a slender column subject to a compressive load Awhich may cause it to buckle. The critical buckling load is determined by the smallestnonzero eigenvalue of an equation of the form (1). The eigenfunction y(x) representsthe bending moment of the column in the buckled state. If the load A is applied exactlyon the center line of the column, then the boundary conditions will be y*(0) y*(1) 0.Suppose, however, that the load at x is applied at a small, random distance to awayfrom the center line. If the column is pinned at both ends, then the boundary conditionsare [2], [6]

(12) y(0) 0, y(1) Ato, y*(0) 0, y*(l) 0.

We suppose that the shape of the column is deterministic, so that the coefficients f,g, q do not depend on to. The eigenfunction y* is uniquely determined by (2) and (5)up to a factor of + 1. Suppose first that to > 0. We then take y* to be the eigenfunctionsatisfying y*> 0 for 0< x < so that y*’(l)< 0. However, if to < 0, then we will takey* to satisfy y* < 0 so that y*’(l) > 0. This will insure, at least for small to, that y andy* will have no zeros on 0 < x < so that Ay and Ay’ will be small. It also insures thatthe approximation formulas will be symmetric about to 0, as is the eigenvalue itself,

Since the problem (12) is not homogeneous, the eigenfunction y cannot benormalized at will. We still, however, use the relation

(13) gy2 dx 1,o

in order to force the approximations of y to y*. The three conditions (12), (13) willserve to determine the two constants in the general solution of (1) as well as A. Theseconditions, together with the above sign conventions, will determine the eigenfunctiony uniquely.

Using (4), we find BTA(to)f*(l)y*’(l)to. Since the choice of y* depends onsign(to), we recast BT into the general form (which works for either y*) BT=-A(to)f*(l)ly*’(l)to I. Putting this into (6) and solving the resulting equation for A(to)yields the first order approximation

a*-J(,o)+o a*-J(.,)(14) X(to) t-O2.

1 +f*(/)ly*’(/)to] 1 +f*(/)ly*’(/)to[

The form of this equation, and physical intuition, suggests that A (to) -< A* but this hasnot been proved.

Consider an example of (14). Suppose that the column is uniform. That is,f g 1,q 0, and use nondimensional coordinates, so that

y"+Ay=0, y(0)=0, y(1)=Ato, and y*=+/-x/sinrx, ,*-"7"2.

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936 DAVID C. BARNES

Solving for y and h (to), we find, for given to, that h (to) is the smallest positive rootof the equation

(15) t3to(2t-sin t) =4 sin t, =q- and y 22t-sin 2t)

sin tx.

Using (14) we find that

(16) A(to)1 +rlol + o=.

We will now check the accuracy of this approximation. Rather than attempt to solve(15) for h(to), it is more convenient to use the inverse functions for I,ol as a functionof =x/-. Solving for ]tol, using both (15) and (16) we find

2 sin 7/-2- 2

2t)’Itol =/ta_2t_sin(I,ol- o=.

Now one can easily verify that the two right-hand sides are tangent to each other atthe point =Tr, to 0. Thus the approximation (16) will be good for small to.

The sign conventions used in this example essentially amounted to decomposingthe sample space f/into its positive and negative parts, f+ and f/-, and then usingtwo distinct approximation formulas on each part. Such decompositions can be usefulin other situations as well.

Suppose that either the coefficient functions f, g, q, or the boundary conditionsare not continuous in to. In such an event, the eigenvalue problem may not be wellposed and the methods used here would not apply. Suppose, however, that we candecompose f/into a disjoint union,

N

(17) f/= U f/,, f/,Nf/j=Q foriCji=1

in such a way that the problem is well posed on each f,. We then select approximatingbase problems for each ,,(18) (f*y*i’)’+(h*g*+q*)y*=O, U’)(y*) U2’)(y*) 0

where, for each i, we might select a fixed value of to, f/i and define the approximatingproblems by

fi*(x) f(x, toi), g*(x) g(x, to,), q* (x) q(x, to,).

Now construct the piecewise 02 approximation to A (to) using

(19) A(to)=A* +BT-J,(to)+02.,, toelj, J,(to)= A*gy* +qy* -fy* ’2 dx.

The error terms O2.i will be O((to-toi)2) on I. Even though such a piecewiseapproximation may not be continuous on all , it will still be a global first orderapproximation to A (w).

This also suggests numerical procedures for high accuracy computations. Suppose,for example, that we need to compute (A). We could then decompose as in (17) sothat ()= 1IN. en solve the N base problems (18) for y and take the mean in(19) to obtain

(20) (A)= E (A +B-J,(w)) d + Oi=1

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APPROXIMATION FORMULA 937

Since each error term in (19) is 0(1/N2), the error term in (20) will be O(1/N). Theobvious thing to do at this point, is to use (20) together with a Romberg styleextrapolation scheme to compute more accurate values for (A).

3. More general problems. These methods will also work on more general problemsof the form

L(y) AM(y), Up(y, to, A) O, p 1, 2,..., 2m,(21)

L(y)= Y (-1)’(fy’))’), M(y)= Y (-1)(gy))).=0 -=0

Here,f and gj are random functions, and we will assume that m > m’-> 0. The boundaryconditions may be quite arbitrary, being subject to the same conditions outlined inthe second order case.

Consider any deterministic problem which approximates (21),

L*(y*)=A*M*(y*), Up*(y*, a*)= 0, p=l,2,...,2m,(22)

L*(y*)= Y. (-1)’(fi*y*(’))’), M*(y*)= 2 (-1)J(g*Y*())))=0 i=0

We have the following theorem.THEOREM 2. Let y, A and y*, A* be eigenpairs for (21) and (22). Define a random

variable J to bym’

,(j)2(23) J(to) ’s,Y f(y.(i))2 dx.=0 =0

Define boundary terms BT1, BT2 and BT3 by the following equations:

(24) (L*(y), y*)= BT1 +(y, L*(y*)),

(25) (M*(y), y*) BT2+ (y, M*(y*)),

(26) (L(y*), y*)-A*(M(y*), y*)= BT3-J(to).

Suppose that y* is normalized so that

M*(y*), y*) 1, L*(y*), y*) A *.(27)

Then

(28)

(29)

A (to) A * + BT1 A *BT2+ (L(y*), y*) A *(M(y*), y*) + 02,

A (to) A * + BT1 A*BT2+ BT3 J(to) + 02.The error term 02 is given by

(30) O_ (AL(Ay) A;tM*(Ay) AAAM(y*) ;t AM(Ay), y*)

where A(. (.)-(. )*.The proof consists of a rather straightforward, but lengthy, computation. First

multiply (22) by y and (21) by y*. Then subtract the equations and integrate to find

(L(y), y*)-(L*(y*), y)= A (M(y), y*)-A*(M*(y*), y).

We now substitute the A notation into this equation, giving

((L* + AL)(y* + Ay), y*)-(L*(y*), y* + Ay)

(,X* + A, )((M* + AM)(y* + Ay), y*)- ;t*(M*(y*), y* + Ay).

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938 DAVID C. BARNES

Now multiply out these expressions, collecting all terms of second order into 02 asgiven by (30). Next, use the relations A(. (.) -(. )* to eliminate all of the remainingterms which involve A. Finally, use (24), (25) and (27) to simplify the formula. Aftersome manipulation, (28) follows. Finally, (26) implies (29).

In the important special case m 2 and m’= 1, we find that

(31)

(32)

(33)

,Ix=lBT1 (f2*y")’y* (fE*y*")’y +f2*y*"y fE*y"y*’ +f*yy*’ f*y’yBT2 g*l y*’y g* y*y’l ’=lx=O,

x=lT3 (fEy*")’y* fEy*’y*"-fy*’y* + ;t *gy*’y*l=o.

As an example, consider a uniform slender column subject to an axial compressiveload h which may cause it to buckle. Suppose that it is pinned at each end and thatit is supported on an elastic foundation which provides, at each point x, a randomrestoring force F(x, tol)Y, which is directly proportional to the displacement y. Thecritical buckling load is determined [6] by the first eigenvalue of the system:

y’’+ F(x, tol)Y A (-y"), y(O) y"(O) y( l) y"(1) O.

Take the base problem to correspond to to1 O, so that

(34) y*"’+ F*(x)y*= A*[-y*"], F*(x) F(x, 0).

Using (29) we see that

A(to) A*- A.(y.,)E_F(x, to,)y.2 + (y,,,)2 dx+02.

Multiplying (34) by y* and integrating shows that

(o=*- ((x, oo-*(xly*ax+O.

We can also deal with random boundary conditions. Suppose, for example, that

y’(O) + a + to2)y"(O) O, (b + to3)y’(l) + y"(1) 0, y(0) 0, y(l) 0.

Using (24)-(26) and the approximations y y*+ o(11 [I), we soon find that

BT1 h *BT2+ T3 b + toa)y*’2(l) (a + to2)y*"2(0) + 02.This approximation can now be used in (29) to obtain an approximation for A(w).

A second example, which has been used to study vibrations of a helicopter roter(see 1 and the references listed there) is given by

y"" ((1/2)a 2(1 x2 + y2)y,), Ay, 0 < x < 1,(35)

y(O)=y’(O)-ay"(O)=y"(1)=O, y’(1)=(1/2)y2a2y’(1)-(1/2)Ay2y(1).

We suppose that a and y are fixed but that a is a random variable, a- a*+ to, withto I. Letting to=0 correspond to the base problem, we use (31)-(33) to find thatBT2- 0 and that

BT1 (A * A )(1/2) y2y*2(1) toy*"2(0) + 02, BT3 -(1/2)A* y2y,2(1).

Substituting this into (29) and solving the resulting equation for A yields the first orderapproximation

(36) A(to)A*- toy*"2(0) J(to) + 02

1 + -12 y2y*2(1)

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APPROXIMATION FORMULA 939

Now J(to) is actually independent of to so that J(to) J(0). Letting to =0 in (26) showsthat J(to)= BT3=-1/2A*y2y*2(1). Substituting into (36) we obtain

y*"2(O)(37) X(to) X*-to

1 +1/23,2y’2(1) - O(to2)"

It follows from this equation that, at to=O, dA(to)/dto<O. Therefore, A(to) is adecreasing function of to. Thus, we see that A* is a decreasing function of a*.

With more involved calculations, we could allow the coefficients in (35) to berandom also. We will not pursue that idea here.

It is interesting to note that Theorem 1 is not, exactly, a special case of Theorem2. To see this, let m 1, m’= 0 in Theorem 2, and take

L(y) -(fy’)’- qy, M(y)=gy,

L*(y*) -(f’y*’)- q’y*, M*(y*) g’y*.

Computing the boundary term given in Theorem 2 we find

(38) BT1 A *BT2+ BT3 f*y*’y f*y’y* fy*’y*l=However, computing the boundary term BT given in Theorem 1 yields a differentresult. But taking the difference between the two boundary terms and doing a directcomputation shows that the difference y*AfAy’lo. Since this term is of second order,we see that either boundary term could be used to get a first order approximation toA(to). This discrepancy arises from the fact that in the work [3] the O2 terms werecollected after the integration by parts whereas in Theorem 2 the 02 terms were collectedbefore the integration by parts. It seems difficult, at this point, to decide which methodshould give a better result. Such questions would have to be dealt with by a morecareful analysis of the O2 terms.

4. Partial differential equations. These methods generalize easily to partialdifferential equations in a deterministic domain . Using Green’s Theorem in placeof integration by parts shows that Theorem 2 can be used when L(.) and M(. arepartial differential operators. Consider the special case ofthe two-dimensional equation

(fu.),+(fUy)y+(hg+q)u=O, in with u+(a+)u=OonO.

Here, g, q, and are random but a is a deterministic function of s, arc length on. For now, is a deterministic domain. Using (29), we see that

a =a*+ (a+) ds+ 1*gu*+qu*-f(u+u) dA+O

where

IIg*u*2da=l.If l, the length ofthe interval in a one-dimensional problem, was a random function

of some , then a simple change of variable s x/l would give a new problemon the fixed domain 0 s 1 with random coecients. Theorem 2 would then apply.In the case of paial differential equations, such a change of variables is much moredicult. One can, however, appeal to the Hadamard variational formula to accomplishmuch the same kind of manipulation.

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940 DAVID C. BARNES

Following Garabedian [5, Chap. 15], we consider a fixed two-dimensional domain

* and let (to) be the region whose boundary, 0(to), is obtained by shifting 0*an "infinitesimal" distance 6 p(s, to) along its inner normal r, so that (0) *.Here, s is arc length on 0 and to is a random variable. We assume that Op/Oto existsnear to O.

Let A((to)) denote the first eigenvalue of V2u+Au=0, with u=0on0(to),and let u and Ul* be the eigenfunctions corresponding to and *. It follows fromthe Hadamard variational formula [5, Chap. 15] that

A/. --/1( (to)) 1() t -[- O(to2), ,1 fo (S, to)(OU:I 2

* \7/ ds.

Thus we obtain the first order approximation formula

(39) )t((to)) AI(*)+*

O(s, to) --! as + 0(o:).

As an example of this, consider a triangular domain * bounded by x 0, y 0, andx+y- or. Suppose the sides x-0, y 0 are fixed but that the diagonal side has arandom variation to. Thus p(s, to) is zero on x=0, y=0 but p(s, to) to on x+y=cr,and (to) is the domain bounded by the lines x 0, y 0, and x +y- rr-x/to. Theminus sign is due to the inward directed normal. There is a problem at the cornerpoints since 0* is not smooth there and p is not continuous. However, there do existanalytic approximations to 0*, 0(to) and p, and the error introduced using themwill be O(toE).

Solving for u*, A *, we find that Ul* 2(sin x sin 2y / sin y sin 2x) and that A 1" 5.Now the change of variables x (1-rrto/v/), y- j(1- rrto/v2) shows that

(40) al( (to))(1 rrto/v/)2"

Using Ul* in (39), we find, after some calculation, that

(41) A 1(o/ (to))---- A1 / toA23/2/7r / O(to2).

One can now easily verify that (40) and (41) agree up to second order terms.

REFERENCES

[1] H. J. AHN, On random transverse vibrations of a rotating beam with a tip mass: Method of integralequations, Quart. J. Mech. Appl. Math., 36 (1983), pp. 97-109.

[2] D. C. BARNES, Buckling ofcolumns and rearrangements offunctions, Quart. Appl. Math., 41 (1983), pp.169-180.

[3] Extremal problems for eigenvaluefunctionals, this Journal, 16 (1985), pp. 1284-1294.[4] W. E. BOYCE, Random eigenvalue problems, in Probabilistic Methods in Applied Mathematics, Vol. 1,

A. T. Barueha-Reid, ed., Academic Press, New York, 1968, pp. 1-73.[5] P. R. GARABEDIAN, Partial Differential Equations, John Wiley, New York, 1964.[6] S. P. TIMOSHENKO AND J. N. GERE, Theory of Elastic Stability, McGraw-Hill, New York, 1961.

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