Chapter COEFFICIENT IDENTIFICATION IN ELLIPTIC DIFFERENTIAL EQUATIONS Ian Knowles Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 Abstract An outline is given for new variational approach to the problem of computing the (possibly discontinuous) coefficient functions p, q, and f in elliptic equations of the form (p(x) u)+λq(x)u f, x R n, from a knowledge of the solutions u.. INTRODUCTION Consider the differential equation Lv (p(x) v) + λq(x)v f(x), x, (.) where is an open bounded set in R n, are real and p satisfies f L 2 (), q L (), and p L (), (.2) p(x) ν >, x. (.3) In addition, we assume that the real constant λ, and q, are chosen so that the homogeneous Dirichlet operator L L p,q (i.e. L acting on W,2 ()) satisfies L is a positive operator in L 2 (); (.4) Supported in part by US National Science Foundation grant DMS-95547. This is an expanded version of a lecture given at the ISAAC97 Conference, University of Delaware, June 3 7, 997
2 for later use we note that as q L (), (.4) is true for all λ small enough. It is known [6, Chapter 8] that the generalized Dirichlet problems associated with (.) are uniquely solvable, and that the solutions v lie in the Sobolev space W,2 (). Our main concern here is the corresponding collection of inverse problems: given v (for one or more values of λ), find one or more of the coefficient functions p, q, and f. These inverse problems arise in connection with groundwater flow (and oil reservoir simulation); in such cases the flow in the porous medium is governed by the diffusion equation: [p(x) w(x, t)] S(x) w t R(x, t), (.5) in which w represents the piezometric head, p the hydraulic conductivity (or sometimes, for a two dimensional aquifer, the transmissivity), R the recharge, and S the storativity of the aquifer (see, for example, [2, p. 24]). It has long been recognized among hydro-geologists [, Chapter 8] that the inability to obtain reliable values for the coefficients in (.5) is a serious impediment to the confident use of such models. Methods that have been employed range from educated guesswork (referred to as trial and error calibration in the hydrology literature the method preferred by most practitioners at this time [, p. 226]) to various attempts at automatic calibration (see [3, 4, 5,,, 2] as part of the extensive literature on these inverse problems). In the steady-state case we have that w t, and R R(x); if R is presumed known, the inverse problem reduces to finding p from a knowledge of w. In [7, 9] a new approach to this reduced problem was given in terms of finding unique minima for certain convex functionals. This approach was shown to be effective in the presence of mild discontinuities in the coefficients, an important practical consideration in view of the fact that fractures in the porous media are commonly encountered. We present here an extension of these ideas to cover the general equation (.). The basic value of such an extension can be seen by looking at (.5) in the case R R(x). In this situation, (.5) can be transformed to (.) by (for example) applying a Laplace transform to the time variable. The raw data needed consists of head measurements taken over both space and time, as well as hydraulic conductivity (or transmissivity) values on the spatial boundary; such data is in general readily available, and reasonably accurate. In principle, one can then use an appropriate functional (of the type discussed below) to recover p, S, and R. We note in passing that while there are other methods to obtain p (mainly from steady-state data on the heads), it has been observed ([, p. 52 and p. 97]) that there are essentially no universally
Coefficient identification 3 applicable methods for estimating R and S and most practitioners use quite rough estimates of these parameters. This leads to instabilities in the model, especially when transient simulations are involved. The functionals used in [7, 9] may be generalized as follows. Let a solution u of (.) be given for which P, Q, and F are the coefficients corresponding to p, q, and f, respectively, that we seek to compute. For functions p, q, f satisfying (.2,.3,.4), let v u p,q,f denote the solution of the boundary value problem determined by (.) and Thus u u P,Q,F. Define v u. (.6) D G {(p, q, f) : p, q, f satisfy (.2), (.3), (.4) and p Γ P Γ } where Γ is a hypersurface in transversal to u. It is convenient to take Γ to be the boundary of the bounded region, and we henceforth assume this to be so. For (p, q, f) in D G define G(p, q, f) p(x)( u 2 u p,q,f 2 )+λq(x)(u 2 u 2 p,q,f ) 2f(x)(u u p,q,f ) dx (.7) The functional studied in [8, 9] corresponds to p and f, while that in [7] corresponds to q f. 2. PROPERTIES OF THE FUNCTIONAL G Some of the properties of G are summarized in the following theorem: Theorem (a) For any c (p, q, f) in D G, G(c) p(x) (u u c ) 2 +λq(x)(u u c ) 2 dx (L p,q (u u c ), u u c ). (.8) (b) G(c) for all c (p, q, f) in D G, and G(c) if and only if u u c. (c) For c (p, q, f ) and c 2 (p 2, q 2, f 2 ) in D G we have G(c ) G(c 2 ) (p p 2 ) ( u 2 ) u c u c2 + +λ(q q 2 )(u 2 u c u c2 ) 2(f f 2 )(u u c + u c2 ).(.9) 2
4 (d) The first Gâteaux differential for G is given by G ( (p, q, f)[h, h 2, h 3 ] u 2 u c 2) h + +λ(u 2 u 2 c)h 2 2(u u c )h 3, (.) for h, h 2 L () with h, and h 3 L 2 (), and G (p, q, f) if and only if u u c. (e) The second Gâteaux differential of G is given by G (p, q, f)[h, k] 2 ( L p,q(e(h)), e(k) ), (.) where h (h, h 2, h 3 ), k (k, k 2, k 3 ), and the functions h, h 2, k, k 2 L (), with h k, h 3, k 3 L 2 (), e(h) (h u p,q,f ) + λh 2 u p,q,f h 3, and (, ) denotes the usual inner product in L 2 (). Proof. If v is a solution of the generalized Dirichlet problem (.,.6), by the standard theory (see for example [6, Chapter 8]) we have (Lv, φ) p(x) v φ + λq(x)vφ dx, (.2) and hence, by (.), φ (p(x) v) dx p(x) v φ dx, (.3) for any function v W,2 () and any φ W,2 (). The latter formula is essentially Green s formula for this situation ( integration by parts ) and will be much used in the rest of this proof. In the sequel, it will be convenient to set c (p, q, f). Observe that G(c) p (u u c ) 2 + 2p u c (u u c ) + λq(u 2 u 2 c) 2f(u u c ) p (u u c ) 2 2(u u c ) (p u c ) + λq(u 2 u 2 c) 2f(u u c ) from (.3), using φ u u c W,2 (), p (u u c ) 2 2(u u c )(λqu c f) + λq(u 2 u 2 c) 2f(u u c ),
Coefficient identification 5 from (.), and the first result in (.8) follows after some rearrangement; the remaining part of (.8) follows by a further integration by parts, and the results in (b) follow from this and (.4). For c (p, q, f ) and c 2 (p 2, q 2, f 2 ) in D G we have from (.7) that G(c ) G(c 2 ) p ( u 2 u c 2 ) p 2 ( u 2 u c2 2 ) + λq (u 2 u 2 c ) λq 2 (u 2 u 2 c 2 ) 2f (u u c ) + 2f 2 (u u c2 ) p (u c2 + u c ) (u c2 u c ) + λq (u 2 c 2 u 2 c ) + +(p p 2 )( u 2 u c2 2 ) + λ(q q 2 )(u 2 u 2 c 2 ) 2f (u u c ) + 2f 2 (u u c2 ) (u c u c2 )( (p (u c ) + (p u c2 )) + λq (u 2 c 2 u 2 c ) + +(p p 2 )( u 2 u c2 2 ) + λ(q q 2 )(u 2 u 2 c 2 ) 2f (u u c ) + 2f 2 (u u c2 ) (u c u c2 ){ p u c + p 2 u c2 + ((p p 2 ) u c2 )} + +(p p 2 )( u 2 u c2 2 ) + λq (u 2 c 2 u 2 c ) + +λ(q q 2 )(u 2 u 2 c 2 ) 2f (u u c ) + 2f 2 (u u c2 ) (u c u c2 )(λq u c + λq 2 u c2 f f 2 ) (p p 2 ) u c2 (u c u c2 ) + (p p 2 )( u 2 u c2 2 ) + +λq (u 2 c 2 u 2 c ) + λ(q q 2 )(u 2 u 2 c 2 ) 2f (u u c ) + 2f 2 (u u c2 ), by (.) and an integration by parts, using u c u c2 W,2 (). Part (c) now follows after some rearrangement. In order to prove (d) and (e) we need two ancillary results. First we note that, for c and h as above (and fixed), lim u c+ɛh u c (.4) ɛ in W,2 (). To see this, we subtract the equations (p u c ) + λqu c f (.5) ((p + ɛh ) u c+ɛh ) + λ(q + ɛh 2 )u c+ɛh f + ɛh 3 (.6)
6 to obtain L p,q (u c+ɛh u c ) ɛ( (h u c+ɛh ) λh 2 u c+ɛh + h 3 ) (.7) If this equation is multiplied on both sides by u c+ɛh u c and integrated over we arrive (after the usual integration by parts) at p (u c+ɛh u c ) 2 + λq(u c+ɛh u c ) 2 ɛ h u c+ɛh (u c+ɛh u c ) λh 2 u c+ɛh (u c+ɛh u c ) + h 3 (u c+ɛh u c ) ɛ h (u c+ɛh u c ) 2 + h u c (u c+ɛh u c ) λh 2 (u c (u c+ɛh u c ) + (u c+ɛh u c ) 2 ) + h 3 (u c+ɛh u c ) ɛ h (u c+ɛh u c ) 2 + h /2 ( u c 2 + (u c+ɛh u c ) 2 ) + + λ h 2 /2 (u 2 c + (u c+ɛh u c ) 2 ) + λ h 2 (u c+ɛh u c ) 2 + +(/2)( h 3 2 + (u c+ɛh u c ) 2 ), after repeated use of the inequality ab (a 2 + b 2 )/2. Now, for λ small enough (this is where the need for assumptions (.3,.4) becomes apparent), the term on the left of the above inequality is bounded below by a constant multiple of u c+ɛh u c W,2 (). For ɛ small enough all the terms in u c+ɛh u c on the right side of the inequality can be moved to the left, and the resulting left side then can be bounded below by a (smaller) constant multiple of u c+ɛh u c W,2 (). As the remaining terms on the right are O(ɛ), the result (.4) now follows. We also need to know that for any function η L () (η u c+ɛh ) W,2 () K (.8) where the constant K does not depend on ɛ. To see this, note that the functional F defined on W,2 () by F (φ) η u c+ɛh φ satisfies F (φ) K φ W,2 (), (.9) where it follows from (.4) that the constant does not depend on ɛ. Consequently, F (W,2 ()). If we use the Riesz representation theorem to identify (W,2 ()) with W,2 (), F is identified with a unique element of W,2 () which we may take to be (η u c+ɛh ), and F (η u c+ɛh ) W,2 (); the estimate (.8) then follows from (.9).
Coefficient identification 7 Now, from (.7) and some algebra, (G(c + ɛ) G(c))/ɛ h ( u 2 u c+ɛh 2 ) + λh 2 (u 2 u 2 c+ɛh ) 2h 3(u u c+ɛh ) +ɛ p( u c 2 u c+ɛh 2 ) + +λq(u 2 c u 2 c+ɛh ) 2f(u c u c+ɛh ) (.2) By (.4) it is sufficient to show that the second integral expression above tends to zero as ɛ. But, ɛ p( u c 2 u c+ɛh 2 ) ɛ p (u c u c+ɛh ) (u c + u c+ɛh ) ɛ (u c+ɛh u c ) (p (u c + u c+ɛh )), after an integration by parts, ɛ (u c+ɛh u c ){(λqu c f) + +λ(q + ɛh 2 )u c+ɛh (f + ɛh 3 ) ɛ (h u c+ɛh )}, using (.5) and (.6), (u c+ɛh u c ){ (h u c+ɛh ) + λh 2 u c+ɛh h 3 } + +ɛ λq(u 2 c+ɛh u2 c) 2f(u c+ɛh u c ) Consequently, the second integral expression in (.2) equals (u c+ɛh u c ){ (h u c+ɛh ) + λh 2 u c+ɛh h 3 } (.2) and this tends to zero as ɛ by (.4) and (.8); (d) is thus established. Finally, the second Gâteaux differential is given by From (.9) and some algebra G (c)[h, k] lim ɛ G (c + ɛh)[k] G (c)[k] ɛ (G (c + ɛh)[k] G (c)[k])/ɛ
8 ɛ ɛ ɛ 2 + ( u c 2 u c+ɛh 2 )k + λ(u 2 c u 2 c+ɛh )k 2 2(u c u c+ɛh )k 3 k (u c u c+ɛh ) (u c + u c+ɛh ) + +λ(u 2 c u 2 c+ɛh )k 2 2(u c u c+ɛh )k 3 (u c+ɛh u c ){ (k (u c + u c+ɛh )) λ(u c + u c+ɛh )k 2 + 2k 3 }, after an integration by parts, L p,q( (h u c+ɛh ) + λh 2 u c+ɛh h 3 ){ (k (u c + u c+ɛh )) + by (.7), +λ(u c + u c+ɛh )k 2 2k 3 }, L p,q( (h u c ) + λh 2 u c h 3 ){ (k u c ) + λu c k 2 k 3 } + L p,q( (h (u c+ɛh u c )) + λh 2 (u c+ɛh u c )) { (k (u c + u c+ɛh )) + λ(u c + u c+ɛh )k 2 2k 3 } + + L p,q( (h u c ) + λh 2 u c h 3 ) { (k (u c+ɛh u c )) + λ(u c+ɛh u c )k 2 } (.22) It remains to show that the second and third integrals in (.22) tend to zero as ɛ. As the operator L p,q is self-adjoint, if we set w ɛ (k (u c + u c+ɛh )) + λ(u c + u c+ɛh )k 2 2k 3 the second integral may be rewritten as ( (h (u c+ɛh u c )) + λh 2 (u c+ɛh u c ))L p,qw ɛ h (u c+ɛh u c ) (L p,qw ɛ ) + λh 2 (u c+ɛh u c ))L p,qw ɛ Now, from (.8), w ɛ is uniformly bounded in ɛ in L 2 (), and as L p,q may be extended uniquely as a bounded linear operator from L 2 () to W,2 (), L p,qw ɛ is bounded independently of ɛ in W,2 (). From the boundedness of on W,2 () to L 2 () L 2 () it follows that (L p,qw ɛ ) is bounded independently of ɛ in L 2 (). From (.4) it now follows that the second integral in (.22) tends to zero with ɛ. Finally, note that L p,q( (h u c ) + λh 2 u c h 3 ) lies in W,2 (); that the third integral vanishes as ɛ follows via (.4) after an integration by parts. This completes the proof of the theorem
Coefficient identification 9 Some comments are in order. First, the differentials listed in (d) and (e) are actually Fréchet differentials, but we omit the proofs. From (.4) and (.) the functional G is convex, but not necessarily strictly convex (see below). The functional introduced in [9] corresponds to the case p, f, and h h 3 k k 3, while the functional in [7] corresponds to the case q f and h 2 h 3 k 2 k 3. As can be seen from the theorem, most of the properties that were important in the special cases are also present in the general result. In particular, part (c) generalizes [8, eq. (2.7)]. One notable exception concerns property (b). As in the earlier cases, the non-negativity of G is a direct consequence of the Dirichlet principle for the elliptic equation. But the uniqueness connection between the minimum of G and the coefficients p, q, f is the subject of on-going work. It will be shown elsewhere that if the coefficient f is presumed to be known, and if one is given solutions u u P,Q,λ and u 2 u P,Q,λ2, where λ λ 2, then under certain conditions the functional H(p, q) G(p, q, λ ) + G(p, q, λ 2 ) satisfies H(p, q) if and only if p P and q Q; furthermore, H is strictly convex in that for any p, q the second differential H satisfies H (p, q)[h, h 2 ] if and only h h 2. It is conjectured that in the general case if one had three solutions of (.), corresponding to different values of λ, then one could form a functional H containing three terms, and recover, uniquely, by minimization of this H, all three coefficients p, q, f. 3. SOME APPLICATIONS While the recovery of multiple coefficients has not as yet been tested computationally, the recovery of single coefficients has been extensively investigated in [7, 9]. The method is remarkably effective, especially so in the case of coefficients (including principal coefficients) with mild discontinuities. Some results additional to those shown in [7] are given in Figure., with various choices of the function p, listed below: P (x, y) P 2 (x, y) { 2, if y >,.5, otherwise; 6y + 5, if.5 y.75 and x y, 6x + 5, if.75 x.5 and y x, 6y + 5, if.75 y.5 and x y, 6x + 5, if.5 x.75 and y x, 2, if y <.5 and x <.5,.5, otherwise;
Figure. Computed examples; q f 2.5 z.5 2.4.8 z.2.6.5 y - x.5.5 y - x.5 (a) z P (x, y), 3 3 grid 2.5 z.5.8.2 z.6.5 y - x.5.5 y - x.5 (b) z P 2 (x, y), 49 49 grid 2.5 z.5 2..4 z.7.5 y - x.5.5 y - x.5 (c) z P 3 (x, y), 3 3 grid
Coefficient identification P 3 (x, y) { 2, if y < x +.95 and x y <.95.5, otherwise. In each case, the correct p is on the left, and the p computed with a preconditioned steepest descent algorithm (see [7] for details) using the functional G (with q f ) appears on the right. The function P was recovered after about steepest descent iterations, while P 2 and P 3 were recovered after 65 and 4 iterations, respectively. References [] Mary P. Anderson and William W. Woessner. Applied Groundwater Modeling. Academic Press, New York, 992. [2] J. Bear. Dynamics of Fluids in Porous Media. American Elsevier, New York, 972. [3] J. Carrera. State of the art of the inverse problem applied to the flow and solute equations. In E. Custodio, editor, Groundwater Flow and Quality Modeling, pages 549 583. D. Reidel Publ. Co., 988. [4] J. Carrera and S.P. Neumann. Adjoint state finite element estimation of aquifer parameters under steady-state and transient conditions. In Proceedings of the 5th International Conference on Finite Elements in Water Resources. Springer-Verlag, 984. [5] R.L. Cooley and R.L. Naff. Regression modeling of groundwater flow. In Techniques of Water-Resources Investigations, number 3- B4. USGS, 99. [6] David Gilbarg and Neil S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York, 977. [7] Ian Knowles. Parameter identification for elliptic problems with discontinuous principal coefficients. preprint, 997. [8] Ian Knowles and Robert Wallace. A variational method for numerical differentiation,. Numerische Mathematik, 7:9, 995. [9] Ian Knowles and Robert Wallace. A variational solution of the aquifer transmissivity problem. Inverse Problems, 2:953 963, 996. [] W. Menke. Geophysical Data Analysis: Discrete Inverse Theory. Academic Press, New York, 989. [] A. Peck, S.M. Gorelick, G. De Marsily, S. Foster, and V. Kovalevsky. Consequences of Spatial Variability in Aquifer Properties and Data Limitations for Groundwater Modeling Practice. Number 75. International Association of Hydrologists, 988.
2 [2] William W-G. Yeh. Review of parameter identification procedures in groundwater hydrology: The inverse problem. Water Resources Research, 22(2):95 8, 986.