ITERATIVE METHODS FOR SOLVING A NONLINEAR BOUNDARY INVERSE PROBLEM IN GLACIOLOGY S. AVDONIN, V. KOZLOV, D. MAXWELL, AND M. TRUFFER

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1 ITERATIVE METHODS FOR SOLVING A NONLINEAR BOUNDARY INVERSE PROBLEM IN GLACIOLOGY S. AVDONIN, V. KOZLOV, D. MAXWELL, AND M. TRUFFER Abstract. We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem. 1. Introduction The realities of data availability make many problems in geophysics ill-posed. For example, if it were possible to measure basal velocities and deformation parameters of a glacier, the surface velocities could then be calculated. Instead, there is no known method to measure basal velocities, but surface velocities can be measured directly on the ground or by a variety of remote sensing methods. Basal velocities must then be inferred through inverse methods (e.g. [21]). In this paper we will consider an ice flow model suggested in [6]. They treated a first order model of planar ice flow along a longitudinal cross section of a glacier, and showed that the longitudinal velocity component obeys a non-linear Poisson equation. It can easily be shown that the same model also describes the out-of-plane velocity component for full order Stokes flow in a transverse cross-section with no out-of-plane gradients and no in-plane velocity components. We are treating the problem of inferring a basal velocity field from surface measurements by deriving several useful properties of an operator that maps a (hypothetical) basal velocity field to the surface of a glacier. These properties can be useful for deriving a convergence proof for an inverse algorithm. This has already been accomplished in the linear case ([12] and [3]) Model Description. Let be the domain in the xy plane with Lipschitz boundary, which has a geometry shown in Figure 1. The upper boundary S = (, l) is interpreted as a surface of the ice sheet; B is a bottom (Lipschitz curve); Γ 1,2 are sides which may or may not be present. We consider the following system of equations in : Sergei Avdonin is with Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK , USA. ffsaa@uaf.edu. Vladimir Kozlov is with Department of Mathematics, University of Linköping, Linköping, Sweden. vlkoz@mai.liu.se. David Maxwell is with Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK , USA. david.maxwell@uaf.edu. Martin Truffer is with Department of Physics and Geophysical Institute, University of Alaska Fairbanks, Fairbanks, AK 99775, USA. truffer@gi.alaska.edu. 1

2 Figure 1. Domain geometry; the parts Γ i of the boundary may be empty. (G( u ) u) = f in, (1) u Γ1 = g 1, u Γ2 = g 2, (2) y u S =, (3) where g 1,2 and f are given functions. In the case of the domain presented in figure 1 right, conditions (2) are not prescribed. We suppose that G(t) = F (t), (4) t where F is a convex function for t, F () =, F C 1,1 [, ) (i.e. F is Lipschitz continuous, F (t) F (τ) C t τ ). For such functions F, the second derivative F (t) exists almost everywhere and we suppose that ν(1 + t) 1 n 1 F (t) µ(1 + t) 1 n 1 (5) for some positive ν, µ and n 1. A finite viscosity version of Glen s flow law, which is often used in glaciology (e.g. [19]), gives an example of the function G(t) as the solution of the equation 1 G(s) = [ T 2 + G 2 n 1 (s) s 2] 2 with T. Then F (t) = t G(τ) τ d τ. In [6] it is shown that such F is strictly convex and satisfies (5). Another commonly used regularized form of Glen s flow law given by G(t) = (ɛ + t 2 ) n+1 2n also satisfies (5). Conditions (2) (3) do not distinguish the unique solution of equation (1), so we add an additional condition on S which can be interpreted as a surface measurements: u S = ϕ. (6) Problem (1) (3), (6) is ill-posed: its solution exists not for every (even smooth) functions ϕ and one cannot expect continuous dependence of the solution on ϕ. If u is sufficiently 2

3 smooth in S then using the unique continuation property for elliptic equations (see, e.g., [9], Theorem ) one can show that this problem has a unique solution. That is the information about ϕ is sufficient to find a solution of (1) (3), (6). The goal of this paper is twofold. First, we present an equivalent reformulation of the Cauchy problem as an operator equation for functions on B and S. This new problem is nonlinear and ill-posed. In the case n = 1 (the operator is still nonlinear) we prove some important properties for it, such as: the operator is Lipschitz, existence of the Frechét derivative and an estimate for the linear part of the Taylor s development of the operator, Hölder continuity of the Frechét derivative. Usually these properties are basic for constructing of algorithms for solving the above nonlinear operator equation and for proving their convergence. We note that although we largely do not address the case n = 3 (which is of primary importance for Glen s flow law), the nonlinearities of the operators we do consider allow us to treat Glen-type flow laws that correspond to the n = 3 case except for extreme values of the parameter t (in applications this means that we know a priori that u is a bounded function), and that are therefore also physically relevant. Second, we give an overview of two iterative methods for solving the similar Cauchy problem for a linear operator and which admit generalizations to the nonlinear case. We believe they can give reasonable results for solving the nonlinear Cauchy problem. The first method is the Landweber iterative procedure for solving the above non-linear operator equation. Convergence analysis of such methods shows the importance of the choice of the first approximation (see [2]). Concerning the second method, convergence of the procedure is proved only in the linear case [12, 3]. We present in Section 6 numerical results which show that one can use this procedure even for the non-linear problem (1) (3). 2. Formulation of the (Cauchy-Dirichlet) ill posed problem (1) (3), (6) as an operator equation Consider problem (1)-(3) with additional boundary condition u B = ψ. (7) Let us set Γ = Γ 1 B Γ 2 and suppose that the function g defined as g 1 (γ), γ Γ 1 g(γ) = ψ(γ), γ B (8) g 2 (γ), γ Γ 2 belongs to the space W 1 1 p,p (Γ), p = n+1. Notice that if p < 2 (i.e., n > 1), g W 1 1 p,p (Γ) n for g j W 1 1 p,p (Γ j ) and ψ W 1 1 p,p (B). If p = 2 (n = 1), then to describe the space of admissible ψ s we have to extend g j to B in a such way that the extension g belongs to W 1/2,2 (Γ) and set ψ B = g B. Then the set of admissible ψ in (7) is ψ = ψ B + ψ, ψ W 1/2,2 (B) (the latter is a subspace of W 1/2,2 (Γ) of functions equal to zero on Γ j, j = 1, 2). In the case when Γ 1 = Γ 2 =, the set of admissible functions ψ is W 1 1/p,p (B). p p 1 We suppose also that f L p (), p =. Problem (1) (3), (7) is well posed and its solution is defined as a solution of the minimization problem min (F ( u ) fu)dxdy, (9) 3

4 where the minimum is taken over all u W 1,p (), u Γ = g. (1) The Euler equation for this problem coincides with equation (1) with boundary conditions (2), (3), (7). One can check that the variational problem (9), (1) satisfies conditions of Theorem 1.3 of [14] (see also [22]). Hence it has a unique solution. Actually, problem (1) (3), (7) should be understood in a weak sense: (G( u ) u v fv)dxdy =, (11) which must be valid for u W 1,p (), u Γ = g, and for all v W 1,p (; Γ). Here W 1,p (; Γ) is a subspace of W 1,p () of the functions equal to zero on Γ. We approach solving the ill-posed problem (1)-(3), (6) by finding Dirichlet data ψ on B such that the solution u of (1)-(3), (7) satisfies (6). Let D A denote the set of admissible data ψ; we define the map A : D A W 1 1/p,p (S) taking Dirichlet data on B to Dirichlet data on S as follows. Let ψ D A and u W 1,p () be the solution of the minimization problem (9), (1). Then A(ψ) = u S W 1 1/p,p (S). Hence the Cauchy (ill-posed) problem (1) (3), (6) is formulated as A(ψ) = ϕ. (12) 3. Lipschitz continuity of the operator A In order to discuss differentiability properties of the map A, we consider the linearization of the PDE (1). To do so, we introduce the matrix L: { } 2 (G( η )ηi ) L(η) = = {L ij (η)}, (13) η j where η = (η 1, η 2 ). Direct calculations show that Since G(t) = F (t)/t, we have From (5) it follows that Using that L(η) = G( η )I + G ( η ) η i,j=1 ( η 2 1 η 1 η 2 η 1 η 2 η 2 2 ). (14) L(η)ξ, ξ = F η, ξ 2 ( η ) + F ( η ) ξ 2 η 2 η, ξ 2. (15) η 2 η η 2 νn[(1 + t) 1 n 1] F (t) µn[(1 + t) 1 n 1]. (16) n (1 + t)1/n 1 (1 + t) 1/n 1 (17) t together with (5), (16), we derive from (15): ν(1 + η ) 1 n 1 ξ 2 L(η)ξ, ξ µn[(1 + η ) 1 n 1] 1 η ξ 2. (18) 4

5 In what follows we consider n = 1. All constructions make sense also for n > 1 but rigorous justifications are much more difficult. Lemma 3.1. The operator A is Lipschitz. That is there exists c > such that A(ψ 2 ) A(ψ 1 ) W 1/2,2 (S) c ψ 2 ψ 1 W 1/2,2 (B). (19) Proof. Let u j solve (11) with ψ j in (7), respectively. Let also For n = 1, Since where G( u 2 ) u 2 G( u 1 ) u 1 = U = u 2 u 1. (2) ν ξ 2 L(η)ξ, ξ µ ξ 2. (21) 1 L = L(u 1, U) = d dt [G( (u 1 + tu) ) (u 1 + tu)]dt = L U, 1 L( (u 1 + tu)) dt, (22) we derive from (11) that (L U) v dxdy = for all v W 1,p (; Γ). (23) Due to estimate (21) problem (23) can be considered as a linear elliptic problem with respect to U with boundary conditions U B = ψ 2 ψ 1, U Γj =, j = 1, 2, and y U = on S. (24) Then in a standard way we get the estimate 1 : U 2 dxdy C ψ 2 ψ 1 2 W 1/2,2 (B). (25) Since U S = A(ψ 2 ) A(ψ 1 ) the trace theorem leads to (19). Lemma 3.2. Let Γ 1 and Γ 2 be not empty, Γ 1 = { [m 1, ]}, Γ 1 = {l [m 2, ]}. Then there exists p > 2 such that for any p 1 [2, p ] the solution U (introduced by (2)) satisfies the estimate U W 1,p 1 (m) c ψ 2 ψ 1 W 1/2,2 (B), (26) where m = (, l) (m, ), max{m 1, m 2 } < m <, and p depends only on the ellipticity constants ν and µ, and Γ. Proof. Let ζ = ζ(y) be a smooth function such that { 1, m < y < ζ(y) =, y < 1(m + max{m 2 1, m 2 }). 1 Everywhere in this paper by C and c we denote (generally distinct) constants independent of the variables appeared in the factors of these constants. 5

6 Then the function ζu satisfies the equation (L (ζu)) = (UL ζ) + ( ζ) (L U). (27) We denote the right-hand side of (27) by Q and now verify that Q (W 1,q (; Γ)) for every q (1, ), (the latter is the space of linear functionals on W 1,q (; Γ))). Indeed let w W 1,q (; Γ), then Q w = ( U(L ζ) ( w) + w ( ζ) (L U) c( U L q w L q + U L 2 w L 2). By the Sobolev embedding theorem, U L q c U L 2 and w L 2 c w L q. Thus Q (W 1,q (; Γ)) with any q (1, ) and (28) Q (W 1,q (;Γ)) c U L 2 (). (29) The operator (L ) corresponding to the left hand side of (27) maps continuously W 1,q (; Γ) into ( W 1,q (; Γ) ) and invertible for q = 2. Using the interpolation result of [8] and together with results of [2] we conclude that this operator is an isomorphism for q close to 2. This fact together with (29) gives for p > 2. Taking into account (25) we get (26). Lemma 3.2 in conjunction with Lemma 3.1 implies ζu W 1,p c U L 2 () (3) Corollary 3.3. Under the hypotheses of Lemma 3.2 there exists p > 2 such that A(ψ 2 ) A(ψ 1 ) W 1 1/p,p (S) c ψ 2 ψ 1 W 1/2,2 (B). (31) 4. Directional Derivative of the operator A Put u B = ψ + th D A (32) and differentiate (1) (3) and (32) with respect to t; w := du dt t=. We get or, in a weak sense, (L( u) w) = in, (33) y w S =, w B = h, w Γ1 Γ 2 =, (34) (L( u) w) ( v) =, v W 1,2 (; Γ). (35) Let us define the Gâteaux derivative of the operator A in a standard way: A (ψ)h = d dt [A(ψ + th)] t=. (36) Clearly, A (ψ)h = w S. (37) 6

7 The operator A acts continuously from W 1/2,2 (B) to W 1/2,2 (S). This can be checked similarly to Lemma 3.1. The following lemma is similar to Lemma 3.2. Its proof is the same. Lemma 4.1. Let Γ j, j = 1, 2, and let w be solution of (35) (34). Then w W 1,p 1 (m) C h W 1/2,2 (B), 2 p 1 p, (38) where p and m are the same as in Lemma 3.2. Corollary 4.2. Using the definition of the operator A and the above Lemma we get for a certain p > 2, independent of ψ and h. A (ψ)h W 1 1/p,p (S) C h W 1/2,2 (B), (39) 4.1. Frechét derivative. We show now that A (ψ) is actually the Frechét derivative of A(ψ). More exactly, the following statement is true. Lemma 4.3. Suppose F C 2,1 [, ). Then A(ψ 2 ) A(ψ 1 ) A (ψ 1 )(ψ 2 ψ 1 ) W 1/2,2 (S) c ψ 2 ψ 1 1+ε W 1/2,2 (B), (4) where ε = 1 2 ( p 2 1) > independent of ψ 1,2. Proof. Let u j the solution of (1) (3) and (7) with ψ = ψ j, and let U = u 2 u 1. Then U satisfies the equation (L(u 1, U) U) = in, (41) U = on Γ 1 Γ 1, y U = on S, U = ψ 2 ψ 1 on B. Let also w solve the problem (33)-(34) with h = ψ 2 ψ 1 and u = u 1. Then V = U w solves the problem (L(u 1, ) V ) = ((L(u 1, ) L(u 1, U)) U) in (42) with the homogeneous boundary conditions Let us set Using (22), we have V Γ1 = V Γ2 = y V S = V B =. (43) Q := L(u 1, U) L(u 1, ) = Q = = j= d dτ L( (u 1 + τtu)dtdτ d dτ L(u 1, τu) dτ. (44) L (j) ( (u 1 + τtu) U xj t dt dτ, where L (j) (η) = η j L(η) and, temporarily, (x, y) = (x 1, x 2 ). One can check that the functions L (j) (η) are bounded. Thus we get Q = (L(u 1, U) L(u 1, ) C U. 7

8 Since L(u 1, U), L(u 1, ) are bounded, we have Q = (L(u 1, U) L(u 1, ) C U α, α 1. (45) Let ζ(y) be a smooth function such that { 1, m/2 y ζ(y) =, y < m. Equation (42) implies (L (ζv )) = (Q (ζu)) + (QU ζ) + ζ Q U + ζ L V + (LV ζ), (46) where L = L(u 1, ). Let us represent where V solves ζv = V + V 1, (47) (L V ) = (Q (ζu)) + (QU ζ) + ζ Q U, (48) and V 1 solves (L V 1 ) = ζ L V + (LV ζ). (49) The both functions V and V 1 satisfy zero Dirichlet boundary condition on Γ and zero Neumann condition on S. Multiplying the both sides of equation (48) by V and integrating by parts and using (45), we get 1/2 1/2 V 2 C U 2(1+α) V 2 m m 1/2 1/2 + C U 2α U 2 V 2. (5) m We use now that, by the Hölder inequality and Sobolev embedding theorem, ( U 2α U 2 C U 2α+2, p = 2α + 2 2α, p = 2α + 2 ). 2 m m Therefore, V 2 C U 2α+2. (51) m Choosing 2α + 2 = p and using Lemma 3.2, we get V 2 C ψ 2 ψ 1 p. (52) W 1/2,2 (B) Similarly to the corresponding part of the proof of Lemma 3.2 (see (28) and (29)) one can check that the right hand side of (49) belongs to (W 1,p (; Γ)), and ζ y L V + (Lζ y V ) (W 1,p (;Γ)) C V L 2 (). (53) 8

9 The same reasoning as in the proof of Lemma 3.2 shows that the operator for the V 1 is an isomorphic from W 1,p (; Γ) to (W 1,p (; Γ)). Therefore, Now we multiply (42) by V and integrate. We get V 2 C Q U V + Using the Hölder inequality and (45) we have V 2 C U 2(α+1) V 1 L p () C V L 2 (). (54) 1/2 Q U V 1. (55) V 2 1/2 + C V 1 p 1/p U (1+α 1)p 1 1/p 1. (56) Choosing α =, α 1 = 1 2/p 1 and using (51), (52), (54), and (25), we have V 2 C ψ 2 ψ 1 p /2+1 + C V W 1/2,2 (B) L 2 ψ 2 ψ 1 1+α 1 W 1/2,2 (B) Finally, using Lemma 3.2 and the Cauchy inequality with q, we get [ ] V 2 C ψ 2 ψ 1 p /2+1 + ψ W 1/2,2 (B) 2 ψ 1 4(1 1/p 1). W 1/2,2 (B) Choose p 1 = 8 6 p. Since ( V 2 ) 1/2 estimates above the left hand side of (4), we arrive at (4) with ε = 1(p 2 /2 1) Continuity of A (ψ). Theorem 4.4. The following inequality holds: (A (ψ 2 ) A (ψ 1 ))h W 1 2,2 (S) C ψ 2 ψ 1 ε W 1 2,2 (B) h W 1 2,2 (B), (57) where ε = 1 2 p if 2 < p 4 and ε = 1/2 if p > 4. Proof. Introduce u j as in the proof of Lemma 4.3 and put U = u 2 u 1. Also let w j solves (33), (34) with u j, and W = w 2 w 1. Then by (36) Furthermore, W satisfies the problem W S = (A (ψ 2 ) A (ψ 1 ))h. (58) (L( u 1 ) W ) = (Q w 2 ), (59) where Q = L( u 1 ) L( u 2 ), with the boundary conditions y W S =, W Γ =. 9

10 Similarly to (45) one can show that Q C U α, α 1. (6) Introduce the same continuous function ζ as in the proof of Lemma 3.2, and rewrite equation (59) in the form (L (ζw )) = (Q (ζw 2 )) (Qw 2 ζ) ζ Q w 2 + ζ L W + (LW ζ), where L = L( u 1 ). Similar to (47) we represent ζw = W + W 1, see (48), (49). The equation for W is: (L( u 1 ) (ζw )) = (Q (ζw 2 )) (Qw 2 ζ) ζ Q w 2. Analog of estimate (5) has now the form ( ) 1/2 ( ) 1/2 W 2 C U 2α ( w w 2 2 ) W 2. m Using the Hölder inequality and Sobolev Embedding Theorem we have U 2α ( w w 2 2 ) C m ( w 2 p 1 m ) 2/p1 ( Choosing α = p 1 2, we arrive at the estimate 2 W 2 C m ( w 2 p 1 m ) 2/p1 ( Using Lemma 4.1 and Lemma 3.2 we obtain Equation for W 1 : U p 1 U 2αp 1 p 1 2 ) p 1 2 p 1. ) p 1 2 p 1. m W 2 C h 2 W 1/2,2 (B) ψ 2 ψ 1 p 1 2 W 1/2,2 (B). (61) (L( u 1 ) W 1 ) = ζ L W + (LW ζ). (62) Analog of (53) in our case can be written as ζ L W + (LW ζ) (W 1,p (;Γ)) C W L 2 () and we arrive at the estimate (compare with 54) Analog of (56) has the form: ( ) 1/2 ( W 2 C U 2α w 2 2 m ( ) 1/p ( + C W 1 L p () C W L 2 (). (63) W 1 p 1 W 2 ) 1/2 1/p U α 1p w2 ) p. (64)

11 Choosing α = and α 1 = 1 2/p and using the Hölder inequality in the last integral in (64) we obtain ( m W 2 C + C ) 1/2 ( w 2 2 ( W 1 p ) 1/p ( W 2 ) 1/2 ) 1/2 ( ) α1 /2 w 2 2 U 2. Using ( ) 1/2 w 2 2 C h W 1/2,2 (B) (65) and (61), (63), (25) together with Lemma 3.2, we get W 2 C h 2 ψ 2 ψ 1 p1/2 1 + C W L 2 () h ψ 2 ψ 1 1 2/p, (66) where = W (B). Hence 1/2,2 W 2 C h 2 [ ψ 2 ψ 1 p1/2 1 + ψ 2 ψ 1 2 4/p ]. (67) Choosing p 1 from p 1 = p in the case p 4 (one can check that 2 p 1 p ) and p 1 = 4 in the case p > 4, we get W 2 C h 2 ψ 2 ψ 1 2ε, (68) where ε = min(1 2/p, 1/2). This implies (57) Adjoint operator to A (ψ) with respect to L 2 -duality on B and S. Let us consider the boundary value problem (for the sake of definiteness we suppose that Γ j, j = 1, 2) (L( u) v) = in, (69) n L( u) v = H on S, (7) v = on Γ, (71) (, where u is the solution of (9), (1), H W 1/2,2 (S)) n is an outward unit normal to. ( ) Notice that since y u = on S and n = on S, n L( u) v = G( u ) 1 y v. Problem (69) (71) should be understood in a weak sense: the integral identity for v W 1,2 (; Γ) L( u) v V = HV (72) is valid for all V W 1,2 (; Γ). Clearly this linear problem has a unique solution v W 1,2 (; Γ). For the function v satisfying (72) we can define a normal derivative N v := n L( u) v on as follows. 11 S

12 Consider the formula ( N v)q = L( u) v Q (73) where Q W 1,2 () and Q = q on. The left hand side of this equality does not depend on the extension of q into the domain because of formula (72). Thus equality (73) can be taken as a definition of N v for v W 1,2 () satisfying (72). Since Q can be chosen such that with c independent of q, then Q W 1,2 () c q W 1/2,2 ( ) N v (W 1/2,2 ( )) and N v (W 1/2,2 ( )) c v W 1,2 (). (74) Restriction of the N v on the portion B of is defined as a restriction of the functional N v to the subspace W 1/2,2 (B) of W 1/2,2 ( ). So, N v B (W 1/2,2 (B)). Remark 4.5. For the case Γ j =, j = 1, 2, it is important to note that if N v S (W 1/2,2 (S)) then N v B (W 1/2,2 (B)) (see [12]). Let us show that the adjoint operator to A with respect to L 2 inner products on B and S is defined by (A (ψ)) H = N v B, (75) where v solves problem (69) (69). Indeed, by definition of N v (73), we have ( N v)h + ( N v)w = L( u) v w. B S Let us represent w as w = w 1 + w 2, where w 1, w 2 W 1,2 (; Γ) and w 1 B =, w 2 S =. Then L( u) v w 2 = because of (72), and because of (33) and (34). Thus or B L( u) v w 1 = ( N v)h = ( N v)w B S A (ψ) H h dγ = S HA (ψ)h dγ, which proves (75). Clearly, (A (ψ)) is a linear and bounded operator mapping (W 1 2,2 (S)) into (W 1 2,2 (B)). 12

13 5. Iterative Methods for solving (1) (3) 5.1. Landweber iteration method. One of the standard methods for solving the equation Aψ = ϕ in Hilbert spaces is the Landweber method (see, e.g., [7]). One choose an initial approximation ψ W 1 2,2 (B) and a positive constant γ, then further iterations are given by ψ k+1 = ψ k γ A(ψ k )(Aψ k ϕ), k =, 1,..., (76) where A(ψ k ) is a linear bounded operator mapping W 1 2,2 (S) into W 1 2,2 (B) and which is adjoint to A (ψ k ) with respect to inner products in W 1 2,2 (S) and W 1 2,2 (B). In what follows, we shall use the notations H B and H S for the spaces W 1 2,2 (B) and W 1 2,2 (S) respectively. In order to use (76) let us express the operator A(ψ k ) through the operator A (ψ k ), constructed in Sect Denote by (, ) HB and (, ) HS the inner products in the spaces H B and H S respectively. Let us introduce the canonical isomorphisms (see, e.g. [14, Sec ]) Then T B : W 1 2,2 (B) (W 1 2,2 (B)), T S : W 1 2,2 (S) (W 1 2,2 (S)). B S (T B ψ 1 ) ψ 2 = (ψ 1, ψ 2 ) HB, (T S ϕ 1 ) ϕ 2 = (ϕ 1, ϕ 2 ) HS. One can verify now that the adjoint operator A to A with respect to these inner products is given by A(ψ) = T 1 B (A (ψ)) T S. The convergence analysis of the iterative procedures of the form (76) for non-linear operators can be found in [2] (see also [1, Ch. 7]). Our nonlinear operator satisfies all conditions from [2] except the Lipschitz continuity of the Frechét derivative, instead we have only Hölder continuity (see Theorem 4.4). The convergence analysis in [2] shows in particular the importance of the choice of the first approximation Linear Case. Here we suppose that F (t) = t 2 /2. In this case equation (1) turns to the Poisson equation u = f in. (77) We suppose that Γ 1 and Γ 2 are not empty and the function and f L 2 (). In this case, by using local regularity estimates, one can show that the operator A can be extended to a bounded operator A : L 2 (B) L 2 (S) (even L 2 (B) H S ). Then the Cauchy (ill-posed) problem is formulated as Aψ = ϕ. (78) 13

14 Furthermore, operator A is determined by w = in, w y =, w B = h, S w = on Γ 1 Γ 2, and A h = w S. This operator can also be extended to the operator L 2 (B) L 2 (S). It coincides with A if we put g j =, f =. Therefore, (A ) is also a bounded linear operator L 2 (S) L 2 (B). To solve the problem (78) one can use Landweber operator iterative procedure in L 2 : ψ k+1 = ψ k γ (A ) (Aψ k ϕ) (79) with constant γ, < γ < 1/ A 2. In this operator procedure the initial approximation ψ can be chosen arbitrary. It is known [7], Sec. 6.9, that ψ n converges in L 2 to the exact solution of (78) provided measurement ϕ is a performed without error. If we have measurement ϕ δ with an error such that ϕ δ ϕ L 2 < δ (8) one have to supply the above iterative procedure with a stopping rule. For example one can use the discrepancy principle: the iteration is terminated with k = k(δ, ϕ δ ) when for the first time ϕδ Aψ k(δ,ϕδ) τδ (81) with τ > 1 fixed. This principle guarantees that ψ k(δ,ϕδ ) ψ as δ. The above realization of the Landweber iterative procedure through the boundary value problems and convergence proof for L 2 space, and in L 2 space with a weight in the case when one of pieces Γ 1 or Γ 2 (or may be both) is empty, has been done in [3, 11, 1] Alternating Method. Another method for solving the Cauchy problem (1) (3), (6) is an alternative method which was introduced in [12] for solving ill-posed linear problem. In the case of equation (77) this method can be presented as follows. We alternate boundary conditions on B and S and keep boundary conditions u = g j on Γ j, j = 1, 2. We choose an initial approximation (guess) ψ and consider equation (77) supplied with boundary conditions u B = ψ, u y =. (82) S On the next step we solve the boundary value problem with boundary conditions y u 1 = y u on B, u 1 = ϕ on S. (83) Then u 2 = u 1 on B, y u 2 = on S, (84) and so on. The convergence of this procedure for the Laplace equation and for the Lamé system was proved in [13], for strongly elliptic formally self-adjoint linear equations and systems in [12] and for the Stokes system in [4]. This method was further developed, notably by Baumeister and Leitao [5]. Its numerical implementation using the boundary element 14

15 method has recently been developed in Mera et al. [18] for the Laplace Beltrami equation in steady-state anisotropic heat conduction, Marin et al. [16] for the Lamé system in isotropic elasticity, and Marin et al. [15] for the Helmholtz equation in acoustics. The above iteration procedure can be applied for the nonlinear case if we make the following changes. In order to find the (k + 1)th iteration one must solve the nonlinear equation (1) for u = u k+1 supplied with the Dirichlet-Neumann (or Neumann-Dirichlet, depending on k is odd or even) boundary conditions on S and B, keeping the same Dirichlet boundary conditions on Γ 1 and G 2. The boundary condition on B is taken from the given Cauchy data and the boundary condition on S is taken from the previous iteration step. It is possible some variations in this procedure, for example, on the iteration step k + 1, instead of the nonlinear equation (1) we can solve the linear equation (G( u k ) u k+1 ) = f (85) with the same boundary conditions. Our numerical experiments with the nonlinear problem (1) (3), (6) indicate that this procedure appears to have similar convergence properties as for the linear case. However, the convergence is sometimes too slow to be practical. In [17] we applied an accelerated version of this technique to inverting glacier basal velocities. 6. Numerical Results In this section we present numerical results to illustrate the viability of the two proposed iterative methods. We constructed numerical solutions of the PDE on two sample domains with known Dirichlet data ψ specified on B. We then perturbed the values of these solutions on S with varying amounts of noise and applied the iterative algorithms to solve the Cauchy problems and reconstruct the solutions. The two domains in these examples are a rectangular region and a parabolic cross-section. In both cases, we used a nonlinearity of the form G(t) = (ε + t 2 ) where ε = 1 7. This is a regularized variation of Glen s flow law. The remaining prescribed data were also the same in all cases: f 1 and g 1 = g 2. The rectangular domain is the region with 1 x 1 and.5 y. The surface S is the part of the boundary with y =, Γ 1 and Γ 2 are the vertical walls at x = 1 and x = 1, and the base B is the remainder of the boundary at y =.5. The parabolic cross-section is the region bounded by the graph of y = 1/2(x 1)(x + 1) and the line y =. In this case, S is the part of the boundary with y =, B is the remainder of the boundary, and Γ 1 and Γ 2 are empty. Note that the parabolic cross-section is inscribed in the rectangular region. On a given domain, the solution u to reconstruct was obtained by solving the (forward) numerical problem with a known Dirichlet condition ψ on B, namely ψ(x, y) = M 2 (x 1)(x + 1) 5/2 The scaling constant M was picked to be equal to the maximum value of an analogous solution with ψ = on B. We used standard finite-element techniques for generating all numerical solutions of PDEs, and we used a finer grid for the forward problem than we used for the inverse problem. Given a true (numerical) solution u, we perturbed the values of φ = u S to φ α by adding Gaussian noise having mean zero and standard deviation M α/1. The noise levels of α% 15

16 ranged over α = 2, 1, and.5 in our tests. For the Landweber method, we measured the discrepancy δ = φ α φ H 1/2 (S) and then ran the algorithm using the stopping principle φ α φ k < τδ whith τ = 1.2. For the alternating method we used the stopping criterion described in [4], modified slightly to accommodate the nonlinearity. Let K be the operator that takes φ H 1/2 (B) and computes φ 2 = u 2 B H 1/2 (B) via equations (82) (84). Let K α be the analogous map with φ = φ α in (83). We then set δ = K α (ψ) ψ H 1/2 (B), where again ψ = u B is the exact solution. The algorithm stops at the first index k such that K α (ψ k ) ψ k H 1/2 (B) < τδ. We note that our tests do not determine how to pick the error threshold δ in practice, which can be difficult u u a) x b) x Figure 2. True and reconstructed basal velocities for the rectangular domain using the Landweber (a) and alternating (b) methods. Thin solid line is u B as a function of x. Reconstructions are shown for α = 2 (dotted), α = 1 (dashed), and α =.5 (heavy solid). In all cases, we started the iterative methods with an initial guess ψ =, which is an a-priori hypothesis that the glacier is frozen to the base. Figure 2 shows the reconstructions for the rectangular domain. Both algorithms obtained solutions that increased in accuracy as α decreased. The solutions from the alternating method were more accurate, having a better defined peak velocity and lower velocities at the edges. Figure 3 shows the reconstructions for the parabolic domain. Again, both algorithms obtain increasingly accurate solutions as the noise level decreased. Since this domain is comparatively shallower than the rectangular domain, we expect a better reconstruction, although we only saw an improvement for the Landweber algorithm. Both algorithms showed oscillations at the edges which correlate with oscillations in the low-frequency eigenmodes of the maps (A ) A. These oscillations 16

17 u.4 u a) x b) x Figure 3. True and reconstructed basal velocities for the parabolic crosssection domain using the Landweber (a) and alternating (b) methods. Thin solid line is u B as a function of x. Reconstructions are shown for α = 2 (dotted), α = 1 (dashed), and α =.5 (heavy solid). decreased with increasing iterations however, as can be seen for the level of accuracy achieved with α =.5 using the alternating algorithm. It should be noted that the higher accuracy of the alternating method corresponded to a higher number of iterations needed to meet the stopping principle. For the parabolic crosssection and α =.5, only 6 Landweber iterations were required by the stopping principle, whereas 23 alternating iterations were needed. The alternating solution for α = 2 and the Landweber solution for α =.5 were quite similar, and these corresponded to 8 alternating iterations and 6 Landweber iterations respectively. Acknowledgments This research is supported in part by the National Science Foundation, grants OPP and ARC References [1] A. Bakushinsky and A. Goncharsky. Ill-posed problems: theory and applications, volume 31. Kluwer Academic Publishers Group, Dordrecht, Translated from the Russian by I. V. Kochikov. [2] A. B. Bakushinsky and M. Yu. Kokurin. Iterative methods for approximate solution of inverse problems., volume 577 of Mathematics and Its Applications. Springer-Verlag, 24. [3] G. Bastay, V. A. Kozlov, and B. O. Turesson. Iterative methods for an inverse heat conduction problem. J. Inverse Ill-Posed Probl., 9(4): , 21. [4] George Bastay, Tomas Johansson, Vladimir A. Kozlov, and Daniel Lesnic. An alternating method for the stationary Stokes system. Z. Angew. Math. Mech., 86(4):268 28, 25. [5] J. Baumeister and A. Leitao. On iterative methods for solving ill-posed problems modeled by partial differential equations. J. Inverse Ill-Posed Probl., 9(1):13v29,

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