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]). 1.1. Model Description. Let Ω be the domain in the xy−plane with Lipschitz boundary, which has a geometry shown in Figure 1. The upper boundary S = (0, l) is interpreted as a surface of the ice sheet; B is a bottom (Lipschitz curve); Γ1,2 are sides which may or may not be present. We consider the following system of equations in Ω: Sergei Avdonin is with Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99775-6660, USA. e-mail:
[email protected]. Vladimir Kozlov is with Department of Mathematics, University of Link¨oping, Link¨oping, Sweden. e-mail:
[email protected]. David Maxwell is with Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99775-6660, USA. e-mail:
[email protected]. Martin Truffer is with Department of Physics and Geophysical Institute, University of Alaska Fairbanks, Fairbanks, AK 99775, USA. e-mail:
[email protected]. 1
Figure 1. Domain geometry; the parts Γi of the boundary may be empty.
−∇·(G(|∇u|)∇u) = f u|Γ1 = g1 ,
in Ω ,
u|Γ2 = g2 ,
∂y u|S = 0 ,
(1) (2) (3)
where g1,2 and f are given functions. In the case of the domain presented in figure 1 right, conditions (2) are not prescribed. We suppose that F 0 (t) , (4) t where F is a convex function for t ≥ 0, F 0 (0) = 0, F ∈ C 1,1 [0, ∞) (i.e. F 0 is Lipschitz continuous, |F 0 (t) − F 0 (τ )| ≤ C|t − τ |). For such functions F , the second derivative F 00 (t) exists almost everywhere and we suppose that 1 1 ν(1 + t) n −1 ≤ F 00 (t) ≤ µ(1 + t) 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 n−1 1 = T02 + G2 (s) s2 2 G(s) with T0 6= 0. Then Z G(t) =
t
F (t) =
G(τ ) τ d τ . 0
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) = ( + t2 )
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
smooth in Ω ∪ S then using the unique continuation property for elliptic equations (see, e.g., [9], Theorem 17.2.6) 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 Frech´et derivative and an estimate for the linear part of the Taylor’s development of the operator, H¨older continuity of the Frech´et 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 = Γ1 ∪ B ∪ Γ2 and suppose that the function g defined as g1 (γ), γ ∈ Γ1 g(γ) = ψ(γ), γ ∈ B g (γ), γ ∈ Γ 2 2 1
(8) 1
belongs to the space W 1− p ,p (Γ) , p = n+1 . Notice that if p < 2 (i.e., n > 1), g ∈ W 1− p ,p (Γ) n 1 1 for gj ∈ W 1− p ,p (Γj ) and ψ ∈ W 1− p ,p (B). If p = 2 (n = 1), then to describe the space of admissible ψ’s we have to extend gj to B in a such way that the extension g˜ belongs to W 1/2,2 (Γ) and set ψB = g˜|B . Then the set of admissible ψ in (7) is ψ = ψB + ψ0 , 1/2,2 ψ0 ∈ W00 (B) (the latter is a subspace of W 1/2,2 (Γ) of functions equal to zero on Γj , j = 1, 2). In the case when Γ1 = Γ2 = ∅ , the set of admissible functions ψ is W 1−1/p,p (B) . 0 p We suppose also that f ∈ Lp (Ω), p0 = p−1 . Problem (1) – (3), (7) is well posed and its solution is defined as a solution of the minimization problem Z min (F (|∇u|) − f u)dxdy , (9) Ω
3
where the minimum is taken over all u ∈ W 1,p (Ω),
u|Γ = g .
(10)
The Euler equation for this problem coincides with equation (1) with boundary conditions (2), (3), (7). One can check that the variational problem (9), (10) 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: Z (G(|∇u|)∇u · ∇v − f v)dxdy = 0 , (11) Ω
which must be valid for u ∈ W 1,p (Ω), u|Γ = g, and for all v ∈ W 1,p (Ω; Γ). Here W 1,p (Ω; Γ) is a subspace of W 1,p (Ω) 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 DA denote the set of admissible data ψ; we define the map A : DA 7→ W 1−1/p,p (S) taking Dirichlet data on B to Dirichlet data on S as follows. Let ψ ∈ DA and u ∈ W 1,p (Ω) be the solution of the minimization problem (9), (10). 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) = = {Lij (η)} , (13) ∂ηj i,j=1 where η = (η1 , η2 ). Direct calculations show that G0 (|η|) L(η) = G(|η|)I + |η|
η12 η1 η2 η1 η2 η22
.
(14)
Since G(t) = F 0 (t)/t, we have hL(η)ξ, ξi = F 00 (|η|)
hη, ξi2 F 0 (|η|) |ξ|2 |η|2 − hη, ξi2 + · . |η|2 |η| |η|2
(15)
From (5) it follows that 1
1
νn[(1 + t) n − 1] ≤ F 0 (t) ≤ µn[(1 + t) n − 1] .
(16)
Using that (1 + t)1/n − 1 ≥ (1 + t)1/n−1 t together with (5), (16), we derive from (15): n
(17)
1
1
ν(1 + |η|) n −1 |ξ|2 ≤ hL(η)ξ, ξi ≤ µn[(1 + |η|) n − 1] 4
1 2 |ξ| . |η|
(18)
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 > 0 such that kA(ψ2 ) − A(ψ1 )kW 1/2,2 (S) ≤ c kψ2 − ψ1 kW 1/2,2 (B) .
(19)
Proof. Let uj solve (11) with ψj in (7), respectively. Let also U = u2 − u1 .
(20)
ν|ξ|2 ≤ hL(η)ξ, ξi ≤ µ|ξ|2 .
(21)
For n = 1, Since Z
1
G(|∇u2 |)∇u2 − G(|∇u1 |)∇u1 = 0
d [G(|∇(u1 + tU )|)∇(u1 + tU )]dt = L∇U , dt
where Z
1
L = L(u1 , U ) =
L(∇(u1 + tU )) dt ,
(22)
0
we derive from (11) that Z (L∇U ) · ∇v dxdy = 0 for all v ∈ W 1,p (Ω; Γ) .
(23)
Ω
Due to estimate (21) problem (23) can be considered as a linear elliptic problem with respect to U with boundary conditions U |B = ψ2 − ψ1 , U |Γj = 0, j = 1, 2, and ∂y U = 0 on S. Then in a standard way we get the estimate1: Z |∇U |2 dxdy ≤ C||ψ2 − ψ1 ||2W 1/2,2 (B) .
(24)
(25)
Ω
Since U |S = A(ψ2 ) − A(ψ1 ) the trace theorem leads to (19).
Lemma 3.2. Let Γ1 and Γ2 be not empty, Γ1 = {0 × [m1 , 0]} , Γ1 = {` × [m2 , 0]}. Then there exists p0 > 2 such that for any p1 ∈ [2, p0 ] the solution U (introduced by (20)) satisfies the estimate kU kW 1,p1 (Ωm ) ≤ c kψ2 − ψ1 kW 1/2,2 (B) , (26) where Ωm = (0, l) × (m, 0), max{m1 , m2 } < m < 0, and p0 depends only on the ellipticity constants ν and µ, Ω and Γ. Proof. Let ζ = ζ(y) be a smooth function such that ( 1, m < y < 0 ζ(y) = 0 , y < 21 (m + max{m1 , m2 }) . 1Everywhere
in this paper by C and c we denote (generally distinct) constants independent of the variables appeared in the factors of these constants. 5
Then the function ζU satisfies the equation ∇ · (L∇(ζU )) = ∇ · (U L∇ζ) + (∇ζ) · (L∇U ) .
(27) ∗
We denote the right-hand side of (27) by Q and now verify that Q ∈ (W 1,q (Ω; Γ)) for every q ∈ (1, ∞), (the latter is the space of linear functionals on W 1,q (Ω; Γ))). Indeed let w ∈ W 1,q (Ω; Γ), then Z Z Q w = (−U (L∇ζ) · (∇w) + w (∇ζ) · (L∇U ) ≤ c(||U || q0 ||w||Lq + ||∇U ||L2 ||w||L2 ). L Ω Ω (28) By the Sobolev embedding theorem, ||U ||Lq0 ≤ c||∇U ||L2 and ||w||L2 ≤ c||∇w||Lq . Thus Q ∈ (W 1,q (Ω; Γ))∗ with any q ∈ (1, ∞) and ||Q||(W 1,q (Ω;Γ))∗ ≤ c||∇U ||L2 (Ω) .
(29)
The operator ∇ · (L∇) corresponding to the left hand side of (27) maps continuously ∗ 1,q 1,q 0 W (Ω; Γ) into W (Ω; Γ) and invertible for q = 2. Using the interpolation result of [8] and together with results of [20] we conclude that this operator is an isomorphism for q close to 2. This fact together with (29) gives ||ζU ||W 1,p0 ≤ c||∇U ||L2 (Ω)
(30)
for p0 > 2. Taking into account (25) we get (26).
Lemma 3.2 in conjunction with Lemma 3.1 implies Corollary 3.3. Under the hypotheses of Lemma 3.2 there exists p0 > 2 such that kA(ψ2 ) − A(ψ1 )kW 1−1/p0 ,p0 (S) ≤ c kψ2 − ψ1 kW 1/2,2 (B) .
(31)
4. Directional Derivative of the operator A Put u|B = ψ + th ∈ DA and differentiate (1)–(3) and (32) with respect to t; w :=
(32) du | . dt t=0
We get
∇ · (L(∇u)∇w) = 0 in Ω , ∂y w|S = 0,
w|B = h,
w|Γ1 ∪Γ2 = 0 ,
(33) (34)
or, in a weak sense, Z
(L(∇u)∇w) · (∇v) = 0 , ∀v ∈ W 1,2 (Ω; Γ).
(35)
Ω
Let us define the Gˆateaux derivative of the operator A in a standard way: d A0 (ψ)h = [A(ψ + th)]|t=0 . dt Clearly, A0 (ψ)h = w|S . 6
(36) (37)
1/2,2
1/2,2
The operator A0 acts continuously from W00 (B) to W00 (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 Γj 6= 0, j = 1, 2, and let w be solution of (35)–(34). Then kwkW 1,p1 (Ωm ) ≤ C khkW 1/2,2 (B) , 2 ≤ p1 ≤ p0 ,
(38)
where p0 and Ωm are the same as in Lemma 3.2. Corollary 4.2. Using the definition of the operator A0 and the above Lemma we get kA0 (ψ)hkW 1−1/p0 ,p0 (S) ≤ C khkW 1/2,2 (B) ,
(39)
for a certain p0 > 2, independent of ψ and h. 4.1. Frech´ et derivative. We show now that A0 (ψ) is actually the Frech´et derivative of A(ψ). More exactly, the following statement is true. Lemma 4.3. Suppose F ∈ C 2,1 [0, ∞). Then kA(ψ2 ) − A(ψ1 ) − A0 (ψ1 )(ψ2 − ψ1 )kW 1/2,2 (S) ≤ c kψ2 − ψ1 k1+ε W 1/2,2 (B) ,
(40)
where ε = 21 ( p20 − 1) > 0 independent of ψ1,2 . Proof. Let uj the solution of (1)–(3) and (7) with ψ = ψj , and let U = u2 − u1 . Then U satisfies the equation ∇ · (L(u1 , U )∇U ) = 0 in Ω , U = 0 on Γ1 ∪ Γ1 , ∂y U = 0 on S , U = ψ2 − ψ1 on B .
(41)
Let also w solve the problem (33)-(34) with h = ψ2 − ψ1 and u = u1 . Then V = U − w solves the problem ∇ · (L(u1 , 0)∇V ) = ∇ · ((L(u1 , 0) − L(u1 , U ))∇U ) in Ω
(42)
with the homogeneous boundary conditions V |Γ1 = V |Γ2 = ∂y V |S = V |B = 0 . Let us set Z Q := L(u1 , U ) − L(u1 , 0) = 0
1
d L(u1 , τ U ) dτ . dτ
(43) (44)
Using (22), we have Z
1
1
Z
d L(∇(u1 + τ tU )dtdτ 0 0 dτ 2 Z 1Z 1 X = L(j) (∇(u1 + τ tU ) Uxj t dt dτ ,
Q=
j=1
0
0
where L(j) (η) = ∂η∂ j L(η) and, temporarily, (x, y) = (x1 , x2 ). One can check that the functions L(j) (η) are bounded. Thus we get |Q| = |(L(u1 , U ) − L(u1 , 0)| ≤ C|∇U | . 7
Since L(u1 , U ), L(u1 , 0) are bounded, we have |Q| = |(L(u1 , U ) − L(u1 , 0)| ≤ C|∇U |α , 0 ≤ α ≤ 1 . Let ζ(y) be a smooth function such that ( 1, ζ(y) = 0,
(45)
m/2 ≤ y ≤ 0 y < m.
Equation (42) implies ∇ · (L∇(ζV )) = −∇ · (Q∇(ζU )) + ∇ · (QU ∇ζ) + ∇ζ · Q∇U + ∇ζ · L∇V + ∇(LV ∇ζ) , (46) where L = L(u1 , 0). Let us represent ζV = V0 + V1 ,
(47)
∇ · (L∇V0 ) = −∇ · (Q∇(ζU )) + ∇ · (QU ∇ζ) + ∇ζ · Q∇U ,
(48)
where V0 solves and V1 solves ∇ · (L∇V1 ) = ∇ζ · L∇V + ∇ · (LV ∇ζ) . (49) The both functions V0 and V1 satisfy zero Dirichlet boundary condition on Γ and zero Neumann condition on S. Multiplying the both sides of equation (48) by V0 and integrating by parts and using (45), we get 1/2 1/2 Z Z Z |∇V0 |2 ≤C |∇U |2(1+α) |∇V0 |2 Ω
Ω
Ωm
1/2
Z +C
|∇U |2α |U |2
Ωm
1/2 Z
|∇V0 |2
.
(50)
Ω
We use now that, by the H¨older inequality and Sobolev embedding theorem, Z Z 2α + 2 2α + 2 2α 2 2α+2 0 |∇U | |U | ≤ C |∇U | , p= , p = . 2α 2 Ωm
Ωm
Therefore, Z
Z
2
|∇V0 | ≤ C Ω
|∇U |2α+2 .
(51)
Ωm
Choosing 2α + 2 = p0 and using Lemma 3.2, we get Z |∇V0 |2 ≤ C kψ2 − ψ1 kpW0 1/2,2 (B) .
(52)
Ωm
Similarly to the corresponding part of the proof of Lemma 3.2 (see (28) and (29)) one can 0 check that the right hand side of (49) belongs to (W 1,p0 (Ω; Γ))∗ , and kζy L∇V + ∇ · (Lζy V )k(W 1,p00 (Ω;Γ))∗ ≤ C k∇V kL2 (Ω) . 8
(53)
The same reasoning as in the proof of Lemma 3.2 shows that the operator for the V1 is an 0 isomorphic from W 1,p0 (Ω; Γ) to (W 1,p0 (Ω; Γ))∗ . Therefore, k∇V1 kLp0 (Ω) ≤ C k∇V kL2 (Ω) .
(54)
Now we multiply (42) by V and integrate. We get Z Z Z 2 |∇V | ≤ C Q∇U ∇V0 + Q∇U ∇V1 . Ω
Ω
Ω
(55)
Using the H¨older inequality and (45) we have 1/2 1/2 Z Z Z |∇V |2 ≤C |∇U |2(α+1) |∇V0 |2 Ω
Ω
Ω
1/p01
1/p0
Z +C
|∇V1 |p0
Ω
Z
0
|∇U |(1+α1 )p1
.
(56)
Ω
Choosing α = 0, α1 = 1 − 2/p1 and using (51), (52), (54), and (25), we have Z p /2+1 1 |∇V |2 ≤ C kψ2 − ψ1 kW0 1/2,2 (B) + C||∇V ||L2 kψ2 − ψ1 k1+α W 1/2,2 (B) Ω
Finally, using Lemma 3.2 and the Cauchy inequality with q, we get Z h i p /2+1 4(1−1/p1 ) |∇V |2 ≤ C kψ2 − ψ1 kW0 1/2,2 (B) + kψ2 − ψ1 kW 1/2,2 (B) . Ω
Choose p1 =
8 6−p0
R . Since ( |∇V |2 )1/2 estimates above the left hand side of (40), we arrive Ω
at (40) with ε = 12 (p0 /2 − 1) .
4.2. Continuity of A0 (ψ). Theorem 4.4. The following inequality holds: k(A0 (ψ2 ) − A0 (ψ1 ))hkW 12 ,2 (S) ≤ C kψ2 − ψ1 kεW 21 ,2 (B) khkW 12 ,2 (B) , where ε = 1 −
2 p0
(57)
if 2 < p0 ≤ 4 and ε = 1/2 if p0 > 4.
Proof. Introduce uj as in the proof of Lemma 4.3 and put U = u2 − u1 . Also let wj solves (33), (34) with uj , and W = w2 − w1 . Then by (36) W |S = (A0 (ψ2 ) − A0 (ψ1 ))h .
(58)
Furthermore, W satisfies the problem ∇ · (L(∇u1 )∇W ) = ∇ · (Q∇w2 ) , where Q = L(∇u1 ) − L(∇u2 ) , with the boundary conditions ∂y W |S = 0, W |Γ = 0 . 9
(59)
Similarly to (45) one can show that |Q| ≤ C|∇U |α , 0 ≤ α ≤ 1 .
(60)
Introduce the same continuous function ζ as in the proof of Lemma 3.2, and rewrite equation (59) in the form ∇ · (L∇(ζW )) = ∇ · (Q∇(ζw2 )) − ∇ · (Qw2 ∇ζ) − ∇ζ · Q∇w2 + ∇ζ · L∇W + ∇ · (LW ∇ζ), where L = L(∇u1 ). Similar to (47) we represent ζW = W0 + W1 , see (48), (49). The equation for W0 is: ∇ · (L(∇u1 )∇(ζW0 )) = ∇ · (Q∇(ζw2 )) − ∇ · (Qw2 ∇ζ) − ∇ζ · Q∇w2 . Analog of estimate (50) has now the form Z 1/2 Z 1/2 Z 2 2α 2 2 2 |∇W0 | ≤ C |∇U | (|∇w2 | + |∇w2 | ) . |∇W0 | Ω
Ωm
Ω
Using the H¨older inequality and Sobolev Embedding Theorem we have p1p−2 Z 2/p1 Z Z 2αp1 1 . |∇w2 |p1 |∇U | p1 −2 |∇U |2α (|∇w2 |2 + |w2 |2 ) ≤ C Ωm
Ωm
Choosing α =
p1 −2 , 2
Ω
we arrive at the estimate
Z
Z
2
p1
2/p1 Z
|∇w2 |
|∇W0 | ≤ C
p1p−2 1
|∇U |
Ωm
Ωm
p1
.
Ω
Using Lemma 4.1 and Lemma 3.2 we obtain Z |∇W0 |2 ≤ C khk2W 1/2,2 (B) kψ2 − ψ1 kpW1 −2 1/2,2 (B) .
(61)
Ωm
Equation for W1 : ∇ · (L(∇u1 )∇W1 ) = ∇ζ · L∇W + ∇ · (LW ∇ζ) .
(62)
Analog of (53) in our case can be written as k∇ζ · L∇W + ∇ · (LW ∇ζ)k(W 1,p00 (Ω;Γ))∗ ≤ C k∇W kL2 (Ω) and we arrive at the estimate (compare with 54) k∇W1 kLp0 (Ω) ≤ C k∇W kL2 (Ω) .
(63)
Analog of (56) has the form: Z 1/2 Z 1/2 Z 2 2α 2 2 |∇W | ≤C |∇U | |∇w2 | |∇W0 | Ωm
Ω
Ω
Z
p0
1/p0 Z
|∇W1 |
+C
α1 p00
|∇U |
Ω
Ω
10
p00
|∇w2 |
1/p00 .
(64)
Choosing α = 0 and α1 = 1 − 2/p0 and using the H¨older inequality in the last integral in (64) we obtain 1/2 1/2 Z Z Z 2 2 2 |∇W0 | |∇W | ≤C |∇w2 | Ωm
Ω
Ω
Z
p0
1/p0 Z
|∇W1 |
+C
2
1/2 Z
|∇w2 |
Ω
Ω
2
|∇U |
α1 /2 .
Ω
Using Z
2
1/2 ≤ C khkW 1/2,2 (B)
|∇w2 |
(65)
Ω
and (61), (63), (25) together with Lemma 3.2, we get Z |∇W |2 ≤ C khk2 kψ2 − ψ1 kp1 /2−1 + C k∇W kL2 (Ω) khk kψ2 − ψ1 k1−2/p0 ,
(66)
Ω
where || · || = || · ||W 1/2,2 (B) . Hence Z |∇W |2 ≤ C khk2 [kψ2 − ψ1 kp1 /2−1 + kψ2 − ψ1 k2−4/p0 ].
(67)
Ω
Choosing p1 from p21 = 3 − p40 in the case p0 ≤ 4 (one can check that 2 ≤ p1 ≤ p0 ) and p1 = 4 in the case p0 > 4, we get Z |∇W |2 ≤ C khk2 kψ2 − ψ1 k2ε , (68) Ω
where ε = min(1 − 2/p0 , 1/2). This implies (57).
4.3. Adjoint operator to A0 (ψ) with respect to L2 -duality on B and S. Let us consider the boundary value problem (for the sake of definiteness we suppose that Γj 6= ∅, j = 1, 2) ∇ · (L(∇u)∇v) = 0 in Ω , ~n · L(∇u)∇v = H
on S ,
(69) (70)
v = 0 on Γ , (71) ∗ 1/2,2 where u is the solution of (9), (10), H ∈ W00 (S) , ~n is an outward unit normal to ∂Ω. 0 Notice that since ∂y u = 0 on S and ~n = on S, ~n · L(∇u)∇v = G(|∇u|)∂y v . 1 Problem (69)–(71) should be understood in a weak sense: the integral identity for v ∈ W 1,2 (Ω; Γ) Z Z L(∇u)∇v · ∇V = HV (72)
Ω
S
1,2
is valid for all V ∈ W (Ω; Γ). Clearly this linear problem has a unique solution v ∈ W 1,2 (Ω; Γ). For the function v satisfying (72) we can define a normal derivative ∂N v := ~n · L(∇u)∇v on ∂Ω as follows. 11
Consider the formula Z
Z L(∇u)∇v · ∇Q
(∂N v)q =
(73)
Ω
∂Ω 1,2
where Q ∈ W (Ω) and Q = q on ∂Ω. The left hand side of this equality does not depend on the extension of q into the domain Ω because of formula (72). Thus equality (73) can be taken as a definition of ∂N v for v ∈ W 1,2 (Ω) satisfying (72). Since Q can be chosen such that kQkW 1,2 (Ω) ≤ c kqkW 1/2,2 (∂Ω) with c independent of q, then ∂N v ∈ (W 1/2,2 (∂Ω))∗ and k∂N vk(W 1/2,2 (∂Ω))∗ ≤ c kvkW 1,2 (Ω) .
(74)
Restriction of the ∂N v on the portion B of ∂Ω is defined as a restriction of the functional 1/2,2 ∂N v to the subspace W00 (B) of W 1/2,2 (∂Ω). So, 1/2,2
∂N v|B ∈ (W00
(B))∗ .
Remark 4.5. For the case Γj = ∅, 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 A0 with respect to L2 inner products on B and S is defined by (A0 (ψ))∗ H = −∂N v|B , (75) where v solves problem (69)–(69). Indeed, by definition of ∂N v (73), we have Z Z Z L(∇u)∇v · ∇w . (∂N v)h + (∂N v)w = Ω
S
B
Let us represent w as w = w1 + w2 , where w1 , w2 ∈ W 1,2 (Ω; Γ) and w1 |B = 0, w2 |S = 0. Then Z L(∇u)∇v · ∇w2 = 0 Ω
because of (72), and Z L(∇u)∇v · ∇w1 = 0 Ω
because of (33) and (34). Thus Z
Z
−
(∂N v)h =
(∂N v)w
B
S
or Z
0
Z
∗
A (ψ) H h dΓ = B
HA0 (ψ)h dΓ ,
S 1
,2
which proves (75). Clearly, (A0 (ψ))∗ is a linear and bounded operator mapping (W002 (S))∗ 1
,2
into (W002 (B))∗ . 12
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 1
,2
approximation ψ0 ∈ W002 (B) and a positive constant γ, then further iterations are given by ψk+1 = ψk − γ A(ψk )(Aψk − ϕ) ,
k = 0, 1, . . . , 1
(76) 1
,2
,2
where A(ψk ) is a linear bounded operator mapping W002 (S) into W002 (B) and which is 1
1
,2
,2
adjoint to A0 (ψk ) with respect to inner products in W002 (S) and W002 (B). In what follows, 1
,2
1
,2
we shall use the notations HB and HS for the spaces W002 (B) and W002 (S) respectively. In order to use (76) let us express the operator A(ψk ) through the operator A0 (ψk )∗ , constructed in Sect. 4.3. Denote by (·, ·)HB and (·, ·)HS the inner products in the spaces HB and HS respectively. Let us introduce the canonical isomorphisms (see, e.g. [14, Sec. 2.1.3]) 1
,2
1
1
,2
1
,2
TB : W002 (B) 7→ (W002 (B))∗ , ,2
TS : W002 (S) 7→ (W002 (S))∗ . Then Z (TB ψ1 ) ψ2 = (ψ1 , ψ2 )HB , ZB (TS ϕ1 ) ϕ2 = (ϕ1 , ϕ2 )HS . S
One can verify now that the adjoint operator A to A0 with respect to these inner products is given by A(ψ) = TB−1 (A0 (ψ))∗ TS . 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 Frech´et derivative, instead we have only H¨older continuity (see Theorem 4.4). The convergence analysis in [2] shows in particular the importance of the choice of the first approximation. 5.2. Linear Case. Here we suppose that F (t) = t2 /2. In this case equation (1) turns to the Poisson equation −∆u = f
in Ω .
(77)
We suppose that Γ1 and Γ2 are not empty and the function and f ∈ L2 (Ω). In this case, by using local regularity estimates, one can show that the operator A can be extended to a bounded operator A : L2 (B) 7→ L2 (S) (even L2 (B) 7→ HS ). Then the Cauchy (ill-posed) problem is formulated as Aψ = ϕ . 13
(78)
Furthermore, operator A0 is determined by −∆w = 0 in Ω , ∂w = 0 , w|B = h , ∂y S w = 0 on Γ1 ∪ Γ2 , and A0 h = w|S . This operator can also be extended to the operator L2 (B) 7→ L2 (S). It coincides with A if we put gj = 0, f = 0. Therefore, (A0 )∗ is also a bounded linear operator L2 (S) 7→ L2 (B). To solve the problem (78) one can use Landweber operator iterative procedure in L2 : ψk+1 = ψk − γ (A0 )∗ (Aψk − ϕ)
(79)
with constant γ, 0 < γ < 1/ kA0 k2 . In this operator procedure the initial approximation ψ0 can be chosen arbitrary. It is known [7], Sec. 6.9, that ψn converges in L2 to the exact solution of (78) provided measurement ϕ is a performed without error. If we have measurement ϕδ with an error such that kϕδ − ϕkL2 < δ (80) 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, ϕδ ) when for the first time
ϕδ − Aψk(δ,ϕ ) ≤ τ δ (81) δ with τ > 1 fixed. This principle guarantees that ψk(δ,ϕδ ) → ψ as δ → 0. The above realization of the Landweber iterative procedure through the boundary value problems and convergence proof for L2 space, and in L2 space with a weight in the case when one of pieces Γ1 or Γ2 (or may be both) is empty, has been done in [3, 11, 10]. 5.3. 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 = gj on Γj , j = 1, 2. We choose an initial approximation (guess) ψ0 and consider equation (77) supplied with boundary conditions ∂u0 u0 |B = ψ0 , = 0. (82) ∂y S On the next step we solve the boundary value problem with boundary conditions ∂y u1 = ∂y u0 on B ,
u1 = ϕ on S .
(83)
Then u2 = u1 on B ,
∂y u2 = 0 on S ,
(84)
and so on. The convergence of this procedure for the Laplace equation and for the Lam´e 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
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 Lam´e 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 = uk+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 Γ1 and G2 . 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(|∇uk |)∇uk+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 + t2 ) where ε = 10−7 . This is a regularized variation of Glen’s flow law. The remaining prescribed data were also the same in all cases: f ≡ 1 and g1 = g2 ≡ 0. The rectangular domain is the region with −1 ≤ x ≤ 1 and −0.5 ≤ y ≤ 0. The surface S is the part of the boundary with y = 0, Γ1 and Γ2 are the vertical walls at x = −1 and x = 1, and the base B is the remainder of the boundary at y = −0.5. The parabolic cross-section is the region bounded by the graph of y = 1/2(x − 1)(x + 1) and the line y = 0. In this case, S is the part of the boundary with y = 0, B is the remainder of the boundary, and Γ1 and Γ2 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 M ψ(x, y) = |(x − 1)(x + 1)|5/2 2 The scaling constant M was picked to be equal to the maximum value of an analogous solution with ψ = 0 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 φα by adding Gaussian noise having mean zero and standard deviation M α/100. The noise levels of α% 15
ranged over α = 2, 1, and 0.5 in our tests. For the Landweber method, we measured the discrepancy δ = ||φα − φ||H 1/2 (S) and then ran the algorithm using the stopping principle ||φα − φk || < τ δ 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 φ0 ∈ H 1/2 (B) and computes φ2 = u2 |B ∈ H 1/2 (B) via equations (82)–(84). Let Kα be the analogous map with φ = φα 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 (ψk ) − ψk ||H 1/2 (B) < τ δ. We note that our tests do not determine how to pick the error threshold δ in practice, which can be difficult.
0.8
0.8
0.6
0.6 u
1
u
1
0.4
0.4
0.2
0.2
a)
0 −1
−0.5
0 x
0.5
1
b)
0 −1
−0.5
0 x
0.5
1
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 = 0.5 (heavy solid). In all cases, we started the iterative methods with an initial guess ψ0 = 0, 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 (A0 )∗ A0 . These oscillations 16
0.8
0.8
0.6 u
u
0.6
0.4
0.4
0.2 0.2 0 a)
0 −1
−0.5
0 x
0.5
1
b)
−1
−0.5
0 x
0.5
1
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 = 0.5 (heavy solid). decreased with increasing iterations however, as can be seen for the level of accuracy achieved with α = 0.5 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 = 0.5, 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 = 0.5 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 0414128 and ARC 0724860. References [1] A. Bakushinsky and A. Goncharsky. Ill-posed problems: theory and applications, volume 301. Kluwer Academic Publishers Group, Dordrecht, 1994. 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, 2004. [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):375–388, 2001. [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):268–280, 2005. [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, 2001. 17
[6] J. Colinge and J. Rappaz. A strongly nonlinear problem arising in glaciology. Mathem. Modelling Numer. Analysis, 33:395–406, 1999. [7] Heinz W. Engl, Martin Hanke, and Andreas Neubauer. Regularization of inverse problems. Kluwer Academic Publishers Group, Dordrecht, 1996. [8] J. A. Griepentrog, K G¨ oger, H.-Chr. Kaiser, and J. Rehberg. Interpolation for function spaces related to mixed boundary value problems. Math. Nachr., 241:110–120, 2002. [9] L. H¨ ormander. The analysis of linear partial differential operators. III. Pseudodifferential operators. Springer-Verlag, Berlin, 1985. [10] T. Johansson and L. Marin. A procedure for the temperature reconstruction in corner domains from cauchy data. Inverse Problems, 23(1):357–372, 2007. [11] Tomas Johansson. An iterative procedure for solving a cauchy problem for second order elliptic equations. Math. Nachr., 272:46–54, 2004. [12] V. A. Kozlov and V. G. Maz’ya. On iterative procedures for solving ill-posed boundary value problems that preserve differential equations. Lenningr. Math. J., 1(5):1207–1228, 1990. [13] V. A. Kozlov, V. G. Maz’ya, and A. V. Fomin. An iterative method for solving the cauchy problem for elliptic equations. Comput. Math. and Math. Phys., 31(1):45–52, 1992. [14] J.-L. Lions. Optimal control of systems governed by partial differential equations. Springer-Verlag, New York-Berlin, 1971. [15] L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen. An alternating iterative algorithm for the cauchy problem associated to the helmholtz equation. Comput. Methods Appl. Mech. Eng., 192(5-6):709v722, 2003. [16] L. Marin, L. Elliott, D. B. Ingham, and D. Lesnic. An iterative boundary element algorithm for a singular cauchy problem in linear elasticity. Comput. Mech., 28(6):479–488, 2002. [17] D. Maxwell, M. Truffer, S. Avdonin, and M. Stuefer. Determining glacier velocities and stresses with inverse methods: an iterative scheme. To appear, J. Glaciol., 2008. [18] N. S. Mera, L. Elliott, D. B. Ingham, and D. Lesnic. A comparison of different regularization methods for a cauchy problem in anisotropic heat conduction. Int. J. Numer. Methods Heat Fluid Flow, 13(56):528–546., 2003. [19] W.S.B. Paterson. The physics of glaciers. Pergamon, New York, 1994. 3rd edition. [20] I.Ya. Shneiberg. On the solvability of linear equations in interpolation families of banach spaces. Soviet Math. Dokl., 14:1328–1331, 1973. [21] Martin Truffer. The basal speed of valley glaciers: An inverse approach. J. Glaciol., 50(169):236–242, 2004. [22] E. Zeidler. Applied functional analysis. Main principles and their applications. Springer-Verlag, New York, 1995.
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