A posteriori error estimates for Fredholm integral equations of the
Transcript of A posteriori error estimates for Fredholm integral equations of the
PREPRINT 2012:21
A posteriori error estimates for Fredholm integral equations of the first kind
N. KOSHEV L. BEILINA
Department of Mathematical Sciences Division of Mathematics
CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG Gothenburg Sweden 2012
Preprint 2012:21
A posteriori error estimates for Fredholm integral equations of the first kind
N. Koshev and L. Beilina
Department of Mathematical Sciences Division of Mathematics
Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden
Gothenburg, October 2012
A posteriori error estimates for Fredholm integral equations of the rst kind 3
A(z) = u (3)
with an operatorA : H ! L2(W) dened as
A(z) :=Z
WK(x y)z(x)dx: (4)
Ill-posed problem.Let the functionz(x) 2 H1 of the equation (2) be unknown in the domainW. De-
termine the functionz(x) for x2 W assuming the functionsK(x y) 2 CkW
;k 0
andu(x) 2 L2(W) in (2) are known.Let d > 0 be the error in the right-hand side of the equation (2):
A(z ) = u ; ku u kL2(s ) d: (5)
whereu is the exact right-hand side corresponding to the exact solution z .To nd the approximate solution of the equation (2) in our numerical tests of
Section 9 we will minimize the functional
Ma (z) = kAz uk2L2(W) + a kzk2
H1(W) ; (6)
Ma : H1 ! R;
wherea = a (d) > 0 is the small regularization parameter.Our goal is to solve the equation (2) on the rather coarse meshwith some regular-
ization parametera and then construct the sequence of the approximated solutionszk on the rened meshesTk with the same regularization parametera . The regular-ization parametera can be chosen using one of the methods, described in [16] (forexample, the method of generalized discrepancy).
We consider now more general form of the Tikhonov functional(6). LetW1,W2;Qbe three Hilbert spaces,Q W1 as a set, the norm inQ is stronger than the normin W1 andQ = W1; where the closure is understood in the norm ofW1: We denotescalar products and norms in these spaces as
(; ) ;kk for W1;
(; )2 ;kk2 for W2
and [; ] ; [] for Q.
Let A : W1 ! W2 be a bounded linear operator. Our goal is to nd the functionz(x) 2 Q which minimizes the Tikhonov functional
Ea (z) : Q ! R; (7)
Ea (z) =12
kAz uk22 +
a2
[z z0]2 ;u 2 W2;z;z0 2 Q; (8)
4 N.Koshevā and L. Beilinaāā
whereĪ± ā (0,1) is the regularization parameter. To do that we search for a stationarypoint of the above functional with respect tozsatisfyingābā Q
Eā²Ī±(z)(b) = 0. (9)
The following lemma is well known [1] for the caseW1 = W2 = L2.Lemma 1. Let A : L2 ā L2 be a bounded linear operator. Then the Frechet
derivative of the functional (6) is
Eā²Ī± (z) (b) = (AāAzāAāu,b)+ Ī± [zāz0,b] ,ābā Q. (10)
In particular, for the integral operator (2) we have
Eā²Ī± (z) (b) =
ā«
Ī©
b(s)
ā«
Ī©
z(y)
ā«
Ī©
K(xāy)K(xās)dx
dyā
ā«
Ī©
K(xās)u(x)dx
ds
(11)+Ī± [zāz0,b] ,ābā Q.
Lemma 2 is also well known, sinceA : W1 āW2 is a bounded linear operator. Weformulate this lemma only for our specific case and refer to [16] for a more generalcase. For the case of a nonlinear operator we refer to [3].
Lemma 2.Let the operator A: W1 āW2 satisfies conditions of Lemma 1. Thenthe functional EĪ± (z) is strongly convex on the space Q with the convexity parameterĪŗ such that (
Eā²Ī± (x)āEā²
Ī± (z) ,xāz)ā„ Īŗ [xāz]2,āx,zā Q. (12)
Similarly, the functionalMĪ±(z) is also strongly convex on the Sobolev spaceH1:(Mā²
Ī± (x)āMā²Ī± (z) ,xāz
)H1
ā„ Īŗ ||xāz||2H1,āx,zā H1, (13)
Remark.We assume in (2) thatuā L2
(Ī©)
since this function can be given with a noise.This is done despite to thatA(z) āCk
(Ī©),kā„ 0.
3 The finite element spaces
Let Ī© āRm,m= 2,3, be a bounded domain with a piecewise-smooth boundaryāĪ© .
Following [11] we discretize the domainĪ© by an unstructured meshT using non-overlapping tetrahedral elements inR
3 and triangles inR2 such thatT = K1, ...,Kl ,wherel is the number of elements inĪ© , and
Ī© = āŖKāTK = K1āŖK2...āŖKl .
A posteriori error estimates for Fredholm integral equations of the first kind 5
We associate with the triangulationT the mesh functionh = h(x) which is apiecewise-constant function such that
h(x) = hK āK ā T,
wherehK is the diameter ofK which we define as the longest side ofK.Let r ā² be the radius of the maximal circle/sphere contained in the elementK. We
make the following shape regularity assumption for every elementK ā T
a1 ā¤ hK ā¤ r ā²a2; a1,a2 = const. > 0. (14)
We introduce now the finite element spaceVh as
Vh =
v(x) ā H1(Ī©) : vāC(Ī©), v|K ā P1(K) āK ā T, (15)
whereP1(K) denote the set of piecewise-linear functions onK. The finite dimen-sional finite element spaceVh is constructed such thatVh ā V. The finite elementmethod which uses piecewise-linear test functions we call CG(1) method. CG(1)can be applied on the conforming meshes.
In a general case we allow also meshes in space with hanging nodes and assumethat the local mesh size has bounded variation in such meshes. This means thatthere exists a constantĪ³ > 0 such thatĪ³hK+ ā¤ hK ā¤ Ī³ā1hK+ for all the neighboringelementsKā andK+. Let S be the internal face of the non-empty intersection ofthe boundaries of two neighboring elementsK+ andKā. We denote the jump of thefunctionvh computed from the two neighboring elementsK+ andKā sharing thecommon sideSas
[vh] = v+h āvāh . (16)
We introduce the discontinuous finite element spaceWh on such meshes as
Wh =
v(x) āV : v|K ā DP1(K) āK ā T, (17)
whereDP1(K) denote the set of discontinuous linear functions onK. The finiteelement spaceWh is constructed such thatWh āV. The finite element method whichuses discontinuous linear functions we call DG(1) method.
Let Pk : V ā M for āM ā V, be the operator of the orthogonal projection. Letthe function f ā H1 (Ī©)ā©C(Ī©) andāxi fxi ā Lā (Ī©) . We define byf I
k the stan-dard interpolant [8] on triangles/tetrahedra of the function f ā H. Then by one ofproperties of the orthogonal projection
ā f āPk fāL2(Ī©) ā¤ā„ā„ f ā f I
k
ā„ā„L2(Ī©)
. (18)
It follows from formula 76.3 of [8] that
ā f āPk fāL2(Ī©) ā¤CI āh ā fāL2(Ī©) ,ā f āV. (19)
whereCI = CI (Ī©) is positive constant depending only on the domainĪ© .
6 N.Koshevā and L. Beilinaāā
4 A finite element method
To formulate a CG(1) for equation (9) we recall the definitionof the spaceVh. TheCG(1) finite element method then reads: findzh āVh such that
Eā²Ī±(zh)(b) = 0 ābāVh. (20)
Similarly, for DG(1) for equation (9) we recall the definition of the spaceWh. TheDG(1) finite element method then reads: findzh āWh such that
Eā²Ī±(zh)(b) = 0 ābāWh. (21)
5 A posteriori error estimate for the regularized solution onlocally refined meshes
In this section we will formulate theorems for accuracy of the regularized solutionfor the case of the more general functionalEĪ± defined in (7).
From the theory of convex optimization it is known, that Lemma 2 claims exis-tence and uniqueness of the global minimizer of the functional EĪ± defined in (7) forzĪ± ā Q such that
EĪ±(zĪ±) = infzāQ
EĪ±(z).
It is well known that the operatorF is Lipschitz continuous
āF(z1)āF(z2)ā ā¤ ||A|| Ā· āz1āz2ā āz1,z2 ā H.
Because of the boundedness of the operatorA there exists the constant
D = 2(āAā2+ Ī±) = const. > 0 (22)
such that the following inequality holds [1]ā„ā„Eā²
Ī± (z1)āEā²Ī± (z2)
ā„ā„ā¤ Dāz1āz2ā ,āz1,z2 ā H. (23)
Similarly, the functionalMĪ±(z) is twice Frechet differentiable [16] and the fol-lowing inequality holds [1]:
ā„ā„Mā²Ī± (z1)āMā²
Ī± (z2)ā„ā„ā¤ Dāz1āz2ā ,āz1,z2 ā H. (24)
Let zk be the computed solution (minimizer) of the Tikhonov functional andzĪ± āH be the regularized solution on the finally refined mesh. LetPk be the operator ofthe orthogonal projection defined in section 3. Then the following theorem is validfor the functional (8):
Theorem 1a
A posteriori error estimates for Fredholm integral equations of the first kind 7
Let zk be a minimizer of the functional (8). Assume that (12) holds.Then thereexists a constant D defined by (22) such that the following estimate holds
[zkāzĪ± ] ā¤DĪŗ||PkzĪ± āzĪ± ||W1. (25)
In particular, if PkzĪ± = zĪ± , then zk = zĪ± , which means that the regularized solutionis reached after k mesh refinements.
Proof.Through the proof the Frechet derivativeEā²
Ī± and the scalar product(Ā·, Ā·) are givenin W1 norm.
Sincezk is a minimizer of the functional (8) then by (12) the minimizer zk isunique and the functional (8) is strongly convex with the strong convexity constantĪŗ . This implies that
Īŗ [zkāzĪ± ]2 ā¤(Eā²
Ī± (zk)āEā²Ī± (zĪ±) ,zkāzĪ±
). (26)
Sincezk is the minimizer of the functional (7), then
(Eā²
Ī± (zk) ,y)
= 0, āyāW1. (27)
Next, sincezĪ± is the minimizer on the setQ, then(Eā²
Ī± (zĪ± ) ,z)
= 0, āzā Q.
Using (26) with the splitting
zkāzĪ± = (zkāPkzĪ±)+ (PkzĪ± āzĪ±) ,
together with the Galerkin orthogonality principle (27) weobtain(Eā²
Ī± (zk)āEā²Ī± (zĪ± ) ,zkāPkzĪ±
)= 0 (28)
and thusĪŗ [zkāzĪ± ]2 ā¤
(Eā²
Ī± (zk)āEā²Ī± (zĪ± ) ,PkzĪ± āzĪ±
). (29)
It follows from (23) that
(Eā²
Ī± (zk)āEā²Ī± (zĪ± ) ,PkzĪ± āzĪ±
)ā¤ D[zkāzĪ± ]||PkzĪ± āzĪ± ||W1.
Substituting above equation into (29) we obtain (25).
The following theorem is valid for the functional (6) when the operatorA :H1(Ī©) ā L2(Ī©):
Theorem 1bLet zk be a minimizer of the functional (6). Assume that (13) holds and that
the regularized solution zĪ± is not yet coincides with the minimizer zk after k meshrefinements. Then there exists a constant D defined by (22) such that the following
8 N.Koshevā and L. Beilinaāā
estimate holds
||zkāzĪ± ||H1 ā¤DĪŗ||PkzĪ± āzĪ± ||H1. (30)
In particular, if PkzĪ± = zĪ± , then zk = zĪ± , which means that the regularized solutionis reached after k mesh refinements.
Proof.In this proof the Frechet derivativeEā²
Ī± and the scalar product(Ā·, Ā·) are given inH1 norm. Sincezk is a minimizer of the functional (6) then by (13) the minimizer zk
is unique and the functional (6) is strongly convex on the spaceH1 with the strongconvexity constantĪŗ . This implies that
Īŗ āzkāzĪ±ā2H1 ā¤
(Mā²
Ī± (zk)āMā²Ī± (zĪ±) ,zkāzĪ±
). (31)
Sincezk is the minimizer of the functional (6), then(Mā²
Ī± (zk) ,y)
= 0, āyā H1. (32)
Next, sincezĪ± is the minimizer on the setH, then(Mā²
Ī± (zĪ±) ,z)
= 0, āzā H1.
Using (31) with the splitting
zkāzĪ± = (zkāPkzĪ±)+ (PkzĪ± āzĪ±) ,
together with the Galerkin orthogonality principle (32) weobtain(Mā²
Ī± (zk)āMā²Ī± (zĪ±) ,zkāPkzĪ±
)= 0 (33)
and thusĪŗ āzkāzĪ±ā
2H1 ā¤
(Mā²
Ī± (zk)āMā²Ī± (zĪ± ) ,PkzĪ± āzĪ±
). (34)
It follows from (24) that(Mā²
Ī± (zk)āMā²Ī± (zĪ± ) ,PkzĪ± āzĪ±
)ā¤ D||zkāzĪ± ||H1||PkzĪ± āzĪ± ||H1.
Substituting above equation into (34) we obtain (30).
In Theorem 2 we derive a posteriori error estimates for the error in the Tikhonovfunctional (6) on the mesh obtained afterk mesh refinements.
Theorem 2Let conditions of Lemma 2 hold. Suppose that there exists minimizer zĪ± āH2(Ī©)
of the functional MĪ± on the set V and mesh T. Suppose also that there exists finiteelement approximation zk of MĪ± on the set Wh and mesh T. Then the followingapproximate a posteriori error estimate for the error in theTikhonov functional (6)holds
A posteriori error estimates for Fredholm integral equations of the first kind 9
|MĪ±(zĪ± )āMĪ±(zk)| ā¤CIā„ā„Mā²
Ī±(zk)ā„ā„
H1(Ī©)
(||hzk||L2(Ī©) +ā[zk]āL2(Ī©) +ā
K
ā[ānzk]āL2(āK))
).
(35)In the case when the finite element approximationzk āVh obtained by CG(1) we
have|MĪ±(zĪ± )āMĪ±(zk)| ā¤CI
ā„ā„Mā²Ī±(zk)
ā„ā„H1(Ī©)
||hzk||L2(Ī©). (36)
ProofBy definition of the Frechet derivative we can write that on the meshT we have
MĪ±(zĪ± )āMĪ±(zk) = Mā²Ī±(zk)(zĪ± āzk)+R(zĪ± ,zk), (37)
where by Lemma 1R(zĪ± ,zk) = O((zĪ± āzk)2), (zĪ± āzk)ā 0 āzĪ± ,zk āV. The term
R(zĪ± ,zk) is small since we assume thatzk is minimizer of the Tikhonov functional onthe meshT and this minimizer is located in a small neighborhood of the regularizedsolutionzĪ± . Thus, we can neglectR in (37), see similar results for the case of ageneral nonlinear operator equation in [1,3]. Next, we use the splitting
zĪ± āzk = zĪ± āzIĪ± +zI
Ī± āzk (38)
and the Galerkin orthogonality [8]
Mā²Ī±(zk)(z
IĪ± āzk) = 0 āzI
Ī± ,zk āWh (39)
to getMĪ± (zĪ±)āMĪ±(zk) ā¤ Mā²
Ī±(zk)(zĪ± āzIĪ±), (40)
wherezIĪ± is a standard interpolant ofzĪ± on the meshT [8]. We have that
||MĪ±(zĪ± )āMĪ±(zk)||H1(Ī©) ā¤ ||Mā²Ī±(zk)||H1(Ī©)||zĪ± āzI
Ī± ||H1(Ī©), (41)
where the term||zĪ± āzIĪ± ||H1(Ī©) in the right hand side of the above inequality can be
estimated through the interpolation estimate with the constantCI
||zĪ± āzIĪ± ||H1(Ī©) ā¤CI ||h zĪ± ||H2(Ī©).
Substituting above estimate into (41) we get
||MĪ± (zĪ±)āMĪ±(zk)||L2(Ī©) ā¤CIā„ā„Mā²
Ī±(zk)ā„ā„
L2(Ī©)||h zĪ± ||H2(Ī©). (42)
Using the facts that [10]
|āzĪ± | ā¤|[zk]|
hK
and
|D2zĪ± | ā¤|[ānzk]|
hK,
we can estimate||h zĪ± ||H2(Ī©) in a following way:
10 N.Koshevā and L. Beilinaāā
||h zĪ± ||H2(Ī©) ā¤ āK||hKzĪ± ||H2(K) ā¤ ā
K||(zĪ± + āzĪ± +D2zĪ± )hK ||L2(K)
ā¤ āK
(||zkhK ||L2(K) +
ā„ā„ā„ā„[zk]
hKhK
ā„ā„ā„ā„L2(K)
+
ā„ā„ā„ā„[ānzk]
hkhK
ā„ā„ā„ā„L2(āK)
)
ā¤ ||hzk||L2(Ī©) +ā[zk]āL2(Ī©) +āKā[ānzk]āL2(āK) .
(43)
HereD2zĪ± denotes the second order derivatives ofzĪ± . Substituting above estimateinto right hand side of (42) we get estimate (35).
6 A posteriori error estimates for the functional (44) in DG(1)
We now provide a more explicit estimate for the weaker normāzkāzĪ±āL2(Ī©) whichis more efficient for practical computations since it does not involves computation ofterms likeā[ānzk]āL2(āK) which are included in estimate (35). To do this, we replace
in (6) the normāzāz0ā2H1(Ī©) with the weaker normāzāz0ā
2L2(Ī©) .
Below in Theorems 3, 4 and 5 we will consider the following Tikhonov func-tional
EĪ±(z) : H ā R,
EĪ± (z) =12āAzāuāL2(Ī©) +
Ī±2āzāz0ā
2L2(Ī©) . (44)
Theorem 3.Let Ī± ā (0,1) and A: L2 ā L2 be a bounded linear operator. Letzk āWh be the minimizer of the functional EĪ± (z) obtained by DG(1) on T . Assumethat the regularized solution zĪ± is not yet reached on the mesh T and is not coincidedwith the minimizer zk. Let jump of the function zk computed from the two neighboringelements K+ and Kā sharing the common side S on the mesh T is defined by
[zk] = z+k āzāk . (45)
Then there exist constants D,CI defined by (24),(19), correspondingly, such that thefollowing estimate holds
āzkāzĪ±āL2(Ī©) ā¤CI D
Ī±ā[zk]āL2(Ī©)
Proof.Conditions (24) imply that
ā„ā„Eā²Ī± (zk)āEā²
Ī± (zĪ±)ā„ā„
L2(Ī©)ā¤ DāzkāzĪ±āL2(Ī©) (46)
with a constantD(||A||,Ī±) > 0. By (19)
āzĪ± āPkzĪ±āL2(Ī©) ā¤CI āh āzĪ±āL2(Ī©) . (47)
A posteriori error estimates for Fredholm integral equations of the first kind 11
Using the Cauchy-Schwarz inequality as well as (46) and (47), we obtain from (30)
āzkāzĪ±āL2(Ī©) ā¤CI D
Ī±āh āzĪ±āL2(Ī©) . (48)
We can estimate||h āzĪ± || in a following way. Using the fact that [10]
|āzĪ± | ā¤|[zk]|
hK(49)
we have
||h āzĪ± ||L2(Ī©) ā¤ āK
||hKāzĪ± ||L2(K)
ā¤ āK||hK
|[zk]|
hK||L2(K) = ā
K||[zk]||L2(K).
(50)
Substituting above estimate in (48) we get
āzkāzĪ±āL2(Ī©) ā¤CI D
Ī± āKā[zk]āL2(K) =
CI DĪ±
ā[zk]āL2(Ī©). (51)
7 A posteriori error estimate for the error in the Tikhonovfunctional (44)
The proof of Theorem 4 is modification of the proof given in [5]. In the proof of thisTheorem we used the fact thezk is obtained using CG(1) onT. Theorem 5 followsfrom the proof of Theorem 4 in the case of DG(1) method.
Theorem 4Let conditions of Lemma 2 hold and A: L2 ā L2 be a bounded linear operator.
Suppose that there exists minimizer zĪ± of the functional EĪ± on the set V and meshT . Suppose also that there exists approximation zk ā Vh of EĪ± . Then the followingapproximate a posteriori error estimate for the error in theTikhonov functional (6)holds
|EĪ±(zĪ± )āEĪ±(zk)| ā¤CI ||Eā²Ī±(zk)||L2(Ī©) Ā· ||h āzĪ± ||L2(Ī©). (52)
ProofBy definition of the Frechet derivative we can write that on every meshTk
EĪ±(zĪ± )āEĪ±(zk) = Eā²Ī±(zk)(zĪ± āzk)+R(zĪ± ,zk), (53)
where by Lemma 1R(zĪ± ,zk) = O(r2), r ā 0 āzĪ± ,zk āV, r = |zĪ± āzk|.Now we neglectR, use the splitting
12 N.Koshevā and L. Beilinaāā
zĪ± āzk = zĪ± āzIĪ± +zI
Ī± āzk (54)
and the Galerkin orthogonality [8]
Eā²Ī±(zk)(z
IĪ± āzk) = 0 āzI
Ī± ,zk āVh (55)
with the space DG(1) for approximation of functionszĪ± , to get
EĪ±(zĪ±)āEĪ±(zk) ā¤ Eā²Ī±(zk)(zĪ± āzI
Ī±), (56)
wherezIĪ± is a standard interpolant ofzĪ± on the meshT [8]. Applying interpolation
estimate (19) tozĪ± āzIĪ± we get
|EĪ±(zĪ± )āEĪ±(zk)| ā¤CI ||Eā²Ī±(zk)||L2(Ī©) Ā· ||h āzĪ± ||L2(Ī©). (57)
Theorem 5Let conditions of Lemma 2 hold and A: L2 ā L2. Suppose that there exists mini-
mizer zĪ± of the functional EĪ± on the set V and mesh T. Suppose also that there existsapproximation zk āWh of EĪ± obtained by DG(1). Then the following approximate aposteriori error estimate for the error in the Tikhonov functional (6) holds
|EĪ±(zĪ± )āEĪ±(zk)| ā¤CI ||Eā²Ī±(zk)||L2(Ī©) Ā· ā[zk]āL2(Ī©). (58)
ProofIn the case of CG(1) by Theorem 4 we have the following a posteriori error
estimate for the error in the Tikhonov functional (44)
|EĪ±(zĪ± )āEĪ±(zk)| ā¤CI ||Eā²Ī±(zk)||L2(Ī©) Ā· ||h āzĪ± ||L2(Ī©). (59)
Using now for||h āzĪ± ||L2(Ī©) the estimates (49)-(50) in the case of DG(1) we get thefollowing a posteriori error estimate
|EĪ±(zĪ± )āEĪ±(zk)| ā¤CI ||Eā²Ī±(zk)||L2(Ī©) Ā· ā[zk]āL2(Ī©). (60)
Using the Theorems 2 - 5 we can now formulate our mesh refinement recom-mendations in CG(1) and DG(1) for a Fredholm integral equation of the first kindused in practical computations. Let us define a posteriori error indicator
Eh(zk) =
ā«
Ī©
zk(y)
ā«
Ī©
K(x,y)K(xās)dx
dyā
ā«
Ī©
K(xās)u(x) dx. (61)
We note that a posteriori error indicator (61) is approximation of the function|Eā²
Ī±(zk)| which is used in the proofs of Theorems 2-5. We neglect the computa-tion of the regularization term in the function|Eā²
Ī±(zk)| since this term is very small,
A posteriori error estimates for Fredholm integral equations of the first kind 13
and obtain a posteriori error indicator (61). Such approximation does not affect onthe refinement of the mesh.
The First Mesh Refinement Recommendation.Using the Theorems 4 and 5we can conclude that we should refine the mesh in neighborhoods of those points inĪ© where the function|Eā²
Ī±(zk)| or the function|Eh(zk)| attains its maximal values.More precisely, letĪŗ ā (0,1) be the tolerance number which should be chosen incomputational experiments. Refine the mesh in such subdomains of Ī© where
ā£ā£Eā²Ī±(zk)
ā£ā£ā„ Īŗ maxĪ©
ā£ā£Eā²Ī±(zk)
ā£ā£
or|Eh(zk)| ā„ Īŗ max
Ī©|Eh(zk)| .
The Second Mesh Refinement Recommendation.Using the Theorem 3 we canconclude that we should refine the mesh in neighborhoods of those points inĪ©where the function|zk| attains its maximal values. More, precisely in such subdo-mains ofĪ© where
|zk| ā„ ĪŗmaxĪ©
|zk|
whereĪŗ ā (0,1) is the number which should be chosen computationally.
8 The Adaptive Algorithm
In this section for solution of a Fredholm integral equationof the first kind (2) wepresent adaptive algorithms which we apply in numerical examples of section 9. Ouralgorithms use mesh refinement recommendations of section 7. In these algorithmswe also assume that the kernel in (2) is such thatK(xāy) = Ļ(yāx). Next, using theconvolution theorem we can determine the functionsz(x) in (2) and the regularizedsolution zĪ± of (6), correspondingly. For example, for calculation of the functionzĪ±(x) in numerical examples of section 9 we use (69). In our algorithms we definethe minimizer and its approximation byzĪ± andzk, correspondingly.
In Algorithm 1 we apply the first mesh refinement recommendation of Section 7,while in Algorithm 2 are used both mesh refinement recommendations of section 7.These algorithms are successfully tested by numerical examples of section 9.
Algorithm 1
ā¢ Step 0. Choose an initial meshT0 in Ī© and obtain the numerical solutionz0
of (6) on T0 using the finite element discretization of (20) for CG(1) or (21)for DG(1) and discretization of the convolution theorem (69). Compute the se-quencezk,k > 0, via following steps:
14 N.Koshevā and L. Beilinaāā
ā¢ Step 1. Interpolate the given right hand side of (2) and the solution zkā1 fromthe meshTkā1 to the meshTk and obtain the numerical solutionzk of (6) onTk
using the finite element discretization of (20) for CG(1) or (21) for DG(1) anddiscretization of (69).
ā¢ Step 2. Refine the meshTk at all points where
|Bh(zk)| ā„ Ī²k maxĪ©
|Bh(zk)|, (62)
with
Bh (zk) =ā«
Ī©
zk(y)
ā«
Ī©
Ļ(x,y)Ļ(x,s)dx
dyā
ā«
Ī©
Ļ (x,s)u(x) dx. (63)
Here the tolerance numberĪ²k ā (0,1) is chosen by the user.ā¢ Step 3. Construct a new meshTk+1 in Ī© and perform steps 1-3 on the new
mesh. Stop mesh refinements when||zkāzkā1|| < Īµ or ||Bh(zk)|| < Īµ, whereĪµis tolerance chosen by the user.
Algorithm 2
ā¢ Step 0. Choose an initial meshT0 in Ī© and obtain the numerical solutionz0
of (6) on T0 using the finite element discretization of (20) for CG(1) or (21)for DG(1) and discretization of the convolution theorem (69). Compute the se-quencezk,k > 0, via following steps:
ā¢ Step 1. Interpolate the given right hand side of (2) and the solution zkā1 fromthe meshTkā1 to the meshTk and obtain the numerical solutionzk of (6) onTk
using the finite element discretization of (20) for CG(1) or (21) for DG(1) anddiscretization of (69).
ā¢ Step 2. Refine the meshTk at all points where
|Bh (zk) | ā„ Ī²k maxĪ©
|Bh (zk) | (64)
with Bh(zk) defined by (63), and where
|zk (x)| ā„ Īŗk maxĪ©
|zk (x)| . (65)
Here the tolerance numbersĪ²k, Īŗk ā (0,1) are chosen by the user.ā¢ Step 3. Construct a new meshTk+1 in Ī© and perform steps 1-3 on the new
mesh. Stop mesh refinements when||zkāzkā1|| < Īµ or ||Bh(zk) || < Īµ, whereĪµis tolerance chosen by the user.
Remarks
A posteriori error estimates for Fredholm integral equations of the first kind 15
ā¢ 1. We note that the choice of the tolerance numbersĪ²k, Īŗk in (63), (65) dependson the concrete values of maxĪ© |Bh(zk) | and maxĪ© |zk (x)|, correspondingly. Ifwe would chooseĪ²k, Īŗk very close to 1 then we would refine the mesh in verynarrow region of the computational domainĪ© , and if we will chooseĪ²k, Īŗk ā 0then almost all mesh of the domainĪ© will be refined what is unsatisfactory.Thus, the values of the numbersĪ²k, Īŗk should be chosen in optimal way. Ournumerical tests show that the choice ofĪ²k, Īŗk = 0.5 is almost optimal one, how-ever, it can be changed during the iterations in adaptive algorithms from onemesh to other.
ā¢ 2. We also note that we neglect the computation of the regularization term in aposteriori error indicator (63) since this term is very small and does not affectson the refinement procedure. However, this term is included in the minimizationprocedure of the Tikhonovās functional (6).
9 Numerical studies of the adaptivity technique inmicrotomography
Microtomography in the backscattering electron mode allows obtain the picture ofa plain layer which is located at a some depth below the surface of the investigatedobject. Due to the fact that the electron probe (monokineticelectron beam) has thefinite radius the measured signal of this layer is distorted.It was shown in [13]and [14] that there exists connection between the measured signal obtained by theelectron microscope and the real scattering coefficient of the object under investiga-tion. The measured signalu(Ī¾ ,Ī·) can be described by a Fredholm integral equationof the first kind
u(Ī¾ ,Ī·) =
ā«
Ī©z(x,y)Ļ(xā Ī¾ ,yāĪ·)dxdy. (66)
Here, the kernel is given by the relation
Ļ(x,y) =1
2Ļr2 exp(āx2 +y2
2r2 ) (67)
with the variance functionr = r(t) depended on the deptht of the layer under inves-tigation.
To solve the equation (66) we minimize the following Tikhonov functional onSobolev spaceH1
MĪ±(z) = ||
ā«
Ī©
Ļ(xā Ī¾ ,yāĪ·)z(x,y)dxdyāu(Ī¾ ,Ī·)ā2L2(Ī©) + Ī±āz(x,y)ā2
H1. (68)
Using the convolution theorem we can obtain the following expression for the min-imizerzĪ±(x,y) (see, for example, [15]) of the functional (68)
16 N.Koshevā and L. Beilinaāā
zĪ± (x,y) =
ā«
R2eāi(Ī» x+Ī½y) Pā(Ī» ,Ī½)U(Ī» ,Ī½)
|P(Ī» ,Ī½)|2 + Ī±(1+ Ī» 2+ Ī½2)2 dĪ»dĪ½, (69)
where functionsU andP are the Fourier transforms of the functionsu andĻ , re-spectively, andPā denotes the complex conjugated function to the functionP.
The goal of our computational test was to restore image of Figure 1-a) whichrepresents the part of the planar microscheme obtained fromthe experimentallymeasured data by microtomograph [14]. This image was measured on the depth 0.9Āµm of the microscheme with the smearing parameterr = 0.149 mkm in (67). Realarea of the image of Figure 1-a) isĪ© = 16.963 mkm. For restoration of the imageof Figure 1-a) we apply the adaptive algorithm of section 8.
First, we computez0 on the initial coarse meshT0 using the finite elementdiscretization of (69) as described in section 3 with the regularization parameterĪ± = 2e10ā 07 in (68) on the coarse mesh presented in Figure 1-g). Let us definethe function
Bh (zk)=
ā«
Ī©
zk(y)
ā«
Ī©
Ļ (xā Ī¾ ,yāĪ·)Ļ (xā Ī¾ ,sāĪ·)dx
dyā
ā«
Ī©
Ļ (xā Ī¾ ,sāĪ·)u(x) dx,
(70)whereĪ© is our two-dimensional domain. We refine the mesh in all subdomains ofĪ© where the gradient of the functionBh (zk)(x) attains its maximal values, or where
|Bh (zk) | ā„ Ī²k maxĪ©
|Bh (zk) | (71)
with Ī²k = 0.5. Next, we perform all steps of the adaptive algorithm untilthe desiredtolerance||zkāzkā1||< Īµ with Īµ = 10eā05 is achieved or the computedL2- normsof the differences||zkāzkā1|| are started abruptly grow.
Figures 1-b)-f) show results of the reconstruction on the adaptively refinedmeshes which are presented in Figures 1-h)-l). Using the Figure 1 we observe thaton the fifth refined mesh corresponding to the Figure 1-l) we obtain the best restora-tion results. Since the computedL2- norms||zk ā zkā1|| are started abruptly growafter the fifth refinement of the initial mesh we conclude thatthe restoration imageof the Figure 1-f) is the resulting one.
Acknowledgements This research was supported by the Swedish Research Council, the SwedishFoundation for Strategic Research (SSF) in Gothenburg Mathematical Modelling Centre (GMMC)and by the Swedish Institute, Visby Program. The first authoracknowledges also the Russian Foun-dation For Basic Research, the grant RFFI 11-01-00040.
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A posteriori error estimates for Fredholm integral equations of the first kind 17
a) 7938 elements b) 11270 elements c) 15916 elements
d) 24262 elements e) 40358 elements f) 72292 elements
g) 7938 elements h) 11270 elements i) 15916 elements
j) 24262 elements k) 40358 elements l) 72292 elements
Fig. 1 Reconstructed images from the experimental backscattering data obtained by the microto-mograph [14,17]. On a) we present the real measured signal onthe part of the planar microschemeobtained by microtomograph on the depth 0.9Āµm. On b)-f) we show results of the image restora-tion presented on a) on different adaptively refined meshes using the algorithm of section 8. Recon-struction results together with correspondingly adaptively refined meshes are presented on g)-l).
18 N.Koshevā and L. Beilinaāā
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A posteriori error estimates for Fredholm integral equations of the first kind 19
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