Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation
We give a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problem for the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundary surfaces (curves in the case of two-dimensional domains). This Cauch...
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| author | Chapko, R. Johansson, B. T. Чапко, Р. Йоханссон, Б. Т. |
| author_facet | Chapko, R. Johansson, B. T. Чапко, Р. Йоханссон, Б. Т. |
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| datestamp_date | 2019-12-05T09:32:42Z |
| description | We give a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problem
for the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundary
surfaces (curves in the case of two-dimensional domains). This Cauchy problem is reduced to finding the values of a
harmonic function and its normal derivative on one of the two closed parts of the boundary according to the information
about these quantities on the other boundary surface. This is an ill-posed problem in which the presence of noise in the
input data may completely destroy the procedure of finding the approximate solution. We describe and present a results
for a procedure of regularization aimed at the stable determination of the required quantities based on the representation
of the solution to the Cauchy problem in the form a single-layer potential. For the given data, this representation yields
a system of boundary integral equations for two unknown densities. We establish the existence and uniqueness of these
densities and propose a method for the numerical discretization in two- and three-dimensional domains. We also consider
the cases of simply connected domains of solution and unbounded domains. Numerical examples are presented both for
two- and three-dimensional domains. These numerical results demonstrate that the proposed method gives good accuracy
with relatively small amount of computations. |
| first_indexed | 2026-03-24T02:15:57Z |
| format | Article |
| fulltext |
UDC 519.6
R. Chapko (Ivan Franko Nat. Univ. Lviv, Ukraine),
B. T. Johansson (Aston Univ., UK)
BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION
OF THE CAUCHY PROBLEM FOR THE LAPLACE EQUATION
МЕТОД ГРАНИЧНИХ IНТЕГРАЛIВ ДЛЯ ЧИСЕЛЬНОГО РОЗВ’ЯЗУВАННЯ
ЗАДАЧI КОШI ДЛЯ РIВНЯННЯ ЛАПЛАСА
We give a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problem
for the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundary
surfaces (curves in the case of two-dimensional domains). This Cauchy problem is reduced to finding the values of a
harmonic function and its normal derivative on one of the two closed parts of the boundary according to the information
about these quantities on the other boundary surface. This is an ill-posed problem in which the presence of noise in the
input data may completely destroy the procedure of finding the approximate solution. We describe and present a results
for a procedure of regularization aimed at the stable determination of the required quantities based on the representation
of the solution to the Cauchy problem in the form a single-layer potential. For the given data, this representation yields
a system of boundary integral equations for two unknown densities. We establish the existence and uniqueness of these
densities and propose a method for the numerical discretization in two- and three-dimensional domains. We also consider
the cases of simply connected domains of solution and unbounded domains. Numerical examples are presented both for
two- and three-dimensional domains. These numerical results demonstrate that the proposed method gives good accuracy
with relatively small amount of computations.
Наведено огляд прямого методу граничних iнтегральних рiвнянь для чисельного розв’язування задачi Кошi для
рiвняння Лапласа у двозв’язних областях; область розв’язування розмiщена мiж двома замкненими граничними
поверхнями (кривими у випадку двовимiрних областей). Ця задача Кошi полягає у знаходженнi значень гармо-
нiчної функцiї та її нормальної похiдної на однiй iз двох замкнених границь за iнформацiєю про цi величини на
iншiй граничнiй поверхнi. Це є некоректна задача, в якiй шум у вхiдних даних може призвести до непридатного
обчисленого наближеного розв’язку. Ми описуємо i наводимо огляд регуляризуючого методу для стiйкого визна-
чення шуканих величин, ґрунтуючись на поданнi розв’язку задачi Кошi у формi потенцiалу простого шару. Таке
подання приводить до системи граничних iнтегральних рiвнянь вiдносно двох невiдомих густин. Встановлено iсну-
вання i єдинiсть густин та запропоновано спосiб чисельної дискретизацiї у дво- та тривимiрних областях. Також
дискутується випадок однозв’язних областей та випадок необмежених областей. Наведено чисельнi приклади для
дво- та тривимiрних областей, яки засвiдчують, що запропонований пiдхiд дає хорошу точнiсть при економних
обчислювальних затратах.
1. Introduction. Let D2 \subset \mathrm{I}\mathrm{R}d, d = 2, 3, be a bounded domain with boundary surface \Gamma 2. This
surface is assumed to be simple (no self-intersections) closed (the surface has itself no boundary and
is connected) and sufficiently smooth. In the case when d = 2, we have a boundary curve with the
similar properties assumed; we shall not explicitly state each time the word “surface” appear that we
also consider planar domains with boundary curves but ask the reader to keep in mind that the present
work also covers the planar case when surfaces are replaced by curves.
Let then \Gamma 1 be a simple closed (smooth) surface lying wholly within D2 with the interior of \Gamma 1
being denoted D1. The solution domain D is the region between the two surfaces \Gamma 1 and \Gamma 2, thus
D = D2 \setminus \=D1, see further Fig. 1 for examples of the configuration.
Let u \in C2(D) \cap C1( \=D) be a harmonic function, that is a solution to the Laplace equation
\Delta u = 0 in D (1.1)
and suppose additionally that u satisfies the following boundary conditions on the outer surface \Gamma 2 :
c\bigcirc R. CHAPKO, B. T. JOHANSSON, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12 1665
1666 R. CHAPKO, B. T. JOHANSSON
u = f on \Gamma 2 and
\partial u
\partial \nu
= g on \Gamma 2. (1.2)
The linear inverse problem we study is: Given the function values and normal derivative on \Gamma 2, find a
harmonic function u in the domain D, matching this data. In particular, reconstruct the corresponding
data u and
\partial u
\partial \nu
on the interior boundary \Gamma 1. Here, \nu is the outward unit normal varying along the
boundary surfaces.
..
ν
.
Γ2
.
Γ1
.
D
..
.
D
.
Γ2
.
Γ1
a b
Fig. 1. Example of a two-dimensional (a) and three-dimensional (b) solution domain D with the boundary part \Gamma 1
contained within the outer boundary \Gamma 2 .
This type of problem is known as a Cauchy problem, and the given data on one boundary part
is termed as Cauchy data. The problem is known to have a unique solution (a consequence of the
Holmgren theorem), however, the continuous dependence on the data cannot be guaranteed making
it fall into the category of ill-posed problems.
The Cauchy problem has a long history going back to Hadamard [17], and serves as a typical
example of an ill-posed problem. The Cauchy problem has several important applications, for exam-
ple, in cardiology, corrosion detection, electrostatics, geophysics, leak identification, nondestructive
testing and plasma physics. Some works, where references to applications and methods for Cauchy
problems can be found, are [7, 10, 19, 20]. We shall not go further into details or references on
history or properties of Cauchy problems, but only state that we assume that data are compatible such
that there exists a solution.
Two possible strategies for the numerical calculation of the solution to a Cauchy problem are the
following. One can recast the problem as an equation for the missing boundary data, and for the
obtained equation Tikhonov regularization is employed for the stable determination of the solution.
The second strategy is to form a sequence of well-posed problems for the same equation and to
prove that this sequence converges to the sought solution of the ill-posed problem. In either of the
strategies, in terms of numerical calculations, approximations of harmonic functions, and in particular
their boundary values, are needed. We shall outline a method belonging to the first category, a
reference to a method in the second category is [23].
The solution domain of interest can be of different form to the one introduced above, in particular
in some applications it can be unbounded or simply connected. The authors have been involved
in developing methods based on boundary integral equations for Cauchy problems that are flexible
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1667
in the sense that they can be adjusted to various solution domains, and numerically efficient since
only boundary data are needed. In the present work, we shall give an overview of one such method
and results obtained. This method corresponds to the first strategy mentioned above. The results
presented are collected from work done by the authors in [1, 2, 4, 9 – 11].
The method is based on boundary integral techniques involving parameterisations of the boundary
surfaces. The solution to (1.1), (1.2) in the region D is represented as a sum of two single-layer
potentials, one for each boundary surface, with unknown surface densities. Matching the given
Cauchy data, a system of boundary integral equations is derived from which the densities over the
two boundary surfaces can be obtained. We note that single-layer approaches have been used before
for ill-posed problems, for example in inverse acoustic scattering, but seems somewhat overlooked
for the Cauchy problem (for some properties of single-layer potentials and history, see [12]).
We point out that rather than using techniques based on parameterisations of the boundary surfaces,
one can use the boundary element method (BEM) since only boundary data is needed in the Cauchy
problem. However, in the BEM, the boundary surfaces are discretised into simpler ones, such as
planes or quadratics, and this is a nontrivial task in itself for surfaces. If these boundary surfaces are
instead known via given parameterisations, then it becomes advantageous not to use the BEM but
instead make use of the parameterisations and to incorporate further transformations that can render
faster and more accurate numerical results.
We mention that methods for Cauchy problems based on the BEM have been developed (mainly
for bounded planar regions), see for example, [26, 27] (the authors of these works have plenty more
results on techniques based on the BEM for ill-posed problems). Recently, meshless techniques have
been advocated, see the survey [21]. Methods based on finite differences and finite elements have also
been developed, see, for example, [3, 13]. Thus, most of the standard numerical methods for partial
differential equations can be applied for solving the Cauchy problem but they tend to be cumbersome
to adjust to, for example, unbounded domains or are not that efficient for three-dimensional regions.
For the outline of this work, in Section 2, we present the general approach for the Cauchy
problem (1.1), (1.2) of representing the solution as a single-layer potential. The system obtained
by matching the representation against the given data is derived together with properties in terms
of uniqueness of a solution. In Section 3, we show how to discretise the obtained system for two-
dimensional solution domains. In Section 4, we give the corresponding details for the discretisation in
three-dimensional domains based on Weinert’s method [29]. A note is included at the end of Section
4, discussing how to adjust the approach for various other types of domains such as unbounded as
well as simply connected ones; references are given where further details can be found. In the final
section, Section 5, we give some numerical results for two- and three-dimensional solution domains.
2. A direct integral equation approach with Tikhonov regularization for the Cauchy prob-
lem (1.1), (1.2). The solution to the Cauchy problem (1.1), (1.2) is sought in the form of a sum of
single-layer potentials over the two boundary surfaces,
u(x) =
\int
\Gamma 1
\phi 1(y)\Phi (x, y) ds(y) +
\int
\Gamma 2
\phi 2(y)\Phi (x, y) ds(y), x \in D, (2.1)
where \phi 1 \in C(\Gamma 1) and \phi 2 \in C(\Gamma 2) are unknown densities (we enforce to have continuous densities
for simplicity in terms of interpreting the boundary integrals), and
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1668 R. CHAPKO, B. T. JOHANSSON
\Phi (x, y) =
\left\{
1
2\pi
\mathrm{l}\mathrm{n}
1
| x - y| , d = 2,
1
4\pi
1
| x - y| , d = 3,
(2.2)
is the fundamental solution of the Laplace equation in \mathrm{I}\mathrm{R}d.
Using the classical jump properties of the single-layer potential and its normal derivative, the
representation (2.1) satisfies (1.1), (1.2) provided that the two densities form a solution pair of the
following system:\int
\Gamma 1
\phi 1(y)\Phi (x, y) ds(y) +
\int
\Gamma 2
\phi 2(y)\Phi (x, y) ds(y) = f(x), x \in \Gamma 2,
1
2
\phi 2(x) +
\int
\Gamma 1
\phi 1(y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y) +
\int
\Gamma 2
\phi 2(y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y) = g(x), x \in \Gamma 2.
(2.3)
For the moment, we take for granted that it exists a unique pair of densities \phi 1 and \phi 2 to this
equation. Using these densities, the sought Cauchy data on the interior boundary \Gamma 1 can be found
from
u(x) =
\int
\Gamma 1
\phi 1(y)\Phi (x, y) ds(y) +
\int
\Gamma 2
\phi 2(y)\Phi (x, y) ds(y), x \in \Gamma 1,
\partial u
\partial \nu
(x) = - 1
2
\phi 1(x) +
\int
\Gamma 1
\phi 1(y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y) +
\int
\Gamma 2
\phi 2(y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y), x \in \Gamma 1.
(2.4)
Before continuing, we briefly mention that there exists other ways of representing the solution
to the Cauchy problem, which also will render a system of boundary integral equations to solve
for a pair of densities. For example, one can employ Green’s representation formula for harmonic
functions
u(x) =
\int
\Gamma 1
\biggl(
\psi 2(y)
\partial \Phi (x, y)
\partial \nu (y)
- \psi 1(y)\Phi (x, y)
\biggr)
ds(y) +Q(x), x \in D,
with \psi 1(x) =
\partial u
\partial \nu
(x) and \psi 2(x) = u(x) for x \in \Gamma 1, and where
Q(x) =
\int
\Gamma 2
\biggl(
g(y)\Phi (x, y) - f(y)
\partial \Phi (x, y)
\partial \nu (y)
\biggr)
ds(y).
Using jump properties of single- and double-layer potentials, we obtain the system
- 1
2
\psi 2(x) -
\int
\Gamma 1
\psi 1(y)\Phi (x, y) ds(y) +
\int
\Gamma 1
\psi 2(y)
\partial \Phi (x, y)
\partial \nu (y)
ds(y) = - Q(x), x \in \Gamma 1,
- 1
2
\psi 1(x) -
\int
\Gamma 1
\psi 1(y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y) +
\partial
\partial \nu (x)
\int
\Gamma 1
\psi 2(y)
\partial \Phi (x, y)
\partial \nu (y)
ds(y) = - \partial Q
\partial \nu
(x), x \in \Gamma 1.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1669
Clearly, the sought after Cauchy data on \Gamma 1 is the solution pair (\psi 1, \psi 2). This approach however
suffers from the drawback of generating a system with a hypersingular kernel and with a complicated
right-hand side.
In the present work, we therefore only concentrate on the potential approach (2.1). We shall make
use of the following boundary integral operators:
(Sij\mu ) (x) =
\int
\Gamma j
\mu (y)\Phi (x, y) ds(y), x \in \Gamma i, (2.5)
and
(Dij\mu ) (x) =
\int
\Gamma j
\mu (y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y), x \in \Gamma j , (2.6)
with i, j = 1, 2.
We can rewrite the system (2.3) in the operator form
S21\phi 1 + S22\phi 2 = f on \Gamma 2,\biggl(
1
2
I +D22
\biggr)
\phi 2 +D21\phi 1 = g on \Gamma 2.
(2.7)
To investigate solutions to (2.7), we use A : L2(\Gamma 1)\times L2(\Gamma 2) \rightarrow L2(\Gamma 2)\times L2(\Gamma 2), where
A =
\left( S21 S22
D21
1
2
I +D22
\right) . (2.8)
The system (2.7) corresponds to the ill-posed Cauchy problem (1.1), (1.2), and therefore it will
inherit the ill-posedness. Thus, rather than showing well-posedness it is important that the operator
A is such that Tikhonov regularization can be applied for the stable solution. Recalling the steps
in [6] (Theorem 4.1) with a straightforward extension to the three-dimensional setting, the following
result can be established:
Theorem 1. The operator A defined in (2.8) is injective and has a dense range.
We can then write our inverse problem as an operator equation
A\phi = F (2.9)
to be solved for \phi = (\phi 1, \phi 2) given the data F = (f, g). To restore stability, Tikhonov regularization
shall be employed, that is we solve the regularized system
(A\ast A+ \alpha I)\phi \alpha = A\ast F,
where A\ast is the adjoint operator to A, and \alpha > 0 is a regularization parameter to be chosen
appropriately.
We note that other spaces can be considered for the operator A in (2.8). The L2-setting is rather
natural from a practical point of view, since data is typically contaminated with noise destroying any
smoothness assumption on the data. Moreover, the element (2.1) with square integrable densities has
traces in H1(\Gamma ) and L2(\Gamma ) for the function values and normal derivative, respectively. It is typically
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1670 R. CHAPKO, B. T. JOHANSSON
with such data that theoretical properties of the Cauchy problem has been derived; for example
interior regularity and local estimates, see [18] (Theorem 3.3.1). It is possible though to instead
consider properties of the operator A having in mind the natural Sobolev trace spaces H1/2(\Gamma 2) and
H - 1/2(\Gamma 2) for the Cauchy data. An analysis in this direction for the Helmholtz equation is given
in [5] for densities in H - 1/2(\Gamma j), j = 1, 2. In this case, for noisy data, smoothing is in general
required to have the given data belong to the required spaces.
In the case of noisy data, we solve
(A\ast A+ \alpha I)\phi \delta \alpha = A\ast F \delta .
Using the properties of the operator A given in Theorem 1, it is known, see Theorem 16.13 [25],
that one can devise a rule for choosing the regularizing parameter \alpha such that \phi \delta \alpha tends to the
solution of (2.9), when the noise level \delta tends to zero. Employing the densities constituting \phi \delta \alpha in
the representation (2.1), we obtain an element u\delta \alpha in H1(D). Applying estimates for the single-layer
operator in terms of the densities, see [28] (Theorem 7.1), we conclude that u\delta \alpha tends to u, with u
obtained from the densities in (2.9) and the representation (2.1). This in turn via the trace theorem
implies that we also obtain a sequence on \Gamma 1 converging with decreasing noise level to the sought
after Cauchy data. In fact, since the difference u - u\delta \alpha is a harmonic function, local estimates can
be applied to conclude that u - u\delta \alpha converges in H\ell +1(D\prime ), for \ell = 1, 2, . . . , and D\prime a sufficiently
smooth domain with D\prime \subset D.
3. Full discretisation of (2.7) for two-dimensional domains. In the case of a planar solution
domain the two boundary parts \Gamma 1 and \Gamma 2 are simple smooth closed curves, which are assumed given
by the parametric representation
\Gamma i :=
\bigl\{
pi(t) = (xi1(t), xi2(t)), t \in [0, 2\pi ]
\bigr\}
,
where pi : \mathrm{I}\mathrm{R} \rightarrow \mathrm{I}\mathrm{R}2 is 2\pi -periodic with | p\prime i(t)| > 0 for all t \in [0, 2\pi ], pi \in C2([0, 2\pi ] \times [0, 2\pi ]),
i = 1, 2.
Using these parametric representations in (2.5) and (2.6), we have the parameterised integral
operators
(\widetilde Sij\psi )(t) = 1
2\pi
2\pi \int
0
\psi (\tau )Hij(t, \tau ) d\tau
and
( \widetilde Dij\psi )(t) =
1
2\pi
2\pi \int
0
\psi (\tau )Kij(t, \tau ) d\tau ,
where \psi (t) := \mu (pi(t)) | p\prime i(t)| , for t \in [0, 2\pi ] and i, j = 1, 2. Recalling the fundamental solution of
the Laplace equation for planar domains, see (2.2), the kernels can be written
Hij(t, \tau ) = \mathrm{l}\mathrm{n}
1
| pi(t) - pj(\tau )|
, t \not = \tau for i = j
and
Kij(t, \tau ) =
(pj(\tau ) - pi(t))\nu (pi(t))
| pi(t) - pj(\tau )| 2
, t \not = \tau for i = j.
The diagonal values of the functions Kij when i = j are
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1671
Kii(t, t) =
p\prime \prime i (t)\nu (pi(t))
2| p\prime i(t)| 2
.
Elementary calculations reveal that the function Hii can be decomposed as
Hii(t, \tau ) = - 1
2
\mathrm{l}\mathrm{n}
\biggl(
4
e
\mathrm{s}\mathrm{i}\mathrm{n}2
t - \tau
2
\biggr)
+ \widetilde Hii(t, \tau ), t \not = \tau ,
with \widetilde Hii(t, \tau ) =
1
2
\mathrm{l}\mathrm{n}
\biggl(
(4/e) \mathrm{s}\mathrm{i}\mathrm{n}2((t - \tau )/2)
| pi(t) - pi(\tau )| 2
\biggr)
, t \not = \tau ,
having the diagonal term \widetilde Hii(t, t) =
1
2
\mathrm{l}\mathrm{n}
\biggl(
1
e| p\prime i(t)| 2
\biggr)
.
Using the above parameterisations of the boundary curves together with the derived expressions
for the kernels in (2.3), we have a parameterised system of integral equations
1
2\pi
2\pi \int
0
\biggl\{
H12(t, \tau )\psi 1(\tau ) +
\biggl[
- 1
2
\mathrm{l}\mathrm{n}
\biggl(
4
e
\mathrm{s}\mathrm{i}\mathrm{n}2
t - \tau
2
\biggr)
+ \~H22(t, \tau )
\biggr]
\psi 2(\tau )
\biggr\}
d\tau = f(p2(t)),
1
2\pi
2\pi \int
0
\Bigl\{
K12(t, \tau )\psi 1(\tau ) +K22(t, \tau )\psi 2(\tau )
\Bigr\}
d\tau +
\psi 2(t)
2| p\prime 2(t)|
= g(p2(t))
(3.1)
to be solved for \psi i(t) = \phi i(pi(t))| p\prime i(t)| , i = 1, 2, with t \in [0, 2\pi ]. Finding these densities, we can
use (2.4) to find the requested Cauchy data on \Gamma 1.
As explained at the end of the previous section, Tikhonov regularization is applied when solv-
ing (3.1) and this means that that the analogous transformations for the corresponding adjoint operators
to (2.5) and (2.6) are needed. Going down that route will render a method requiring additional com-
putational cost. This can be avoided by the simplistic but common approach of first discretising
the parameterised integral equations (3.1) and then apply regularization. Numerically, the obtained
results with this latter approach tend to match the more computationally demanding strategy. It can
be made rigorous by showing an error estimate between the operator A and the discretised one, and
this can be done following ideas for a similar error estimate for the Symm’s integral equation, see
[22] (Section 3.4.2). One can then devise a parameter choice rule for Tikhonov regularization of the
discretised operator based on the fineness of the discretisation and the error level.
For the discretisation of the involved integrals, we consider two quadrature rules both constructed
via trigonometric interpolation with 2n equidistant nodal points
tj :=
j\pi
n
, j = 0, . . . , 2n - 1. (3.2)
The two quadrature rules are
1
2\pi
2\pi \int
0
f(\tau ) \mathrm{l}\mathrm{n}
\biggl(
4
e
\mathrm{s}\mathrm{i}\mathrm{n}2
t - \tau
2
\biggr)
d\tau \approx
2n - 1\sum
k=0
Rk(t) f(tk) (3.3)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1672 R. CHAPKO, B. T. JOHANSSON
and
1
2\pi
2\pi \int
0
f(\tau ) d\tau \approx 1
2n
2n - 1\sum
k=0
f(tk) (3.4)
with explicit expressions for the weight functions given in [24].
Using the Nyström method with quadratures (3.3) and (3.4) in the integral equations (3.1), we
obtain the following system of linear equations:
1
2n
2n - 1\sum
j=0
\psi 1,jH12(ti, tj) +
2n - 1\sum
j=0
\psi 2,j
\biggl[
1
2n
\~H22(ti, tj) -
1
2
Rj(ti)
\biggr]
= f i,
1
2n
2n - 1\sum
j=0
\psi 1,jK12(ti, tj) +
1
2n
2n - 1\sum
j=0
\psi 2,jK22(ti, tj) +
\psi 2,i
2| p\prime 2(ti)|
= gi
(3.5)
to be solved for \psi 1,j \approx \psi 1(tj) and \psi 2,j \approx \psi 2(tj) with the right-hand side f i = f(p2(ti)) and
gi = g(p2(ti)), for i = 0, . . . , 2n - 1. Rearranging (3.5), we arrive at the following system of linear
algebraic equations:
\bfA \bfx = \bfb , (3.6)
where the matrix \bfA \in \mathrm{I}\mathrm{R}4n\times 4n, and \bfx = [\psi 1, \psi 2]
\top and \bfb = [f, g]\top . The matrix \bfA will have a
large condition number due to the ill-posedness of the Cauchy problem, and to obtain a stable smooth
solution regularization of this system is necessary.
As explained at the end of the previous section, to solve (3.6) in a stable way, we employ Tikhonov
regularization; the standard version of Tikhonov regularization amounts to solve the minimization
problem
\mathrm{m}\mathrm{i}\mathrm{n}
x
\bigl\{
\| \bfA \bfx - \bfb \| 22 + \lambda \| \bfx \| 22
\bigr\}
, (3.7)
where \lambda \in \mathrm{I}\mathrm{R} is a regularization parameter that has to be appropriately chosen. The Tikhonov
regularized solution x\lambda in (3.7) is equivalently given as the solution to the regularized normal
equations
(\bfA \ast \bfA + \lambda I)\bfx \lambda = \bfA \ast \bfb ,
where \bfA \ast is the transpose of the matrix \bfA . Although there are optimal choices for the regularization
parameter (the discrepancy principle), it is often simpler and faster to use a heuristic choice such as
the L-curve rule [14, 15].
Once the discrete (and regularized) densities \psi 1 and \psi 2 have been constructed, the corresponding
discrete approximations for the Cauchy data on the interior boundary curve \Gamma 1 are obtained from
(2.4) using the quadratures
u(p1(ti)) \approx
2n - 1\sum
j=0
\biggl\{ \biggl[
1
2n
H11(ti, tj) -
1
2
Rj(ti)
\biggr]
\psi 1,j +
1
2n
H21(ti, tj)\psi 2,j
\biggr\}
and
\partial u
\partial \nu
(p1(ti)) \approx - 1
2| p\prime 1(ti)|
\psi 1,i +
1
2n
2n - 1\sum
j=0
\Bigl\{
K11(ti, tj)\psi 1,j +K21(ti, tj)\psi 2,j
\Bigr\}
.
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BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1673
4. Full discretisation of (2.7) for three-dimensional domains. For planar domains certain
parameterisations are assumed for the boundary parts. The analogous basic assumption for three-
dimensional domains is that the closed boundary surfaces \Gamma 1 and \Gamma 2 can each be smoothly and
bijectively mapped onto the unit sphere \BbbS 2. This means that there shall exist one-to-one mappings
q1 : \BbbS 2 \rightarrow \Gamma 1 and q2 : \BbbS 2 \rightarrow \Gamma 2
with smoothly varying Jacobians Jq1 and Jq2 , respectively.
We can then rewrite the integral equations (2.7) over the unit sphere and obtain
\widetilde S21\psi 1 + \widetilde S22\psi 2 = \widetilde f on \BbbS 2,\biggl(
1
2
I + \widetilde D22
\biggr)
\psi 2 + \widetilde D21\psi 1 = \widetilde g on \BbbS 2
(4.1)
with the densities \psi \ell (\widehat x) = \phi \ell (q\ell (\widehat x)), \ell = 1, 2, to be determined from the data \widetilde f(\widehat x) = f(q2(\widehat x)),
and \widetilde g(\widehat x) = g(q2(\widehat x)) for \widehat x \in \BbbS 2. The integral operators involved are parameterisations of (2.5), (2.6)
over the unit sphere and given by\Bigl( \widetilde S\ell j\phi \Bigr) (\widehat x) = \int
\BbbS 2
\phi (\widehat y)L\ell j(\widehat x, \widehat y) ds(\^y), \widehat x \in \BbbS 2, (4.2)
and \Bigl( \widetilde D\ell j\phi
\Bigr)
(\widehat x) = \int
\BbbS 2
\phi (\widehat y)M\ell j(\widehat x, \widehat y) ds(\^y), \widehat x \in \BbbS 2. (4.3)
Recalling the fundamental solution to the Laplace equation in \mathrm{I}\mathrm{R}3, see (2.2), the kernels are found
to be
L\ell j(\widehat x, \widehat y) =
\left\{
\Phi (q\ell (\widehat x), qj(\widehat y))Jqj (\widehat y), \ell \not = j,
R\ell (\widehat x, \widehat y)
| \widehat x - \widehat y| , \ell = j,
(4.4)
and
M\ell j(\widehat x, \widehat y) =
\left\{
- \langle q\ell (\widehat x) - qj(\widehat y), \nu (q\ell (\widehat x))\rangle
4\pi | q\ell (\widehat x) - qj(\widehat y)| 3 Jqj (\widehat y), \ell \not = j,\widetilde R\ell (\widehat x, \widehat y)
| \widehat x - \widehat y| , \ell = j,
(4.5)
where
R\ell (\widehat x, \widehat y) = Jq\ell (\widehat y)
\left\{
1
4\pi
| \widehat x - \widehat y|
| q\ell (\widehat x) - q\ell (\widehat y)| , \widehat x \not = \widehat y,
1
4\pi
1
Jq\ell (\widehat x) , \widehat x = \widehat y, (4.6)
and
\widetilde R\ell (\widehat x, \widehat y) = - R\ell (\widehat x, \widehat y)
\left\{
(q\ell (\widehat x) - q\ell (\widehat y), \nu (q\ell (\widehat x)))
| q\ell (\widehat x) - q\ell (\widehat y)| 2 , \widehat x \not = \widehat y,
-
2
\sum 3
j=1
q\prime \ell j(\widehat x)\nu j(\widehat x) - \sum 3
j=1
q\prime \prime \ell j(\widehat x)\nu j(\widehat x)
2J2
q\ell
(\widehat x) , \widehat x = \widehat y. (4.7)
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1674 R. CHAPKO, B. T. JOHANSSON
We used that
\mathrm{l}\mathrm{i}\mathrm{m}\widehat y\rightarrow \widehat x
| \widehat x - \widehat y|
| q\ell (\widehat x) - q\ell (\widehat y)| = 1
Jq\ell (\widehat x) .
Points on the unit sphere are given using the standard spherical coordinates,
\widehat x = p(\theta , \varphi ) = (\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi , \mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi , \mathrm{c}\mathrm{o}\mathrm{s} \theta ), with \theta \in [0, \pi ], \varphi \in [0, 2\pi ]. (4.8)
The integral operators \widetilde S\ell \ell and \widetilde D\ell \ell , \ell = 1, 2, are both weakly singular. For approximation
and numerical implementation of these, we make the singularities explicit. In fact, we can make a
transformation and move the singularities to appear at the north pole \widehat n = (0, 0, 1) of the unit sphere.
Define the orthogonal transformations for \psi \in \mathrm{I}\mathrm{R} by
DF (\psi ) =
\left(
\mathrm{c}\mathrm{o}\mathrm{s}\psi - \mathrm{s}\mathrm{i}\mathrm{n}\psi 0
\mathrm{s}\mathrm{i}\mathrm{n}\psi \mathrm{c}\mathrm{o}\mathrm{s}\psi 0
0 0 1
\right) and DT (\psi ) =
\left(
\mathrm{c}\mathrm{o}\mathrm{s}\psi 0 - \mathrm{s}\mathrm{i}\mathrm{n}\psi
0 1 0
\mathrm{s}\mathrm{i}\mathrm{n}\psi 0 \mathrm{c}\mathrm{o}\mathrm{s}\psi
\right) .
The linear orthogonal transformation
T\widehat x = DF (\varphi )DT (\theta )DF ( - \varphi ) (4.9)
satisfies that T\widehat x\widehat x = \widehat n for every \widehat x \in \BbbS 2. Moreover, | \widehat x - \widehat y| = | T - 1\widehat x (\widehat n - \widehat \eta )| = | \widehat n - \widehat \eta | , and \widehat \eta = T\widehat x\widehat y.
Using this transformation in the operators \widetilde S\ell \ell and \widetilde D\ell \ell , \ell = 1, 2, defined in (4.2), (4.3), these
operators are transformed into
(\widetilde S\ell \ell \phi )(\widehat x) = \int
\BbbS 2
\phi (T - 1\widehat x \widehat \eta )R\ell (\widehat x, T - 1\widehat x \widehat \eta )
| \widehat n - \widehat \eta | ds(\widehat \eta ), \widehat x \in \BbbS 2,
and
( \widetilde D\ell \ell \phi )(\widehat x) = \int
\BbbS 2
\phi (T - 1\widehat x \widehat \eta ) \widetilde R\ell (\widehat x, T - 1\widehat x \widehat \eta )
| \widehat n - \widehat \eta | ds(\widehat \eta ), \widehat x \in \BbbS 2,
for \ell = 1, 2.
The Cauchy data on the interior surface \Gamma 1 can, using the representation (2.4), be written over
the unit sphere to get
u = \widetilde S11\psi 1 + \widetilde S12\psi 2 on \BbbS 2,
\partial u
\partial \nu
=
\biggl(
- 1
2
I + \widetilde D11
\biggr)
\psi 1 + \widetilde D12\psi 2 on \BbbS 2
(4.10)
with the operators defined above.
To discretise (4.1), the following quadrature is used for integrals over the unit sphere having a
continuous integrand \int
\BbbS 2
f(\widehat y) ds(\widehat y) \approx 2n\prime +1\sum
\rho \prime =0
n\prime +1\sum
s\prime =1
\~\mu \rho \prime \~as\prime f(\widehat ys\prime \rho \prime ). (4.11)
Here
\widehat ys\prime \rho \prime = p(\theta s\prime , \varphi \rho \prime ), with \varphi \rho \prime = \rho \prime \pi /(n\prime + 1) and \theta s\prime = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s} zs\prime (4.12)
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BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1675
and zs\prime being the zeros of the Legendre polynomials Pn\prime +1 and with p as in (4.8) expressing an
element on the unit sphere. For the coefficients,
\~as\prime =
2(1 - z2s\prime )
((n\prime + 1)Pn\prime (zs\prime ))2
and
\~\mu \rho \prime =
\pi
n\prime + 1
.
In the case when the integrand has a weak singularity, we use the quadrature rule\int
\BbbS 2
f(\widehat y)
| \widehat n - \widehat y| ds(\widehat y) \approx
2n\prime +1\sum
\rho \prime =0
n\prime +1\sum
s\prime =1
\~\mu \rho \prime \~bs\prime f(\widehat ys\prime \rho \prime ) (4.13)
with weights
\~bs\prime = \~as\prime
n\prime \sum
i=0
Pi(zs\prime ).
The above quadratures are both obtained from approximating the regular part of the integrand
using spherical harmonics and then performing exact integration. According to [16, 29], the chosen
quadrature rules have superalgebraic convergence order.
The ill-posed system of integral equations (4.1) is then discretised using a projection Galerkin
method. In the previous section, trigonometric polynomials were used. To follow the similar idea,
the analogue is to invoke spherical polynomials. We shall therefore search for the densities in terms
of spherical polynomials of degree n. An orthonormal basis for the (n + 1)2-dimensional space of
such spherical polynomials are given by the spherical harmonics.
Thus, we write the approximation of the densities as
\psi \ell (\widehat x) \approx \~\psi \ell (\^x) =
n\sum
k=0
k\sum
m= - k
\psi \ell
k,mY
R
k,m(\widehat x) for \widehat x \in \BbbS 2, \ell = 1, 2, (4.14)
where \psi \ell
k,m are unknown coefficients. Here, the real-valued spherical harmonics are
Y R
k,m =
\left\{ \mathrm{I}\mathrm{m}Yk,| m| , 0 < m \leq k,
\mathrm{R}\mathrm{e}Yk,| m| , - k \leq m \leq 0,
(4.15)
with Yk,m(\theta , \varphi ) = cmk P
| m|
k (\mathrm{c}\mathrm{o}\mathrm{s} \theta )eim\varphi the classical (complex-valued) spherical harmonic functions,
Pm
k the Legendre functions and
cmk = ( - 1)
| m| - m
2
\sqrt{}
2k + 1
4\pi
(k - | m| )!
(k + | m| )! , m = - k, . . . , k, k = 0, 1, . . . .
Define a discrete inner product on the space of spherical polynomials of degree n by
(v, w) =
2n+1\sum
\rho =0
n+1\sum
s=1
\mu \rho asv(\widehat ys\rho )w(\widehat ys\rho ). (4.16)
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1676 R. CHAPKO, B. T. JOHANSSON
The coefficients as and \mu \rho are generated as in (4.11) but with the integer n\prime replaced by a possibly
different integer n. The expression (4.16) is indeed an inner product on the space of spherical
polynomials of degree n and this is due to the fact that (4.11) is exact for spherical polynomials of
degree 2n.
We then employ the inner product (4.16) to the system of integral equations (4.1). This means
first discretising (4.1) by replacing the densities by (4.14) and approximating the integrals via (4.11)
and (4.13), and then employing the discrete inner product to identify the coefficients needed in (4.14)
(multiplying with the basis elements Y R
k,m). This strategy leads to the linear system
n\sum
k=0
k\sum
m= - k
\bigl(
\psi 1
k,mA
21
kk\prime mm\prime + \psi 2
k,mA
22
kk\prime mm\prime
\bigr)
=
2n+1\sum
\rho =0
n+1\sum
s=1
\mu \rho as \widetilde f(\widehat xs\rho )Y R
k,m(\widehat xs\rho ),
n\sum
k=0
k\sum
m= - k
\Bigl(
\psi 1
k,m
\widetilde A21
kk\prime mm\prime + \psi 2
k,m
\widetilde A22
kk\prime mm\prime
\Bigr)
=
2n+1\sum
\rho =0
n+1\sum
s=1
\mu \rho as\widetilde g(\widehat xs\rho )Y R
k,m(\widehat xs\rho ),
(4.17)
where k\prime = 0, . . . , n, m = - k\prime , . . . , k\prime .
To give expressions for the coefficients in this linear system, let
V j
s,k,m(\widehat xs\rho , \widehat ys\prime \rho \prime , \widehat ys\prime \rho \prime s\rho ) =
\left\{ \widetilde a
\prime
sL2j(\widehat xs\rho , \widehat ys\prime \rho \prime )Y R
k,m(\widehat ys\prime \rho \prime ), j = 1,
\widetilde b\prime sR2(\widehat xs\rho , \widehat ys\prime \rho \prime s\rho )Y R
k,m(\widehat ys\prime \rho \prime s\rho ), j = 2,
and
W j
s,k,m(\widehat xs\rho , \widehat ys\prime \rho \prime , \widehat ys\prime \rho \prime s\rho ) =
\left\{
\widetilde a\prime sL2j(\widehat xs\rho , \widehat ys\prime \rho \prime )Y R
k,m(\widehat ys\prime \rho \prime ), j = 1,
\widetilde b\prime sR2(\widehat xs\rho , \widehat ys\prime \rho \prime s\rho )Y R
k,m(\widehat ys\prime \rho \prime s\rho ) +
1
2
Y R
k,m(\widehat xs\rho ), j = 2.
Here the kernels are given by (4.4) – (4.6), \widehat xs\rho and \widehat ys\prime \rho \prime are points on the unit sphere generated as
in (4.12), and \widehat ys\prime \rho \prime sp = T - 1\widehat xs\rho
\widehat ys\prime \rho \prime
with T\widehat xs\rho
given by (4.9).
Then the coefficients in (4.17) can be expressed as
A2j
kk\prime mm\prime =
2n+1\sum
\rho =0
n+1\sum
s=1
2n\prime +1\sum
\rho \prime =0
n\prime +1\sum
s\prime =1
\widetilde \mu \rho \prime \mu \rho asV j
s,k,m(\widehat xs\rho , \widehat ys\prime \rho \prime , \widehat ys\prime \rho \prime s\rho )Y R
k\prime ,m\prime (\widehat xs\rho )
and \widetilde A2j
kk\prime mm\prime =
2n+1\sum
\rho =0
n+1\sum
s=1
2n\prime +1\sum
\rho \prime =0
n\prime +1\sum
s\prime =1
\widetilde \mu \rho \prime \mu \rho asW j
s,k,m(\widehat xs\rho , \widehat ys\prime \rho \prime , \widehat ys\prime \rho \prime s\rho )Y R
k\prime ,m\prime (\widehat xs\rho ),
where j = 1, 2.
Solving the linear system (4.17), using Tikhonov regularization, we obtain an approximation to
the densities in (2.1) via the expression (4.14). Invoking the obtained approximation of the densities
in the expression (4.10) together with quadrature, the sought values on \Gamma 1 are found to be
un(\widehat x) = n\prime +1\sum
s\prime =1
2n\prime +1\sum
\rho \prime =0
\Bigl(
\~bs\prime \~\mu \rho \prime \~\psi 1(T
- 1\widehat x \widehat ys\prime \rho \prime )R1(\widehat x, T - 1\widehat x \widehat ys\prime \rho \prime ) + \~as\prime \~\mu \rho \prime \~\psi 2(\widehat ys\prime \rho \prime )L12(\widehat x, \widehat ys\prime \rho \prime )\Bigr) (4.18)
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BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1677
and
\partial un
\partial \nu
(\widehat x) = n\prime +1\sum
s\prime =1
2n\prime +1\sum
\rho \prime =0
\Bigl(
\~bs\prime \~\mu \rho \prime \~\psi 1(T
- 1\widehat x \widehat ys\prime \rho \prime ) \widetilde R1(\widehat x, T - 1\widehat x \widehat ys\prime \rho \prime ) +
+ \~as\prime \~\mu \rho \prime \~\psi 2(\widehat ys\prime \rho \prime )M12(\widehat x, \widehat ys\prime \rho \prime )\Bigr) - 1
2
\~\psi 1(\widehat x), (4.19)
where \widehat x \in \BbbS 2.
As was mentioned in Section 3 the proposed method is regularizing for planar domains. The same
holds in the three-dimensional case; this follows by noting that also in the three-dimensional setting
there are error estimates between the exact solution to (2.7) and its discretisation, and then using
the similar arguments from Section 3. Rather than writing out and scrutinising the full details for
showing that the method is regularizing, we give numerical evidence when performing the numerical
experiments in Section 5 and present there a basic strategy for choosing the required parameters.
4.1. Remark. The above introduced integral equation method for the elliptic Cauchy prob-
lem (1.1), (1.2) can be applied not only for doubly-connected domains. For example, the suggested
approach was successfully applied in the following cases: a simply connected two-dimensional
domain bounded by a simple closed curve, allowed to be nonsmooth in the sense of having cor-
ners, see [1], to a semiinfinite three-dimensional domain containing a cavity [10], and to a toroidal
domain [2] (for such a domain the boundary surface is not simply connected).
In each of these cases, some adjustment is needed for the numerical implementation: in the case
of a nonsmooth domain with corner points, we need to take into account the possible singularities
that can be present at the corner points; for a semiinfinite domain Green’s functions are incorporated
to obtain integral equations over the cavity (which has a bounded boundary surface); for toroidal
domains several transformations are used to take advantage of the symmetry of such a domain to
obtain integral equations over a planar two-dimensional domain.
The proposed strategy can also be employed for well-posed problems for the Laplace equation
such as mixed ones (which can be viewed as having incomplete Cauchy data). Having an efficient
solver for mixed boundary value problems, it is possible to apply iterative regularizing methods for
the Cauchy problem (1.1), (1.2), which at each iterative step solves such mixed problems. Methods in
this direction, such as [23], which falls under the second category of regularizing methods mentioned
in the introduction, and their numerical implementation for elliptic Cauchy problems in two- and
three-dimensional regions can be found in for example [4, 8].
Based on the above research and results, we can conclude that our proposed approach is
lightweight (in terms of computations) and flexible for elliptic Cauchy problems with boundary
parts consisting of parameterised curves and surfaces isomorphic to the unit circle respectively the
unit sphere.
We point out that the proposed approach can also be used for other Cauchy problems for elliptic
equations or systems occurring in applications such as elasticity, fluid flow and wave propagation, for
example, the Helmholtz equation, the Klein – Gordon equation and stationary Stokes system.
5. Numerical experiments. In this section, we illustrate by numerical examples the robustness
of the proposed integral equation based method for the reconstruction of the harmonic function
satisfying the Cauchy problem (1.1), (1.2), for both exact and noisy data. In the case of noisy data,
random pointwise errors are added to the function values f on the outer boundary with the percentage
given in terms of the L2-norm.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1678 R. CHAPKO, B. T. JOHANSSON
...
.........
..
..
−4
.
−2
.
0
.
2
.
4
.
−2
.
0
.
2
–1
1
0
1
1
0 0
–1 –1
a b
Fig. 2. The solution domain used in Example 1 (a) and Example 2 (b).
Example 1. We use synthetic Cauchy data on the outer boundary part \Gamma 2, constructed as follows:
the Dirichlet boundary-value problem for the Laplace equation with boundary conditions u = f1 on
\Gamma 1 and u = f on \Gamma 2, for given boundary functions f1 and f, is numerically solved by the above
boundary integral equation approach. Then, to generate the required trace of the normal derivative
of the solution on \Gamma 2, the following representation is employed:
\partial u
\partial \nu
(x) =
1
2
\phi 2(x) +
\int
\Gamma 1
\phi 1(y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y) +
\int
\Gamma 2
\phi 2(y)
\partial \Phi (x, y)
\partial \nu (x)
ds(y), x \in \Gamma 2. (5.1)
We consider the case when the outer boundary curve \Gamma 2 is a circle of radius 3 and the interior
boundary \Gamma 1 is given by the parameterisation (see Fig. 2, a)
\Gamma 1 =
\bigl\{
p1(t) = r(t)(\mathrm{c}\mathrm{o}\mathrm{s} t, \mathrm{s}\mathrm{i}\mathrm{n} t), t \in [0, 2\pi ]
\bigr\}
with the radial function
r(t) =
\Biggl( \biggl(
1
2
\mathrm{c}\mathrm{o}\mathrm{s} t
\biggr) 10
+
\biggl(
2
3
\mathrm{s}\mathrm{i}\mathrm{n} t
\biggr) 10\Biggr) - 0.1
.
To generate the required synthetic Cauchy with the above explained strategy, the Dirichlet data
functions are choosen as f1(x1, x2) = x21 on \Gamma 1 and f(x1, x2) = 1 on \Gamma 2. Then (5.1) is used to find
the required normal derivative g.
In the Cauchy problem (1.1), (1.2), it is data on the inner boundary \Gamma 1 that has to be reconstructed
from data on the outer boundary \Gamma 2; the sought function value on \Gamma 1 is thus the above chosen function
f1 restricted to \Gamma 1 and this shall be compared with the one obtained numerically with the proposed
procedure for the Cauchy problem.
The result of the reconstructions of the sought Cauchy data on the interior curve \Gamma 1 are given in
Fig. 3 and Fig. 4 for exact and 3% noisy data, respectively.
The discretization parameter (3.2) controlling the number of mesh points on each boundary curve
was taken as n = 64. The value of the regularization parameter used, \alpha \ast , was chosen by trial and
error; we calculated the numerical solutions for \alpha = 10 - p with p = 1, . . . , 15, and use the value
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BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1679
0 1 2 3 4 5 t
0
1
2
3
4
a
0 1 2 3 4 5 t
0
1
2
3
4
b
Fig. 3. Reconstruction ( ) of the boundary function u(p1(t)) ( ) on \Gamma 1 in Example 1 for exact data (a) and 3%
noisy data (b).
0 1 2 3 4 5 t
− 8
− 6
− 4
− 2
0
2
a
0 1 2 3 4 5 t
−8
−6
−4
−2
0
2
b
Fig. 4. Reconstruction ( ) of the normal derivative
\partial u
\partial \nu
(p1(t)) ( ) on \Gamma 1 in Example 1 for exact data (a) and 3%
noisy data (b).
giving the most accurate result. Note here that we have compared \alpha \ast with the corresponding value
for the regularization parameter obtained with the L-curve rule [15] and this value is near to \alpha \ast .
It was mentioned at the end of Section 4 that the proposed method is regularizing. To exemplify
this, in Table 1 are the discrete L2-errors for the reconstructions of the solution and its normal
derivative on the boundary \Gamma 1 as a function of the parameters \alpha \ast , \delta and n.
From Table 1 it can be seen that for a fixed error level the mesh size and regularizing parameter
can be selected to give an approximation to the solution and its normal derivative, respectively,
that decreases with the mesh size. However, for a fixed error a too fine mesh size will cause the
error to start to increase since the linear system solved will be too ill-conditioned. The error in the
reconstructions decrease as the error level decreases. One could elaborate further on this but we only
note here that the basic strategy of, for a fixed error level, choosing a mesh size that does not render
a too large condition number of the matrix corresponding to the linear system (4.17) and then choose
\alpha \ast as above, will render approximations on \Gamma 1 having errors that decrease with \alpha and which render
pleasing numerical results.
As has been reported in the references mentioned in Section 4.1, one can change the solution
domain and data, and as long as the distance between the boundary curves and the growth of the data
are of the type as in the presented example, results of the similar kind are obtained. It is important to
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1680 R. CHAPKO, B. T. JOHANSSON
Table 1. Errors for Example 1
\delta n \alpha \ast e2 q2
0\% 16 10 - 4 0.07032 0.39314
32 10 - 5 0.00281 0.02223
64 10 - 8 0.00003 0.00029
128 10 - 11 6E - 7 0.00002
3\% 16 10 - 3 0.15164 0.55168
32 10 - 3 0.10286 0.33754
64 10 - 4 0.07055 0.30161
128 10 - 4 0.04409 0.20125
5\% 16 10 - 3 0.23139 0.75139
32 10 - 3 0.18108 0.44474
64 10 - 4 0.13227 0.48922
128 10 - 4 0.07758 0.34147
have Cauchy data on a sufficiently large boundary part, and in general the derivative is reconstructed
with less accuracy compared with the function values, as expected. Moreover, choosing a too fine
mesh (a large number n) the numerical results will deteriorate since the condition number of the
involved matrix of the linear system solved will have a too large condition number then reflecting
the ill-posedness of the Cauchy problem. Thus, if the reader implements the procedure for a similar
example, no surprises is to be expected but results of the same accuracy shall be obtained.
Example 2. We also include an example in a three-dimensional domain. Let the solution domain
D be the region having the outer boundary surface being the sphere
\Gamma 2 =
\bigl\{
\xi 2(\theta , \varphi ) = 1.5(\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi , \mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi , \mathrm{c}\mathrm{o}\mathrm{s} \theta ), 0 \leq \theta \leq \pi , 0 \leq \varphi \leq 2\pi
\bigr\}
and the interior boundary surface being given by the parameterisation
\Gamma 1 =
\bigl\{
\xi 1(\theta , \varphi ) = r(\theta , \varphi )(\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi , \mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi , \mathrm{c}\mathrm{o}\mathrm{s} \theta ), 0 \leq \theta \leq \pi , 0 \leq \varphi \leq 2\pi
\bigr\}
with the radial function
r(\theta , \varphi ) =
1
2
\sqrt{}
1 +
\surd
2
\sqrt{}
\mathrm{c}\mathrm{o}\mathrm{s} 2\theta +
\sqrt{}
2 - \mathrm{s}\mathrm{i}\mathrm{n}2 2\theta ,
see Fig. 2, b. Both these surfaces satisfy, by construction, the assumption of the existence of a smooth
one-to-one map to the unit sphere needed in the proposed method for the Cauchy problem (1.1), (1.2).
We choose as the exact solution of the Laplace equation the function
uex(x) = 2x23 - 2x22 + 3x1
and this then generates the following Cauchy data:
f(x) = uex(x), x \in \Gamma 2 and g(x) =
\partial uex
\partial \nu
(x), x \in \Gamma 2.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
BOUNDARY-INTEGRAL APPROACH FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM . . . 1681
3
–1
–0.5
ϕ
0
θ
0.5
4
1
22 10 0
a
–1
–0.5
ϕ
0
θ
0.5
4
1
22 10 0
b
Fig. 5. The exact (a) values u(\xi 1(\theta , \varphi )) and numerical approximation (b) on the boundary surface \Gamma 1 with 3% noise for
Example 2.
–4
–2
ϕ
0
2
θ4
4
22 10 0
a
–4
–2
ϕ
0
2
θ4
4
22 10 0
b
Fig. 6. The exact (a) normal derivative
\partial u
\partial \nu
(\xi 1(\theta , \varphi )) and numerical approximation (b) on the boundary surface \Gamma 1 with
3% noise for Example 2.
We recall that the integer n is the degree of the spherical harmonic polynomials approximating
the densities via (4.14), n\prime is the number of points chosen in the quadrature (cubature) rules (4.11)
and (4.13); the numbers n and n\prime enter into the approximation via (4.18) and (4.19). We give results
when n = n\prime . Given an integer n the number of discretisation points on each surface is (n + 1)2.
Further improvements can possibly be made by other choices of n\prime .
The result of the reconstructions of the sought Cauchy data on the interior surface \Gamma 1 are given
in Fig. 5 and Fig. 6 for exact and 3% noisy data, respectively.
The similar conclusions as in the planar case can be drawn, see further the references for higher-
dimensional domains given at the end of Section 4.1.
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Received 29.03.16
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| id | umjimathkievua-article-1952 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:57Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/61/368823d8f2e48c1b5113b23908330861.pdf |
| spelling | umjimathkievua-article-19522019-12-05T09:32:42Z Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation Метод граничних iнтегралiв для чисельного розв’язування задачi Кошi для рiвняння Лапласа Chapko, R. Johansson, B. T. Чапко, Р. Йоханссон, Б. Т. We give a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problem for the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundary surfaces (curves in the case of two-dimensional domains). This Cauchy problem is reduced to finding the values of a harmonic function and its normal derivative on one of the two closed parts of the boundary according to the information about these quantities on the other boundary surface. This is an ill-posed problem in which the presence of noise in the input data may completely destroy the procedure of finding the approximate solution. We describe and present a results for a procedure of regularization aimed at the stable determination of the required quantities based on the representation of the solution to the Cauchy problem in the form a single-layer potential. For the given data, this representation yields a system of boundary integral equations for two unknown densities. We establish the existence and uniqueness of these densities and propose a method for the numerical discretization in two- and three-dimensional domains. We also consider the cases of simply connected domains of solution and unbounded domains. Numerical examples are presented both for two- and three-dimensional domains. These numerical results demonstrate that the proposed method gives good accuracy with relatively small amount of computations. Наведено огляд прямого методу граничних iнтегральних рiвнянь для чисельного розв’язування задачi Кошi для рiвняння Лапласа у двозв’язних областях; область розв’язування розмiщена мiж двома замкненими граничними поверхнями (кривими у випадку двовимiрних областей). Ця задача Кошi полягає у знаходженнi значень гармонiчної функцiї та її нормальної похiдної на однiй iз двох замкнених границь за iнформацiєю про цi величини на iншiй граничнiй поверхнi. Це є некоректна задача, в якiй шум у вхiдних даних може призвести до непридатного обчисленого наближеного розв’язку. Ми описуємо i наводимо огляд регуляризуючого методу для стiйкого визначення шуканих величин, ґрунтуючись на поданнi розв’язку задачi Кошi у формi потенцiалу простого шару. Таке подання приводить до системи граничних iнтегральних рiвнянь вiдносно двох невiдомих густин. Встановлено iснування i єдинiсть густин та запропоновано спосiб чисельної дискретизацiї у дво- та тривимiрних областях. Також дискутується випадок однозв’язних областей та випадок необмежених областей. Наведено чисельнi приклади для дво- та тривимiрних областей, яки засвiдчують, що запропонований пiдхiд дає хорошу точнiсть при економних обчислювальних затратах. Institute of Mathematics, NAS of Ukraine 2016-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1952 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 12 (2016); 1665-1682 Український математичний журнал; Том 68 № 12 (2016); 1665-1682 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1952/934 Copyright (c) 2016 Chapko R.; Johansson B. T. |
| spellingShingle | Chapko, R. Johansson, B. T. Чапко, Р. Йоханссон, Б. Т. Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation |
| title | Boundary-integral approach for the numerical solution of the Cauchy
problem for the Laplace equation |
| title_alt | Метод граничних iнтегралiв для чисельного розв’язування задачi Кошi для рiвняння Лапласа |
| title_full | Boundary-integral approach for the numerical solution of the Cauchy
problem for the Laplace equation |
| title_fullStr | Boundary-integral approach for the numerical solution of the Cauchy
problem for the Laplace equation |
| title_full_unstemmed | Boundary-integral approach for the numerical solution of the Cauchy
problem for the Laplace equation |
| title_short | Boundary-integral approach for the numerical solution of the Cauchy
problem for the Laplace equation |
| title_sort | boundary-integral approach for the numerical solution of the cauchy
problem for the laplace equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1952 |
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