Brief Communications

UDC 517.946.9 : 517.544.2

Regularization of Incorrect Problems for Equations of Mathematical Physics

O. A. Liskovets

A. N. Tikhonov’s method [1] of regularizing unstable problems is proposed—apparently for the first time—for use in solving partial differential equations. The convergence of the approximate solutions to the desired one is proved, provided the latter is unique.

1. Let it be required, in the rectangle \(\Pi\{0<x<\pi,\ 0<y<1\}\), to find a harmonic function satisfying the following boundary conditions:

\[ u(0,y)=u_0(y),\quad u(\pi,y)=u_1(y),\quad 0\leq y\leq 1, \]
\[ u(x,0)=\varphi(x),\quad u_y(x,0)=\psi(x),\quad 0\leq x\leq \pi. \tag{1} \]

The solution is assumed to exist, and we shall be interested only in finding it.

Such a problem is unstable, since arbitrarily small variations of the boundary data may correspond to arbitrarily large variations of the solution; this follows from the classical Hadamard example of an incorrect problem.

An analogous problem and a method for solving it, using expansion in a series, were considered repeatedly by V. K. Ivanov (see, for example, [2]).

We shall generalize the problem formulated above by replacing the Laplace equation by an arbitrary linear elliptic equation of second order

\[ L(u)=f(x,y),\quad (x,y)\in \Pi, \tag{2} \]

with coefficients continuous in \(\overline{\Pi}\).

If we introduce the operator
\[ Au=(L(u),\ u(0,y),\ u(\pi,y),\ u(x,0),\ u_y(x,0)) \]
and the vector
\[ F=(f(x,y),\ u_0(y),\ u_1(y),\ \varphi(x),\ \psi(x)), \]
then, in the new notation, the problem takes the form

\[ Au=F. \tag{3} \]

1. The approach proposed by A. N. Tikhonov is based primarily on a topological theorem according to which a one-to-one continuous mapping of a compact metric space into another metric space is continuous also in the inverse direction.

It is not difficult to see that the operator \(L(u)\) acts continuously from \(W_2^{(2)}(\Pi)\) into \(L_2(\Pi)\). At the same time, the boundary values of the function \(u\in W_2^{(2)}(\Pi)\), written in (1), also vary continuously in the \(L_2\) metric of the corresponding manifolds by virtue of the embedding theorems ([3], p. 78) (the applicability of the embedding theorems on the boundary is ensured by its simplicity [3], p. 81).

Thus the operator \(A\) of problem (3) is continuous from \(W_2^{(2)}(\Pi)\) into the space \(L_2\) of 5-dimensional vectors with square norm equal to the sum of the mean-square values of their components. This is all the more true for \(u\in W_2^{(n)}(\Pi)\), \(n>2\). Moreover, the direct correspondence in problem (3), i.e. the operator \(A\), is obviously one-to-one.

The uniqueness of the inverse correspondence (the operator \(A^{-1}\)) is nothing other than the uniqueness of the solution of problem (3) in \(W_2^{(2)}(\Pi)\). To prove it, it is sufficient to establish the triviality of the solution of the corresponding homogeneous problem.

This is easy to do if \(L(u)\equiv \Delta u\). Indeed, let the solution of such a homogeneous problem be a harmonic function \(u(x,y)\in W_2^{(2)}(\Pi)\). Then, analogously to what was said earlier, \(u(x,1)\in L_2(0,\pi)\), and therefore it is representable in the form

\[ u(x,1)=\sum_{m=1}^{\infty} b_m\sin mx,\quad b_m=\frac{2}{\pi}\int_{0}^{\pi} u(x,1)\sin mx\,dx. \]

Since on the remaining part of the boundary the function \(u\) vanishes, we have

\[ u(x,y)=\sum_{m=1}^{\infty} b_m \frac{\operatorname{sh} my}{\operatorname{sh} m}\sin mx,\qquad (x,y)\in \Pi . \]

Finding from this \(u_y(x,0)\) (termwise differentiation of the series is legitimate) and equating it to zero, we see that all \(b_m=0\) and, consequently, \(u\equiv 0\).

For an operator \(L(u)\) of general form it is difficult to give such a proof. Therefore, in what follows we shall restrict ourselves to the assertion that everywhere below uniqueness of the solution of problem (2), (1) will be assumed.

1. The construction of the compact metric space mentioned above, whose limiting element is the desired solution, is carried out by reducing the original problem to a variational one by the method of least squares with specially chosen additional terms playing the role of viscosity (see [1]).

For values of the parameter \(\alpha>0\) we shall seek a function minimizing in \(W_2^{(n)}(\Pi)\), \(n>2\), the functional

\[ J^\alpha(u;F)=\|Au-F\|_{L_2}^2+\alpha N(u), \tag{4} \]

\[ \|Au-F\|_{L_2}^2 = \|L(u)-f\|_{L_2(\Pi)}^2 + \|u|_{x=0}-u_0\|_{L_2(0,1)}^2 + \]

\[ +\|u|_{x=\pi}-u_1\|_{L_2(0,1)}^2 + \|u|_{y=0}-\varphi\|_{L_2(0,\pi)}^2 + \|u_y|_{y=0}-\psi\|_{L_2(0,\pi)}^2, \tag{5} \]

and \(N(u)\) is any quadratic functional of \(u\in W_2^{(n)}(\Pi)\), equivalent to \(\|u\|_{W_2^{(n)}(\Pi)}^2\), for example

\[ N(u)=\|u\|_{L_2(\Pi)}^2+\|u\|_{L_2^{(n)}(\Pi)}^2 . \]

For \(\alpha=0\), the minimum of the functional (4) is attained at the desired solution, if it exists and belongs to \(W_2^{(n)}(\Pi)\).

Theorem 1. For any \(\alpha>0\) and any quadratically summable functions \(f,u_0,u_1,\varphi,\psi\) (i.e., for \(F\in L_2\)), there exists, moreover a unique, function \(u^\alpha\in W_2^{(n)}(\Pi)\) minimizing the functional (4) in the space \(W_2^{(n)}(\Pi)\).

With the aid of the identity

\[ J^\alpha\left(\frac{u_1-u_2}{2};0\right) + J^\alpha\left(\frac{u_1+u_2}{2};F\right) = \frac12 J^\alpha(u_1;F)+\frac12 J^\alpha(u_2;F) \]

the theorem is proved quite analogously to [4].

Theorem 2. If there exists a (unique) solution of problem (2), (1) \(u\in W_2^{(n)}(\Pi)\), \(n>2\), such that \(F\in L_2\), then \(\|u^\alpha-u\|_{C_{n-2}(\Pi)}\to 0\) as \(\alpha\to 0\).

Proof. Denote \(N(u)=H^2\).

\[ J^\alpha(u^\alpha;F)\le J^\alpha(u;F) = \|Au-F\|_{L_2}^2+\alpha N(u) = \alpha N(u) = \alpha H^2 . \]

Consequently, \(N(u^\alpha)\le H^2\), i.e., all \(u^\alpha\), \(\alpha>0\), belong to the bounded set in \(W_2^{(n)}(\Pi)\)

\[ \{\bar u:N(\bar u)\le H^2\}. \]

By virtue of the complete continuity of the embedding operator from \(W_2^{(n)}(\Pi)\) into \(C_{n-2}(\Pi)\), the set \(\{u^\alpha\}\) is compact in \(C_{n-2}(\Pi)\), and therefore the operator \(A^{-1}\) acts continuously on the set \(\{Au^\alpha\}\) from \(L_2\) into \(C_{n-2}(\Pi)\). And since

\[ \|Au^\alpha-Au\|_{L_2} \le \|Au^\alpha-F\|_{L_2} + \|Au-F\|_{L_2} = \]

\[ = \|Au^\alpha-F\|_{L_2} \le \sqrt{J^\alpha(u^\alpha;F)} \le \sqrt{\alpha}\,H \to 0 \]

as \(\alpha\to 0\), it follows at the same time that

\[ \|u-u^\alpha\|_{C_{n-2}(\Pi)}\to 0. \]

However, in practice the initial data \(F\) are obtained, as a rule, from measurements, which are inevitably associated with errors of one kind or another. It is therefore important to make sure that the algorithm can be adapted to this case as well.

Theorem 3. Suppose that the conditions of Theorem 2 are satisfied, and suppose that approximations to the initial data \(F^\delta\) are known such that \(\|F-F^\delta\|_{L_2}\le \delta\), \(\delta\to 0\). If \(\alpha=\alpha(\delta)\) is chosen so that \(O(\delta^2)\le \alpha(\delta)\le o(1)\), then the functions \(u^{\alpha(\delta)}\), minimizing the functionals \(J^{\alpha(\delta)}(u;F^\delta)\), converge as \(\delta\to 0\) to the desired solution:
\[ \|u^{\alpha(\delta)}-u\|_{C_{n-2}(\Pi)}\to 0 . \]

The proof in its main features coincides with the preceding one. It is not difficult to estimate also the difference of the minimizing functions under variation of the right-hand sides.

Theorem 4. If \(F,F^\delta\in L_2\), \(\|F-F^\delta\|_{L_2}\le \delta\), then for the corresponding minimizing functions
\[ \|u^\alpha-u^{\alpha,\delta}\|_{W_2^{(n)}(\Pi)} \le \frac{\delta}{\alpha}\frac{\|A\|}{m} \left( m=\min_{W_2^{(n)}(\Pi)} \frac{N(u)}{\|u\|_{W_2^{(n)}(\Pi)}^2} \right), \]
where \(\|A\|\) is the norm of \(A\) as an operator from \(W_2^{(n)}(\Pi)\) into \(L_2\).

The proof, as in [4], follows from the fact that the quadratic, with respect to \(\lambda\), trinomial \(J^\alpha(u^\alpha+\lambda z;F)\) with \(z\in W_2^{(n)}(\Pi)\) has a minimum at \(\lambda=0\), and therefore the identity
\[ (Au^\alpha-F,\;Az)+\alpha N(u^\alpha,z)=0,\quad z\in W_2^{(n)}(\Pi), \]
is valid; here the first term denotes the scalar product in \(L_2\), and \(N(u^\alpha,z)\) is the bilinear form corresponding to the quadratic functional \(N(u)\). Subtracting from this the analogous identity connecting \(u^{\alpha,\delta}\), \(F^\delta\), and \(z\), then putting \(v=u^\alpha-u^{\alpha,\delta}\) and choosing \(z=v\), we obtain
\[ \alpha N(v)=(F-F^\delta,\;Av)-\|Av\|_{L_2}^{2} \le (F-F^\delta,\;Av)\le \]
\[ \le \|F-F^\delta\|_{L_2}\|Av\|_{L_2} \le \delta\|A\|\|v\|_{W_2^{(n)}(\Pi)}, \]
which proves the theorem.

From it one can easily obtain a somewhat different form [4] of the approximation-convergence theorem, analogous to Theorem 3.

Thus, the algorithm described is stable in \(C_{n-2}(\Pi)\) with respect to errors in the right-hand sides (1), (2), so that, having some way of minimizing the functional \(J^\alpha(u;F^\delta)\) and reasonably choosing the value of \(\alpha\) depending on the accuracy of the data \(\delta\), or conversely, one may approximate the desired solution well (for not too large \(\delta\), or, respectively, \(\alpha\)).

Among the shortcomings of the given regularization method one should include the circumstance that it gives approximations \(u^\alpha\) which are not solutions of the original equation (the latter does not coincide with the Euler equation for the variational functional), whereas, for example, harmonic functions are usually preferably approximated by harmonic functions.

4. The method set forth admits a number of generalizations. With its help one can solve other ill-posed problems for equation (2), for example the following:
\[ \Delta u=f,\quad (x,y)\in \Pi, \]
\[ u_x(0,y)=u_2(y),\quad u_x(\pi,y)=u_3(y),\quad u(x,0)=\varphi(x),\quad u_y(x,0)=\psi(x) \]
(its ill-posedness is shown by the behavior of the harmonic functions
\[ \frac{\operatorname{ch} my\cos mx}{m} \]
for integer
\[ m\to\infty \]
). For this it suffices in the functional (4), instead of the 2nd and 3rd terms (5), to take
\[ \|u_x|_{x=0}-u_2\|_{L_2(0,1)}^2 + \|u_x|_{x=\pi}-u_3\|_{L_2(0,1)}^2 . \]

Moreover, the applicability of the method is by no means constrained by the condition of obligatory instability of the problem. It can be used without concern for establishing the ill-posedness of the problem. Thus, for example, the determination of a harmonic function from the boundary data
\[ u(0,y)=u_0(y),\quad u_x(0,y)=u_2(y),\quad u(x,0)=\varphi(x),\quad u_y(x,0)=\psi(x) \]
or
\[ u(0,y)=u_0(y),\quad u_x(0,y)=u_2(y),\quad u(\pi,y)=u_1(y),\quad u(x,0)=\varphi(x) \]
is unstable only conjecturally. Nevertheless, the algorithm of Sec. 3, with the corresponding replacement of \(\|\ \ \|_{L_2}\), is known to be suitable (continuous dependence of the direct correspondence

and uniqueness of the solution are proved for all the problems of the present item in the same way as in item 2).

The results of item 3 carry over verbatim to linear (not necessarily purely differential) equations of arbitrary order and to finite systems of such equations. In this case their ellipticity is not essential; it is sufficient that, for some \(n\), the properties indicated in item 2 be satisfied.

The same is true for problems of higher dimension, with the sole difference that, by virtue of embedding theorems, the convergence of the approximations will, generally speaking, be weaker than in Theorems 2 and 3.

References

1. Tikhonov A. N. Dokl. Akad. Nauk SSSR, 151, No. 3, 501—504, 1963.
2. Ivanov V. K. Mat. sb., 61: 2, 1963, pp. 211—223.
3. Sobolev S. L. Some Applications of Functional Analysis in Mathematical Physics. LSU Publishing House, L., 1950.
4. Morozov V. A. Vestn. MGU, ser. 1, No. 4, 13—21, 1965.

Received by the editors
November 23, 1965

Institute of Mathematics
Academy of Sciences of the BSSR