UDC 517.948.32/.33

MATHEMATICS

V. P. MASLOV

REGULARIZATION OF ILL-POSED PROBLEMS FOR SINGULAR INTEGRAL EQUATIONS

(Presented by Academician A. N. Tikhonov, 19 XII 1966)

Numerous works on singular integral equations have been devoted to the construction of a regularizer that would reduce a singular integral equation to an equivalent Fredholm equation \((^{1-7})\). At the same time, for a general closed linear operator \(A\) in Hilbert space, a necessary condition for the existence of a regularizer is the closedness of the range \(R(A)\) \((^{5,7})\). This is equivalent to the well-posedness of the problem \(Au=f\), provided that \(u\) is orthogonal to the subspace of solutions of \(Av=0\).

Therefore, for an ill-posed integral equation with a singular kernel we shall pose a different regularization problem—regularization in the sense of A. N. Tikhonov \((^{8,9})\), i.e., we shall abandon the principal requirement: the equivalence of the regularized and the original problems. The discussion will concern only the uniform convergence of solutions of a family of regularized equations to some solution of the original problem. We shall prove that a certain modification of A. N. Tikhonov’s regularization, developed by him for bounded operators, is also applicable to unbounded operators, i.e., to integral equations with singular kernels, and also to ill-posed linear integro-differential equations with partial derivatives.

Let \(H\) be a Hilbert space, and let \(T\) be a closed operator in \(H\) with dense domain \(D(T)\). Consider the equation

\[ Tv=f, \tag{1} \]

where \(v \in D(T)\), \(f \in R(T)\) is given.

In the case of bounded operators, the regularization proposed for this problem by A. N. Tikhonov has the form

\[ (T^*T+\delta)x_\delta = T^* f_\delta;\qquad \delta \leq \|f_\delta-f\|. \tag{2} \]

We shall suppose that the operator \(T\) is unbounded. Assume first that \(f_\delta \in D(T^*)\).

Denote by \(P_T, P_{T^*}\) the projection operators onto the subspaces of solutions of \(Tu=0\) and \(T^*u=0\), respectively; by \(T_1\) the restriction of the operator \(T\) to the domain* \((1-P_T)D(T)\); and by \(T_{1*}\) the restriction of the operator \(T^*\) to the domain \((1-P_{T^*})D(T^*)\). It is obvious that: 1) \(T_1^{-1}\), \(T_{1*}^{-1}\) exist; 2) \(TT_1^{-1}=1\), \(\overline{T_1^{-1}T}=1-P_T\); 3) \(\overline{P_TT_1^{-1}}=0\); \(\overline{P_T^*T_{1*}^{-1}}=0\); 4) \(\overline{P_{T^*}T}=0\), \(\overline{P_TT^*}=0\). (The bar denotes closure.) The operator \(A=T_1^{-1}(1-P_{T^*})\) is defined on an everywhere dense set in \(H\), and therefore the operators \(A^*\) and \(A^*A\), whose domains are dense, exist. It is not difficult to verify that \(A^*=T_{1*}^{-1}(1-P_T)\). We shall prove this assertion.

* By \(P_TH\) we mean the subspace of solutions of \(Tx=0\); \((1-P_T)H\) is its orthogonal complement. By \((1-P_T)D\), where \(D \subset H\), we mean the domain equal to \(D \cap (1-P_T)H\).

Proof. Let \(f \in D[T_{1*}^{-1}(1-P_T)] = R(T^*) \oplus P_T H\), and let \(F=T_{1*}^{-1}(1-P_T)f\); then \(P_{T*}F=0\) and \((1-P_T)f=T^*F\), while
\[ (f,Ag)=\bigl(f,T_1^{-1}(1-P_{T*})g\bigr) =\bigl(f(1-P_T)T_1^{-1}(1-P_{T*})g\bigr) =\bigl([1-P_T]f,T_1^{-1}[1-P_{T*}]g\bigr) \]
\[ =\bigl(T^*F,T_1^{-1}[1-P_{T*}]g\bigr) =\bigl(F,[1-P_{T*}]g\bigr)=(F,g), \]
i.e. \(A^*f=F \Rightarrow T_{1*}^{-1}(1-P_T)\subset A^*\). Let \(g\in D(T)\), \(h\in D(A^*)\), \(A^*h=f\); then
\[ (f,Tg)=(h,ATg)=([1-P_T]h,g)\Rightarrow T^*f=(1-P_T)h \Rightarrow (1-P_T)D(A^*)=R(T^*)\Rightarrow D(A^*)=R(T^*)\oplus P_T H \]
\[ \Rightarrow A^*=T_{1*}^{-1}(1-P_T). \]

As indicated above, the operator
\[ AA^*=T_1^{-1}(1-P_{T*})T_{1*}^{-1}(1-P_T)=T_1^{-1}T_{1*}^{-1}(1-P_T) \]
has an everywhere dense domain of definition. Consequently, the operator
\(T_1^{-1}T_{1*}^{-1}\) is defined on a domain everywhere dense in \((1-P_T)H\).

By the known von Neumann lemma, for any closed operator \(B\) with an everywhere dense domain of definition, the operators \((1+B^*B)^{-1}\) and \(B(1+B^*B)^{-1}\) are defined on all of \(H\) and are bounded by one. Consequently, \((B[1+B^*B])^*=(1+B^*B)B^*\) is bounded by one.

Putting \(B=\delta^{-1/2}T\), we arrive at the conclusion that the operators
\[ C_\delta=(\delta+T^*T)^{-1} \]
and
\[ B_\delta=(\delta+T^*T)^{-1}T^* \]
are bounded respectively by \(\delta^{-1}\) and \(\delta^{-1/2}\).

It is also obvious that the operator
\[ R_\delta=(\delta+T^*T)^{-1}T^*T_1 \]
is defined on \((1-P_T)H\) and is bounded by two. Indeed,
\[ \|R_\delta\|=\|(\delta+T^*T)^{-1}T^*T_1\| \leq \|(\delta+T^*T)^{-1}T^*T\| \]
\[ =\|1-\delta(\delta+T^*T)^{-1}\|\leq 2. \]
Since \(P_{T*}T_1=P_{T*}T=0\), and hence \(T^*T_1=T_{1*}T_1\), we have
\[ R_\delta=(\delta+T^*T)^{-1}T_{1*}T_1. \]
It follows from this that
\[ R_\delta^{-1}=T_1^{-1}T_{1*}^{-1}(\delta+T^*T) =\delta T_1^{-1}T_{1*}^{-1}+1-P_T \]
exists and is defined on a domain dense in \((1-P_T)H\). On elements of this domain, obviously, \(R_\delta^{-1}\to 1\) as \(\delta\to 0\).

Let \(z=(1-P_T)v\).

Since \(\|R_\delta\|\leq 2\), by Theorem 3.2 of Part I of the book \({}^{(10)}\), we obtain that \(R_\delta\) converge as \(\delta\to 0\) to \(1\) on the subspace \((1-P_T)H\). Consequently,
\[ x_\delta-z=B_\delta f_\delta-z =B_\delta(f_\delta-f)+B_\delta f-z \]
\[ =B_\delta(f_\delta-f)+B_\delta T_1z-z =B_\delta(f_\delta-f)+R_\delta z-z\to 0 \quad\text{as }\delta\to 0, \]
since
\[ \|B_\delta(f_\delta-f)\|\leq \frac{1}{\sqrt{\delta}}\|f_\delta-f\| =\sqrt{\delta}\to 0, \]
and \(z\in(1-P_T)H\), and hence \(\|R_\delta z-z\|\to 0\) as \(\delta\to 0\).

In solving the well-posed problem (2), we proceeded from the fact that
\(f_\delta\in D(T^*)\). This restriction can be avoided by means of the following procedure.

Consider the equation
\[ (T^*T+\delta_1)y_\delta=T^*u_\delta,\qquad \delta_1=\|[(1+\delta TT^*)^{-1}-1]f_\delta\|, \tag{3} \]
where \(u_\delta\), in turn, satisfies the equation
\[ (1+\delta TT^*)u_\delta=f_\delta,\qquad f_\delta\in H;\qquad \|f_\delta-f\|\leq \delta,\quad f\in R(T), \tag{4} \]
\(f_\delta, f\) are given.

In this case it is obvious that \(T^*u_\delta\) exists for any \(f_\delta\in H\). Moreover,
\(\delta_1\to 0\) as \(\delta\to 0\), and
\[ \|u_\delta-f\|\leq \delta+\delta_1\to 0 \quad\text{as }\delta\to 0. \]
Indeed,
\[ \delta_1=\|[(1+\delta TT^*)^{-1}-1](f_\delta-f+f)\| \leq 2\delta+\|[(1+\delta TT^*)^{-1}-1]f\|\to 0 \quad\text{as }\delta\to 0, \]
and
\[ \|u_\delta-f\| =\|(1+\delta TT^*)^{-1}f_\delta-f_\delta+(f_\delta-f)\| \leq \|[(1+\delta TT^*)^{-1}-1]f_\delta\|+\delta =\delta_1+\delta. \]
Thus, in view of the preceding considerations, \(y_\delta-z\to 0\) as
\(\delta\to 0\).

Suppose now that \(H=L_2[R^n]\). Consider
\[ v_{\delta_1,\varepsilon} =\varepsilon^{-(n+2)/2}(R_{\delta_1}-1)(1-P_T)e^{-(x-\xi)^2/4\varepsilon}, \qquad x,\xi\in R^n, \]

i.e., \(v_{\delta_1,\varepsilon}\) satisfies the equation

\[ (\delta_1+T^{*}T)v_{\delta_1,\varepsilon} = \delta_1(1-P_T)e^{-(x-\xi)^2/4\varepsilon}\varepsilon^{-(n+2)/2}. \]

By what has been proved, for fixed \(\varepsilon\) the function \(v_{\delta_1,\varepsilon}\) tends to zero in norm as \(\delta_1\to0\). Consequently, there exists a function \(\varepsilon=\varepsilon(\delta_1)\to0\) as \(\delta_1\to0\) such that \(\varepsilon(\delta_1)>\delta_1^{\,2/(4+n)}\) and \(\|v_{\delta,\varepsilon(\delta_1)}\|\) is bounded above as \(\delta_1\to0\).

Let us prove that if \(z(x)\in (1-P_T)L_2(R^n)\cap C^2\) and \(Tz(x)=f\), then

\[ \max_x\left|z(x)-\frac{1}{(2\pi\varepsilon(\delta_1))^{n/2}} \int y_\delta(\xi)e^{-(x-\xi)^2/4\varepsilon(\delta_1)}\,d\xi\right| = O[\varepsilon(\delta_1)]_{\delta\to0}. \tag{5} \]

Indeed,

\[ \begin{aligned} z(x)&-\frac{1}{(2\pi\varepsilon)^{n/2}} \int e^{-(x-\xi)^2/4\varepsilon}y_\delta(\xi)\,d\xi \\ &= \frac{1}{(2\pi\varepsilon)^{n/2}} \int e^{-(x-\xi)^2/4\varepsilon}[z(x)-z(\xi)]\,d\xi + \frac{1}{(2\pi\varepsilon)^{n/2}} \int e^{-(x-\xi)^2/4\varepsilon}[z(\xi)-y_\delta(\xi)]\,d\xi \\ &\le \varepsilon\max z_{x_i x_j} + \left| \frac{1}{(2\pi\varepsilon)^{n/2}} \left(B_{\delta_1}[y_\delta-f]+B_{\delta_1}f-z,\ e^{-(x-\xi)^2/4\varepsilon}\right) \right| \\ &\le \varepsilon\max z_{x_i x_j} + \frac{\delta_1+\delta}{\sqrt{\delta_1}(2\pi\varepsilon)^{n/2}} \left\|e^{-(x-\xi)^2/4\varepsilon}\right\| + \frac{1}{(2\pi\varepsilon)^{n/2}} \left|\left((R_{\delta_1}-1)(1-P_T)z,\ e^{-(x-\xi)^2/4\varepsilon}\right)\right| \\ &\le \varepsilon\max z_{x_i x_j} + \frac{(\delta_1+\delta)(2\pi\varepsilon)^{-n/4}}{\sqrt{\delta_1}} + \|z\|\,\varepsilon\,\|v_{\delta_1,\varepsilon}\| = O(\varepsilon). \end{aligned} \]

Thus we have proved the following theorem:

Theorem. For any closed linear operator \(T\) with everywhere dense domain of definition in \(L_2[R^n]\), problem (3)—(4) is well posed, and its solution \(y_\delta\) satisfies relation (5), if there exists a solution of the equation \(Tz(x)=f\) belonging to \(C^q\cap(1-P_T)D(T)\).

We note that under the additional condition of Tikhonov well-posedness of problem (1), using the effective technique of M. M. Lavrent'ev (11), one can obtain refined estimates for the regularization presented here.

In conclusion I express my deep gratitude to A. N. Tikhonov for the discussion, the result of which was the present note.

Moscow State University
named after M. V. Lomonosov

Received
20 XI 1966

CITED LITERATURE

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