View source for Ill-posed problems

''incorrectly-posed problems, improperly-posed problems''

{{TEX|done}}
{{MSC|47A52|47J06,65F22}}

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Problems  for which at least one of the conditions below, which characterize  well-posed problems, is violated. The problem of determining a solution  $z=R(u)$ in a metric space $Z$ (with metric $\rho_Z(,)$) from "initial  data" $u$ in a metric space $U$ (with metric $\rho_U(,)$) is said to be  well-posed on the pair of spaces $(Z,U)$ if: a) for every $u \in U$  there exists a solution $z \in Z$; b) the solution is uniquely  determined; and c) the problem is stable on the spaces $(Z,U)$, i.e.:  For every $\epsilon > 0$ there is a $\delta(\epsilon) > 0$ such  that for any $u_1, u_2 \in U$ it follows from $\rho_U(u_1,u_2) \leq  \delta(\epsilon)$ that $\rho_Z(z_1,z_2) < \epsilon$, where $z_1 =  R(u_1)$ and $z_2 = R(u_2)$.

The concept of a well-posed  problem is due to J. Hadamard (1923), who took the point of view that  every mathematical problem corresponding to some physical or  technological problem must be well-posed. In fact, what physical  interpretation can a solution have if an arbitrary small change in the  data can lead to large changes in the solution? Moreover, it would be  difficult to apply approximation methods to such problems. This put the  expediency of studying ill-posed problems in doubt.

However,  this point of view, which is natural when applied to certain  time-depended phenomena, cannot be extended to all problems. The  following problems are unstable in the metric of $Z$, and therefore  ill-posed: the solution of integral equations of the first kind;  differentiation of functions known only approximately; numerical  summation of Fourier series when their coefficients are known  approximately in the metric of $\ell_2$; the Cauchy problem for the  Laplace equation; the problem of analytic continuation of functions; and  the inverse problem in gravimetry. Other ill-posed problems are the  solution of systems of linear algebraic equations when the system is  ill-conditioned; the minimization of functionals having non-convergent  minimizing sequences; various problems in linear programming and optimal  control; design of optimal systems and optimization of constructions  (synthesis problems for antennas and other physical systems); and  various other control problems described by differential equations (in  particular, differential games).  Various physical and technological  questions lead to the problems listed (see {{Cite|TiAr}}).

A  broad class of so-called inverse problems that arise in physics,  technology and other branches of science, in particular, problems of  data processing of physical experiments, belongs to the class of  ill-posed problems. Let $z$ be a characteristic quantity of the  phenomenon (or object) to be studied.  In a physical experiment the  quantity $z$ is frequently inaccessible to direct measurement, but what  is measured is a certain transform $Az=u$ (also called outcome). For the  interpretation of the results it is necessary to determine $z$ from  $u$, that is, to solve the equation
\begin{equation}
\label{eq1}
Az = u.
\end{equation}
Problems  of solving an equation \ref{eq1} are often called pattern recognition  problems. Problems leading to the minimization of functionals (design of  antennas and other systems or constructions, problems of optimal  control and many others) are also called synthesis problems.

Suppose  that in a mathematical model for some physical experiments the object  to be studied (the phenomenon) is characterized by an element $z$ (a  function, a vector) belonging to a set $Z$ of possible solutions in a  metric space $\hat{Z}$. Suppose that $z_T$ is inaccessible to direct  measurement and that what is measured is a transform, $Az_T=u_T$, $u_T  \in AZ$, where $AZ$ is the image of $Z$ under the operator $A$.  Evidently, $z_T = A^{-1}u_T$, where $A^{-1}$ is the operator inverse to  $A$. Since $u_T$ is obtained by measurement, it is known only  approximately. Let $\tilde{u}$ be this approximate value. Under these  conditions the question can only be that of finding a "solution" of the  equation
\begin{equation}
\label{eq2}
Az = \tilde{u},
\end{equation}
approximating $z_T$.

In  many cases the operator $A$ is such that its inverse $A^{-1}$ is not  continuous, for example, when $A$ is a completely-continuous operator in  a Hilbert space, in particular an integral operator of the form
$$
\int_a^b K(x,s) z(s) \rd s.
$$
Under  these conditions one cannot take, following classical ideas, an exact  solution of \ref{eq2}, that is, the element $z=A^{-1}\tilde{u}$, as an  approximate "solution" to $z_T$. In fact: a) such a solution need not  exist on $Z$, since $\tilde{u}$ need not belong to $AZ$; and b) such a  solution, if it exists, need not be stable under small changes of  $\tilde{u}$ (due to the fact that $A^{-1}$ is not continuous) and,  consequently, need not have a physical interpretation.  The problem  \ref{eq2} then is ill-posed.

==Numerical methods for solving ill-posed problems.== 

For  ill-posed problems of the form \ref{eq1} the question arises: What is  meant by an approximate solution? Clearly, it should be so defined that  it is stable under small changes of the original information. A second  question is: What algorithms are there for the construction of such  solutions? Answers to these basic questions were given by A.N. Tikhonov  (see {{Cite|Ti}}, {{Cite|Ti2}}).

The selection method.  In some cases an approximate solution of \ref{eq1} can be found by the  selection method. It consists of the following: From the class of  possible solutions $M \subset Z$ one selects an element $\tilde{z}$ for  which $A\tilde{z}$ approximates the right-hand side of \ref{eq1} with  required accuracy. For the desired approximate solution one takes the  element $\tilde{z}$. The question arises: When is this method  applicable, that is, when does
$$
\rho_U(A\tilde{z},Az_T) \leq \delta
$$
imply that
$$
\rho_Z(z,z_T) \leq \epsilon(\delta),
$$
where  $\epsilon(\delta) \rightarrow 0$ as $\delta \rightarrow 0$? This holds  under the conditions that the solution of \ref{eq1} is unique and that  $M$ is compact (see {{Cite|Ti3}}). On the basis of these arguments one  has formulated the concept (or the condition) of being Tikhonov  well-posed, also called conditionally well-posed (see {{Cite|La}}). As  applied to \ref{eq1}, a problem is said to be conditionally well-posed  if it is known that for the exact value of the right-hand side $u=u_T$  there exists a unique solution $z_T$ of \ref{eq1} belonging to a given  compact set $M$. In this case $A^{-1}$ is continuous on $M$, and if  instead of $u_T$ an element $u_\delta$ is known such that  $\rho_U(u_\delta,u_T) \leq \delta$ and $u_\delta \in AM$, then as an  approximate solution of \ref{eq1} with right-hand side $u = u_\delta$  one can take $z_\delta = A^{-1}u_\delta $. As $\delta \rightarrow 0$,  $z_\delta$ tends to $z_T$.

In many cases the  approximately known right-hand side $\tilde{u}$ does not belong to $AM$.  Under these conditions equation \ref{eq1} does not have a classical  solution. As an approximate solution one takes then a generalized  solution, a so-called quasi-solution (see {{Cite|Iv}}). A quasi-solution  of \ref{eq1} on $M$ is an element $\tilde{z}\in M$ that minimizes for a  given $\tilde{u}$ the functional $\rho_U(Az,\tilde{u})$ on $M$ (see  {{Cite|Iv2}}). If $M$ is compact, then a quasi-solution exists for any  $\tilde{u} \in U$, and if in addition $\tilde{u} \in AM$, then a  quasi-solution $\tilde{z}$ coincides with the classical (exact) solution  of \ref{eq1}. The existence of quasi-solutions is guaranteed only when  the set $M$ of possible solutions is compact.

The  regularization method. For a number of applied problems leading to  \ref{eq1} a typical situation is that the set $Z$ of possible solutions  is not compact, the operator $A^{-1}$ is not continuous on $AZ$, and  changes of the right-hand side of \ref{eq1} connected with the  approximate character can cause the solution to go out of $AZ$. Such  problems are called essentially ill-posed. An approach has been worked  out to solve ill-posed problems that makes it possible to construct  numerical methods that approximate solutions of essentially ill-posed  problems of the form \ref{eq1} which are stable under small changes of  the data. In this context, both the right-hand side $u$ and the operator  $A$ should be among the data.

In what follows, for  simplicity of exposition it is assumed that the operator $A$ is known  exactly.  At the basis of the approach lies the concept of a  regularizing operator (see {{Cite|Ti2}}, {{Cite|TiAr}}). An operator  $R(u,\delta)$ from $U$ to $Z$ is said to be a regularizing operator for  the equation $Az=u$ (in a neighbourhood of $u=u_T$) if it has the  following properties: 1) there exists a $\delta_1 > 0$ such that the  operator $R(u,\delta)$ is defined for every $\delta$, $0 \leq \delta  \leq \delta_1$, and for any $u_\delta \in U$ such that  $\rho_U(u_\delta,u_T) \leq \delta$; and 2) for every $\epsilon > 0$  there exists a $\delta_0 = \delta_0(\epsilon,u_T)$ such that  $\rho_U(u_\delta,u_T) \leq \delta \leq \delta_0$ implies  $\rho_Z(z_\delta,z_T) \leq \epsilon$, where $z_\delta =  R(u_\delta,\delta)$.

Sometimes it is convenient to use  another definition of a regularizing operator, comprising the previous  one. An operator $R(u,\alpha)$ from $U$ to $Z$, depending on a parameter  $\alpha$, is said to be a regularizing operator (or regularization  operator) for the equation $Az=u$ (in a neighbourhood of $u=u_T$) if it  has the following properties: 1) there exists a $\delta_1 > 0$ such  that $R(u,\alpha)$ is defined for every $\alpha$ and any $u_\delta \in  U$ for which $\rho_U(u_\delta,u_T) < \delta \leq \delta_1$; and 2)  there exists a function $\alpha = \alpha(\delta)$ of $\delta$ such that  for any $\epsilon > 0$ there is a $\delta(\epsilon) \leq \delta_1$  such that if $u_\delta \in U$ and $\rho_U(u_\delta,u_T) \leq  \delta(\epsilon)$, then $\rho_Z(z_\delta,z_T) < \epsilon$, where  $z_\delta = R(u_\delta,\alpha(\delta))$. In this definition it is not  assumed that the operator $ R(u,\alpha(\delta))$ is globally  single-valued.

If $\rho_U(u_\delta,u_T)$, then as an  approximate solution of \ref{eq1} with an approximately known right-hand  side $u_\delta$ one can take the element $z_\alpha =  R(u_\delta,\alpha)$ obtained by means of the regularizing operator  $R(u,\alpha)$, where $\alpha = \alpha(\delta)$ is compatible with the  error of the initial data $u_\delta$ (see {{Cite|Ti}}, {{Cite|Ti2}},  {{Cite|TiAr}}). This is said to be a regularized solution of  \ref{eq1}. The numerical parameter $\alpha$ is called the regularization  parameter. As $\delta \rightarrow 0$, the regularized approximate  solution $z_\alpha(\delta) = R(u_\delta,\alpha(\delta))$ tends (in the  metric of $Z$) to the exact solution $z_T$.

Thus, the  task of finding approximate solutions of \ref{eq1} that are stable under  small changes of the right-hand side reduces to: a) finding a  regularizing operator; and b) determining the regularization parameter  $\alpha$ from additional information on the problem, for example, the  size of the error with which the right-hand side $u$ is given.

The  construction of regularizing operators. It is assumed that the equation  $Az = u_T$ has a unique solution $z_T$. Suppose that instead of $Az =  u_T$ the equation $Az = u_\delta$ is solved and that  $\rho_U(u_\delta,u_T) \leq \delta$. Since $\rho_U(Az_T,u_\delta) \leq  \delta$, the approximate solution of $Az = u_\delta$ is looked for in  the class $Z_\delta$ of elements $z_\delta$ such that  $\rho_U(u_\delta,u_T) \leq \delta$. This $Z_\delta$ is the set of  possible solutions. As an approximate solution one cannot take an  arbitrary element $z_\delta$ from $Z_\delta$, since such a "solution" is  not unique and is, generally speaking, not continuous in $\delta$. As a  selection principle for the possible solutions ensuring that one  obtains an element (or elements) from $Z_\delta$ depending continuously  on $\delta$ and tending to $z_T$ as $\delta \rightarrow 0$, one uses the  so-called variational principle (see {{Cite|Ti}}). Let $\Omega[z]$ be a  continuous non-negative functional defined on a subset $F_1$ of $Z$  that is everywhere-dense in $Z$ and is such that: a) $z_1 \in F_1$; and  b) for every $d > 0$ the set of elements $z$ in $F_1$ for which  $\Omega[z] \leq d$, is compact in $F_1$. Functionals having these  properties are said to be stabilizing functionals for problem \ref{eq1}.  Let $\Omega[z]$ be a stabilizing functional defined on a subset $F_1$  of $Z$. ($F_1$ can be the whole of $Z$.) Among the elements of  $F_{1,\delta} = F_1 \cap Z_\delta$ one looks for one (or several) that  minimize(s) $\Omega[z]$ on $F_{1,\delta}$. The existence of such an  element $z_\delta$ can be proved (see {{Cite|TiAr}}). It can be  regarded as the result of applying a certain operator $R_1(u_\delta,d)$  to the right-hand side of the equation $Az = u_\delta$, that is,  $z_\delta=R_1(u_\delta,d)$. Then $R_1(u,\delta)$ is a regularizing  operator for equation \ref{eq1}. In practice the search for $z_\delta$  can be carried out in the following manner: under mild addition
al  restrictions on $\Omega[z] $ (quasi-monotonicity of $\Omega[z]$, see  {{Cite|TiAr}}) it can be proved that $\inf\Omega[z]$ is attained on  elements $z_\delta$ for which $\rho_U(Az_\delta,u_\delta) = \delta$. An  element $z_\delta$ is a solution to the problem of minimizing  $\Omega[z]$ given $\rho_U(Az,u_\delta)=\delta$, that is, a solution of a  problem of conditional extrema, which can be solved using Lagrange's  multiplier method and minimization of the functional
$$
M^\alpha[z,u_\delta] = \rho_U^2(Az,u_\delta) + \alpha \Omega[z].
$$
For  any $\alpha > 0$ one can prove that there is an element $z_\alpha$  minimizing $M^\alpha[z,u_\delta]$. The parameter $\alpha$ is determined  from the condition $\rho_U(Az_\alpha,u_\delta) = \delta$. If there is an  $\alpha$ for which $\rho_U(Az_\alpha,u_\delta) = \delta$, then the  original variational problem is equivalent to that of minimizing  $M^\alpha[z,u_\delta]$, which can be solved by various methods on a  computer (for example, by solving the corresponding Euler equation for  $M^\alpha[z,u_\delta]$). The element $z_\alpha$ minimizing  $M^\alpha[z,u_\delta]$ can be regarded as the result of applying to the  right-hand side of the equation $Az = u_\delta$ a certain operator  $R_2(u_\delta,\alpha)$ depending on $\alpha$, that is, $z_\alpha =  R_2(u_\delta,\alpha)$ in which $\alpha$ is determined by the discrepancy  relation $\rho_U(Az_\alpha,u_\delta) = \delta$. Then $R_2(u,\alpha)$ is  a regularizing operator for \ref{eq1}. Equivalence of the original  variational problem with that of finding the minimum of  $M^\alpha[z,u_\delta]$ holds, for example, for linear operators $A$. For  non-linear operators $A$ this need not be the case (see  {{Cite|GoLeYa}}).

The so-called smoothing functional  $M^\alpha[z,u_\delta]$ can be introduced formally, without connecting it  with a conditional extremum problem for the functional $\Omega[z]$, and  for an element $z_\alpha$ minimizing it sought on the set  $F_{1,\delta}$. This poses the problem of finding the regularization  parameter $\alpha$ as a function of $\delta$, $\alpha = \alpha(\delta)$,  such that the operator $R_2(u,\alpha(\delta))$ determining the element  $z_\alpha = R_2(u_\delta,\alpha(\delta)) $ is regularizing for  \ref{eq1}. Under certain conditions (for example, when it is known that  $\rho_U(u_\delta,u_T) \leq \delta$ and $A$ is a linear operator) such a  function exists and can be found from the relation  $\rho_U(Az_\alpha,u_\delta) = \delta$. There are also other methods for  finding $\alpha(\delta)$.

Let $T_{\delta_1}$ be a class  of non-negative non-decreasing continuous functions on $[0,\delta_1]$,  $z_T$ a solution of \ref{eq1} with right-hand side $u=u_T$, and $A$ a  continuous operator from $Z$ to $U$. For any positive number $\epsilon$  and functions $\beta_1(\delta)$ and $\beta_2(\delta)$ from  $T_{\delta_1}$ such that $\beta_2(0) = 0$ and $\delta^2 /  \beta_1(\delta) \leq \beta_2(\delta)$, there exists a $\delta_0 =  \delta_0(\epsilon,\beta_1,\beta_2)$ such that for $u_\delta \in U$ and  $\delta \leq \delta_0$ it follows from $\rho_U(u_\delta,u_T) \leq  \delta$ that $\rho_Z(z^\delta,z_T) \leq \epsilon$, where $z^\alpha =  R_2(u_\delta,\alpha)$ for all $\alpha$ for which $\delta^2 /  \beta_1(\delta) \leq \alpha \leq \beta_2(\delta)$.

Methods  for finding the regularization parameter depend on the additional  information available on the problem. If the error of the right-hand  side of the equation for $u_\delta$ is known, say $\rho_U(u_\delta,u_T)  \leq \delta$, then in accordance with the preceding it is natural to  determine $\alpha$ by the discrepancy, that is, from the relation  $\rho_U(Az_\alpha^\delta,u_\delta) = \phi(\alpha) = \delta$.

The  function $\phi(\alpha)$ is monotone and semi-continuous for every  $\alpha > 0$. If $A$ is a linear operator, $Z$ a Hilbert space and  $\Omega[z]$ a strictly-convex functional (for example, quadratic), then  the element $z_{\alpha_\delta}$ is unique and $\phi(\alpha)$ is a  single-valued function. Under these conditions, for every positive  number $\delta < \rho_U(Az_0,u_\delta)$, where $z_0 \in \set{ z :  \Omega[z] = \inf_{y\in F}\Omega[y] }$, there is an $\alpha(\delta)$ such  that $\rho_U(Az_\alpha^\delta,u_\delta) = \delta$ (see  {{Cite|TiAr}}).

However, for a non-linear operator $A$ the equation $\phi(\alpha) = \delta$ may have no solution (see {{Cite|GoLeYa}}).

The  regularization method is closely connected with the construction of  splines (cf.  [[Spline|Spline]]). For example, the problem of finding a  function $z(x)$ with piecewise-continuous second-order derivative on  $[a,b]$ that minimizes the functional 
$$
\Omega[z] = \int_a^b (z^{\prime\prime}(x))^2 \rd x 
$$ 
and takes given values $\set{z_i}$ on a grid $\set{x_i}$, is equivalent to the construction of a spline of the second degree.

A  regularizing operator can be constructed by spectral methods (see  {{Cite|TiAr}}, {{Cite|GoLeYa}}), by means of the classical integral  transforms in the case of equations of convolution type (see  {{Cite|Ar}}, {{Cite|TiAr}}), by the method of quasi-mappings (see  {{Cite|LaLi}}), or by the iteration method (see {{Cite|Kr}}). Necessary  and sufficient conditions for the existence of a regularizing operator  are known (see {{Cite|Vi}}).

Next, suppose that not  only the right-hand side of \ref{eq1} but also the operator $A$ is given  approximately, so that instead of the exact initial data $(A,u_T)$ one  has $(A_h,u_\delta)$, where
$$
\rho_U(u_\delta,u_T) \leq \delta, \qquad 
h = \sup_{\text{$z \in F_1$, $\Omega[z] \neq 0$}} \frac{\rho_U(A_hz,Az)}{\Omega[z]^{1/2}} < \infty.
$$
Under  these conditions the procedure for obtaining an approximate solution is  the same, only instead of $M^\alpha[z,u_\delta]$ one has to consider  the functional
$$
M^\alpha[z,u_\delta,A_h] = \rho_U^2(A_hz,u_\delta) + \alpha\Omega[z],
$$
and the parameter $\alpha$ can be determined, for example, from the relation (see {{Cite|TiAr}})
$$
\rho_U^2(A_hz,u_\delta) = \bigl( \delta + h \Omega[z_\alpha]^{1/2} \bigr)^2.
$$

If  \ref{eq1} has an infinite set of solutions, one introduces the concept  of a normal solution. Suppose that $Z$ is a normed space. Then one can  take, for example, a solution $\bar{z}$ for which the deviation in norm  from a given element $z_0 \in Z$ is minimal, that is,
$$
\norm{\bar{z} - z_0}_Z = \inf_{z \in Z} \norm{z - z_0}_Z .
$$
An  approximation to a normal solution that is stable under small changes  in the right-hand side of \ref{eq1} can be found by the regularization  method described above. The class of problems with infinitely many  solutions includes degenerate systems of linear algebraic equations.  So-called badly-conditioned systems of linear algebraic equations can be  regarded as systems obtained from degenerate ones when the operator $A$  is replaced by its approximation $A_h$. As a normal solution of a  corresponding degenerate system one can take a solution $z$ of minimal  norm $\norm{z}$. In the smoothing functional one can take for  $\Omega[z]$ the functional $\Omega[z] = \norm{z}^2$. Approximate  solutions of badly-conditioned systems can also be found by the  regularization method with $\Omega[z] = \norm{z}^2$ (see  {{Cite|TiAr}}).

Similar methods can be used to solve a  Fredholm integral equation of the second kind in the spectrum, that is,  when the parameter $\lambda$ of the equation is equal to one of the  eigen values of the kernel.

==Instability problems in the minimization of functionals.==
 
A  number of problems important in practice leads to the minimization of  functionals $f[z]$. One distinguishes two types of such problems. In the  first class one has to find a minimal (or maximal) value of the  functional. Many problems in the design of optimal systems or  constructions fall in this class.  For such problems it is irrelevant on  what elements the required minimum is attained. Therefore, as  approximate solutions of such problems one can take the values of the  functional $f[z]$ on any minimizing sequence $\set{z_n}$.

In  the second type of problems one has to find elements $z$ on which the  minimum of $f[z]$ is attained.  They are called problems of minimizing  over the argument. E.g., the minimizing sequences may be divergent. In  these problems one cannot take as approximate solutions the elements of  minimizing sequences.  Such problems are called unstable or ill-posed.  These include, for example, problems of optimal control, in which the  function to be optimized (the object function) depends only on the phase  variables.

Suppose that $f[z]$ is a continuous  functional on a metric space $Z$ and that there is an element $z_0 \in  Z$ minimizing $f[z]$. A minimizing sequence $\set{z_n}$ of $f[z]$ is  called regularizing if there is a compact set $\hat{Z}$ in $Z$  containing $\set{z_n}$. If the minimization problem for $f[z]$ has a  unique solution $z_0$, then a regularizing minimizing sequence converges  to $z_0$, and under these conditions it is sufficient to exhibit  algorithms for the construction of regularizing minimizing sequences.  This can be done by using stabilizing functionals $\Omega[z]$.

Let  $\Omega[z]$ be a stabilizing functional defined on a set $F_1 \subset  Z$, let $\inf_{z \in F_1}f[z] = f[z_0]$ and let $z_0 \in F_1$.  Frequently, instead of $f[z]$ one takes its $\delta$-approximation  $f_\delta[z]$ relative to $\Omega[z]$, that is, a functional such that  for every $z \in F_1$,
$$
\abs{f_\delta[z] - f[z]} \leq \delta\Omega[z].
$$
Then for any $\alpha > 0$ the problem of minimizing the functional
$$
M^\alpha[z,f_\delta] = f_\delta[z] + \alpha \Omega[z]
$$
over the argument is stable.

Let  $\set{\delta_n}$ and $\set{\alpha_n}$ be null-sequences such that  $\delta_n/\alpha_n \leq q < 1$ for every $n$, and let  $\set{z_{\alpha_n,\delta_n}} $ be a sequence of elements minimizing  $M^{\alpha_n}[z,f_{\delta_n}]$. This is a regularizing minimizing  sequence for the functional $f_\delta[z]$ (see {{Cite|TiAr}}),  consequently, it converges as $n \rightarrow \infty$ to an element  $z_0$. As approximate solutions of the problems one can then take the  elements $z_{\alpha_n,\delta_n}$.

Similarly approximate solutions of ill-posed problems in optimal control can be constructed.

In  applications ill-posed problems often occur where the initial data  contain random errors. For the construction of approximate solutions to  such classes both deterministic and probability approaches are possible  (see {{Cite|TiAr}}, {{Cite|LaVa}}).

====References====
{|
|-
|  valign="top"|{{Ref|Ar}}||valign="top"| V.Ya. Arsenin, "On a method for  obtaining approximate solutions to convolution integral equations of the  first kind" ''Proc. Steklov Inst. Math.'', '''133''' (1977) pp. 31–48  ''Trudy Mat. Inst. Steklov.'', '''133''' (1973) pp. 33–51
|-
|  valign="top"|{{Ref|Ba}}||valign="top"| A.B.  Bakushinskii, "A general  method for constructing regularizing algorithms for a linear ill-posed  equation in Hilbert space" ''USSR Comp. Math. Math. Phys.'', '''7''' : 3  (1968) pp. 279–287 ''Zh.  Vychisl. Mat. i Mat. Fiz.'', '''7''' : 3  (1967) pp.  672–677
|-
|  valign="top"|{{Ref|GoLeYa}}||valign="top"| A.V.  Goncharskii, A.S.  Leonov, A.G. Yagoda, "On the residual principle for solving nonlinear  ill-posed problems" ''Soviet Math. Dokl.'', '''15''' (1974) pp. 166–168  ''Dokl. Akad. Nauk SSSR'', '''214''' : 3 (1974) pp. 499–500
|-
|  valign="top"|{{Ref|Iv}}||valign="top"| V.K. Ivanov, "On ill-posed  problems" ''Mat. Sb.'', '''61''' : 2 (1963) pp.  211–223 (In Russian)
|-
|  valign="top"|{{Ref|Iv2}}||valign="top"| V.K. Ivanov, "On linear  problems which are not well-posed" ''Soviet Math. Dokl.'', '''3'''  (1962) pp. 981–983 ''Dokl. Akad. Nauk SSSR'', '''145''' : 2 (1962) pp.  270–272
|-
| valign="top"|{{Ref|Kr}}||valign="top"| A.V.  Kryanev, "The solution of incorrectly posed problems by methods of  successive approximations" ''Soviet Math. Dokl.'', '''14''' (1973) pp.  673–676 ''Dokl. Akad. Nauk SSSR'', '''210''' : 1 pp.  20–22
|-
|  valign="top"|{{Ref|La}}||valign="top"| M.M. [M.A.  Lavrent'ev]  Lavrentiev, "Some improperly posed problems of mathematical physics",  Springer (1967) (Translated from Russian)
|-
| valign="top"|{{Ref|LaLi}}||valign="top"| R. Lattes, J.L. Lions, "Méthode de quasi-réversibilité et applications", Dunod (1967)
|-
|  valign="top"|{{Ref|LaVa}}||valign="top"| M.M.  Lavrent'ev, V.G.  Vasil'ev, "The posing of certain improper problems of mathematical  physics" ''Sib. Math. J.'', '''7''' : 3 (1966) pp.  450–463 ''Sibirsk.  Mat. Zh.'', '''7''' : 3 (1966) pp.  559–576
|-
|  valign="top"|{{Ref|Ti}}||valign="top"| A.N. Tikhonov, "Solution of  incorrectly formulated problems and the regularization method" ''Soviet  Math. Dokl.'', '''4''' (1963) pp. 1035–1038 ''Dokl. Akad. Nauk SSSR'',  '''151''' : 3 (1963) pp.  501–504
|-
|  valign="top"|{{Ref|Ti2}}||valign="top"| A.N. Tikhonov, "Regularization  of incorrectly posed problems" ''Soviet Math. Dokl.'', '''4''' (1963)  pp. 1624–1627 ''Dokl. Akad. Nauk SSSR'', '''153''' : 1 (1963) pp. 49–52
|-
|  valign="top"|{{Ref|Ti3}}||valign="top"| A.N. Tikhonov, "On stability of  inverse problems" ''Dokl. Akad. Nauk SSSR'', '''39''' : 5 (1943) pp.  176–179 (In Russian)
|-
|  valign="top"|{{Ref|Ti4}}||valign="top"| A.N. Tikhonov, "On the stability  of the functional optimization problem" ''USSR Comp. Math. Math.  Phys.'', '''6''' : 4 (1966) pp. 28–33 ''Zh.  Vychisl. Mat. i Mat.  Fiz.'', '''6''' : 4 (1966) pp.  631–634
|-
|  valign="top"|{{Ref|TiAr}}||valign="top"| A.N. Tikhonov, V.I. [V.I.  Arsenin] Arsenine, "Solution of ill-posed problems", Winston (1977)  (Translated from Russian)
|-
|  valign="top"|{{Ref|Vi}}||valign="top"| V.A. Vinokurov, "On the  regularization of discontinuous mappings" ''USSR Comp. Math.  Math.  Phys.'', '''11''' : 5 (1971) pp. 1–21 ''Zh. Vychisl. Mat. i Mat. Fiz.'',  '''11''' : 5 (1971) pp.  1097–1112
|-
|}

====Comments====

The  idea of conditional well-posedness was also found by B.L. Phillips  {{Cite|Ph}}; the expression "Tikhonov well-posed" is not widely used in  the West.

Other problems that lead to ill-posed  problems in the sense described above are the [[Dirichlet  problem|Dirichlet problem]] for the wave equation, the  non-characteristic [[Cauchy problem|Cauchy problem]] for the heat  equation, the initial boundary value problem for the backward[[Heat  equation|heat equation]], inverse scattering problems ({{Cite|CoKr}}),  identification of parameters (coefficients) in partial differential  equations from over-specified data ({{Cite|Ba2}}, {{Cite|EnGr}}), and  computerized tomography ({{Cite|Na2}}).

If $A$ is a  bounded linear operator between Hilbert spaces, then, as also mentioned  above, regularization operators can be constructed via[[Spectral  theory|spectral theory]]: If $U(\alpha,\lambda) \rightarrow 1/\lambda$  as $\alpha \rightarrow 0$, then under mild assumptions,  $U(\alpha,A^*A)A^*$ is a regularization operator (cf. {{Cite|Gr}}); for  choices of the regularization parameter leading to optimal convergence  rates for such methods see {{Cite|EnGf}}. For $U(\alpha,\lambda) =  1/(\alpha+\lambda)$, the resulting method is called Tikhonov  regularization: The regularized solution $z_\alpha^\delta$ is defined  via $(\alpha I + A^*A)z = A^*u_\delta$. A variant of this method in  Hilbert scales has been developed in {{Cite|Na}} with parameter choice  rules given in {{Cite|Ne}}. The parameter choice rule discussed in the  article given by $\rho_U(Az_\alpha^\delta,u_\delta) = \delta$ is called  the discrepancy principle ({{Cite|Mo}}), or often the Morozov  discrepancy principle.

====References==== 
{|
|-
| valign="top"|{{Ref|Ba2}}||valign="top"| J. Baumeister, "Stable solution of inverse problems", Vieweg (1986)
|-
| valign="top"|{{Ref|Ba3}}||valign="top"| H.P. Baltes (ed.), ''Inverse source problems in optics'', Springer (1978)
|-
| valign="top"|{{Ref|Ba4}}||valign="top"| H.P. Baltes (ed.), ''Inverse scattering problems in optics'', Springer (1980)
|-
|  valign="top"|{{Ref|BaGi}}||valign="top"| G. Backus, F. Gilbert, "The  resolving power of gross earth data" ''Geophys. J.  R. Astr.  Soc.'',  '''16''' (1968)
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|  valign="top"|{{Ref|BeBlSt}}||valign="top"| J.V. Beck, B.  Blackwell,  C.R. StClair, "Inverse heat conduction: ill posed problems", Wiley  (1985)
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| valign="top"|{{Ref|BoJo}}||valign="top"| W.M.  Boerner, A.K.  (eds.) Jordan, "Inverse methods in electromagnetics"  ''IEEE Trans.  Antennas Propag.'', '''2''' (1981)
|-
| valign="top"|{{Ref|Ca}}||valign="top"| J.R. Cann on, "The one-dimensional heat equation", Addison-Wesley (1984)
|-
| valign="top"|{{Ref|CaSt}}||valign="top"| A. Carasso, A.P. Stone, "Improperly posed boundary value problems", Pitman (1975)
|-
|  valign="top"|{{Ref|Co}}||valign="top"| A.M. Cormak, "Representation of a  function by its line integrals with some radiological applications"  ''J. Appl. Phys.'', '''34''' (1963)
|-
|  valign="top"|{{Ref|Co2}}||valign="top"| L. Colin, "Mathematics of  profile inversion", ''Proc. Workshop Ames Res. Center, June 12–16,  1971'', '''TM-X-62.150''', NASA
|-
| valign="top"|{{Ref|CoKr}}||valign="top"| D.L. Colton, R. Kress, "Integral equation methods in scattering theory", Wiley (1983)
|-
|  valign="top"|{{Ref|EnGf}}||valign="top"| H.W. Engl, H.  Gfrerer, "A  posteriori parameter choice for general regularization methods for  solving linear ill-posed problems" ''Appl. Num. Math.'', '''4''' (1988)  pp. 395–417
|-
| valign="top"|{{Ref|EnGr}}||valign="top"|  H.W. Engl (ed.)  C.W. Groetsch (ed.), ''Inverse and ill-posed  problems'', Acad. Press (1987)
|-
|  valign="top"|{{Ref|Gr}}||valign="top"| C.W. Groetsch, "The theory of  Tikhonov regularization for Fredholm equations of the first kind",  Pitman (1984)
|-
| valign="top"|{{Ref|Gr2}}||valign="top"|  C.W. Groetsch, "The theory of Tikhonov regularization for Fredholm  equations of the first kind", Pitman (1984)
|-
| valign="top"|{{Ref|He}}||valign="top"| G.T. Herman (ed.), ''Image reconstruction from projections'', Springer (1979)
|-
|  valign="top"|{{Ref|HeNa}}||valign="top"| G.T. Herman (ed.)  F. Natterer  (ed.), ''Mathematical aspects of computerized tomography, Proc.  Oberwolfach, February 10–16, 1980'', Springer (1981)
|-
|  valign="top"|{{Ref|Jo}}||valign="top"| F. John, "Continuous dependence  on data for solutions of partial differential equations with a  prescribed bound" ''Comm. Pure Appl. Math.'', '''13''' (1960) pp.  551–585
|-
| valign="top"|{{Ref|Ka}}||valign="top"| M. Kac,  "Can one hear the shape of a drum?"  ''Amer. Math. Monthly'', '''73'''  (1966) pp.  1–23
|-
|  valign="top"|{{Ref|LaRoSh}}||valign="top"| M.H.  Lavrent'ev, V.G.  Romanov, S.P. Shishalskii, "Ill-posed problems of mathematical physics  and analysis", Amer. Math. Soc.  (1986) (Translated from Russian)
|-
|  valign="top"|{{Ref|Mo}}||valign="top"| V.A. Morozov, "Methods for  solving incorrectly posed problems", Springer (1984) (Translated from  Russian)
|-
| valign="top"|{{Ref|Na}}||valign="top"| F.  Natterer, "Error bounds for Tikhonov regularization in Hilbert scales"  ''Applic.  Analysis'', '''18''' (1984) pp.  29–37
|-
| valign="top"|{{Ref|Na2}}||valign="top"| F. Natterer, "The mathematics of computerized tomography", Wiley (1986)
|-
|  valign="top"|{{Ref|Ne}}||valign="top"| A. Neubauer, "An a-posteriori  parameter choice for Tikhonov regularization in Hilbert scales leading  to optimal convergence rates" ''SIAM J. Numer.  Anal.'', '''25''' (1988)  pp. 1313–1326
|-
| valign="top"|{{Ref|Pa}}||valign="top"| L.E. Payne, "Improperly posed problems in partial differential equations", SIAM (1975)
|-
|  valign="top"|{{Ref|Ph}}||valign="top"| B.L. Phillips, "A technique for  the numerical solution of certain integral equations of the first kind"  ''J. ACM'', '''9''' (1962) pp.  84–97
|-
|}