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| − | This is being TeXed by --[[User:Jjg|Jjg]] 12:29, 22 April 2012 (CEST)
| + | Typos fixed during TeXing |
| − | | + | * In paragraph starting "The so-called smoothing functional...", replaced $R_2[u,\alpha]$ by $R_2(u,\alpha)$ |
| − | ''incorrectly-posed problems, improperly-posed problems''
| + | Typographical changes |
| − | | + | * Removed some redundant curly brackets (braces), eg, $\{\Omega[z]\}^{1/2}$ becomes $\Omega[z]^{1/2}$ |
| − | Problems for which at least one of the conditions below, which characterize well-posed problems, is violated. The problem of determining a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501201.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501202.png" /> (with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501203.png" />) from "initial data" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501204.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501205.png" /> (with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501206.png" />) is said to be well-posed on the pair of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501207.png" /> if: a) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501208.png" /> there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i0501209.png" />; b) the solution is uniquely determined; and c) the problem is stable on the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012010.png" />, i.e.: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012011.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012012.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012013.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012014.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012017.png" />.
| + | * Some complicated inline formulae are now displayed, eg, the integral whose minimum is the solution to the spline problem. |
| − | | + | Editorial changes |
| − | 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.
| + | * ''the expression "Tikhonov well-posed" is not used in the West'' becomes ''the expression "Tikhonov well-posed" is not widely used in the West'', since Google gives 900 hits for this exact phrase. |
| − | | + | * Last line of comments, added remark that "the discrepancy principle" of often called "the Morozov discrepancy principle. |
| − | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012018.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012019.png" />; 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|TiArAr}}).
| + | --[[User:Jjg|Jjg]] 13:34, 23 April 2012 (CEST) |
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| − | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012020.png" /> be a characteristic quantity of the phenomenon (or object) to be studied. In a physical experiment the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012021.png" /> is frequently inaccessible to direct measurement, but what is measured is a certain transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012022.png" /> (also called outcome). For the interpretation of the results it is necessary to determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012023.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012024.png" />, that is, to solve the equation
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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| − | Problems of solving an equation (1) 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.
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| − | Suppose that in a mathematical model for some physical experiments the object to be studied (the phenomenon) is characterized by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012026.png" /> (a function, a vector) belonging to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012027.png" /> of possible solutions in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012028.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012029.png" /> is inaccessible to direct measurement and that what is measured is a transform, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012032.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012033.png" /> under the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012034.png" />. Evidently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012036.png" /> is the operator inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012037.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012038.png" /> is obtained by measurement, it is known only approximately. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012039.png" /> be this approximate value. Under these conditions the question can only be that of finding a "solution" of the equation
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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| − | approximating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012041.png" />.
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| − | In many cases the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012042.png" /> is such that its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012043.png" /> is not continuous, for example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012044.png" /> is a completely-continuous operator in a Hilbert space, in particular an integral operator of the form
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012045.png" /></td> </tr></table>
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| − | Under these conditions one cannot take, following classical ideas, an exact solution of (2), that is, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012046.png" />, as an approximate "solution" to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012047.png" />. In fact: a) such a solution need not exist on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012048.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012049.png" /> need not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012050.png" />; and b) such a solution, if it exists, need not be stable under small changes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012051.png" /> (due to the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012052.png" /> is not continuous) and, consequently, need not have a physical interpretation. The problem (2) then is ill-posed.
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| − | ==Numerical methods for solving ill-posed problems.==
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| − | For ill-posed problems of the form (1) 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}}).
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| − | The selection method. In some cases an approximate solution of (1) can be found by the selection method. It consists of the following: From the class of possible solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012053.png" /> one selects an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012054.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012055.png" /> approximates the right-hand side of (1) with required accuracy. For the desired approximate solution one takes the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012056.png" />. The question arises: When is this method applicable, that is, when does
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012057.png" /></td> </tr></table>
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| − | imply that
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012058.png" /></td> </tr></table>
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012059.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012060.png" />? This holds under the conditions that the solution of (1) is unique and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012061.png" /> 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|LaLa}}). As applied to (1), a problem is said to be conditionally well-posed if it is known that for the exact value of the right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012062.png" /> there exists a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012063.png" /> of (1) belonging to a given compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012064.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012065.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012066.png" />, and if instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012067.png" /> an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012068.png" /> is known such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012070.png" />, then as an approximate solution of (1) with right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012071.png" /> one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012072.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012073.png" />, <img al
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| − | ign="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012074.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012075.png" />.
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| − | In many cases the approximately known right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012076.png" /> does not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012077.png" />. Under these conditions equation (1) 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 (1) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012078.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012079.png" /> that minimizes for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012080.png" /> the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012081.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012082.png" /> (see {{Cite|Iv2}}). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012083.png" /> is compact, then a quasi-solution exist for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012084.png" />, and if in addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012085.png" />, then a quasi-solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012086.png" /> coincides with the classical (exact) solution of (1). The existence of quasi-solutions is guaranteed only when the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012087.png" /> of possible solutions is compact.
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| − | The regularization method. For a number of applied problems leading to (1) a typical situation is that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012088.png" /> of possible solutions is not compact, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012089.png" /> is not continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012090.png" />, and changes of the right-hand side of (1) connected with the approximate character can cause the solution to go out of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012091.png" />. 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 (1) which are stable under small changes of the data. In this context, both the right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012092.png" /> and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012093.png" /> should be among the data.
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| − | In what follows, for simplicity of exposition it is assumed that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012094.png" /> is known exactly. At the basis of the approach lies the concept of a regularizing operator (see {{Cite|Ti2}}, {{Cite|TiArAr}}). An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012095.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012096.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012097.png" /> is said to be a regularizing operator for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012098.png" /> (in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i05012099.png" />) if it has the following properties: 1) there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120100.png" /> such that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120101.png" /> is defined for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120103.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120105.png" />; and 2) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120106.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120107.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120108.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120110.png" />.
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| − | Sometimes it is convenient to use another definition of a regularizing operator, comprising the previous one. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120111.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120112.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120113.png" />, depending on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120114.png" />, is said to be a regularizing operator (or regularization operator) for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120115.png" /> (in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120116.png" />) if it has the following properties: 1) there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120117.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120118.png" /> is defined for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120119.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120120.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120121.png" />; and 2) there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120123.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120124.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120125.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120127.png" />, then <img align="absmiddle" border
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| − | ="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120128.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120129.png" />. In this definition it is not assumed that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120130.png" /> is globally single-valued.
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120131.png" />, then as an approximate solution of (1) with an approximately known right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120132.png" /> one can take the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120133.png" /> obtained by means of the regularizing operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120134.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120135.png" /> is compatible with the error of the initial data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120136.png" /> (see {{Cite|Ti}}, {{Cite|Ti2}}, {{Cite|TiArAr}}). This is said to be a regularized solution of (1). The numerical parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120137.png" /> is called the regularization parameter. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120138.png" />, the regularized approximate solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120139.png" /> tends (in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120140.png" />) to the exact solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120141.png" />.
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| − | Thus, the task of finding approximate solutions of (1) that are stable under small changes of the right-hand side reduces to: a) finding a regularizing operator; and b) determining the regularization parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120142.png" /> from additional information on the problem, for example, the size of the error with which the right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120143.png" /> is given.
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| − | The construction of regularizing operators. It is assumed that the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120144.png" /> has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120145.png" />. Suppose that instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120146.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120147.png" /> is solved and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120148.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120149.png" />, the approximate solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120150.png" /> is looked for in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120151.png" /> of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120152.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120153.png" />. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120154.png" /> is the set of possible solutions. As an approximate solution one cannot take an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120155.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120156.png" />, since such a "solution" is not unique and is, generally speaking, not continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120157.png" />. As a selection principle for the possible solutions ensuring that one obtains an element (or elements) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120158.png" /> depending continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120159.png" /> a
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| − | nd tending to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120160.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120161.png" />, one uses the so-called variational principle (see {{Cite|Ti}}). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120162.png" /> be a continuous non-negative functional defined on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120163.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120164.png" /> that is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120165.png" /> and is such that: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120166.png" />; and b) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120167.png" /> the set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120168.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120169.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120170.png" />, is compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120171.png" />. Functionals having these properties are said to be stabilizing functionals for problem (1). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120172.png" /> be a stabilizing functional defined on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120173.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120174.png" />. (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120175.png" /> can be the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120176.png" />.) Among the elements of <img align="absmid
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| − | dle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120177.png" /> one looks for one (or several) that minimize(s) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120178.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120179.png" />. The existence of such an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120180.png" /> can be proved (see {{Cite|TiArAr}}). It can be regarded as the result of applying a certain operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120181.png" /> to the right-hand side of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120182.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120183.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120184.png" /> is a regularizing operator for equation (1). In practice the search for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120185.png" /> can be carried out in the following manner: under mild additional restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120186.png" /> (quasi-monotonicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120187.png" />, see {{Cite|TiArAr}}) it can be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120188.png" /> is attained on elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120189.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120190.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120191.png" /> is a solution to the problem of minimizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120192.png" /> given <img align="absmiddle" bor
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| − | der="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120193.png" />, that is, a solution of a problem of conditional extrema, which can be solved using Lagrange's multiplier method and minimization of the functional
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120194.png" /></td> </tr></table>
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| − | For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120195.png" /> one can prove that there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120196.png" /> minimizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120197.png" />. The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120198.png" /> is determined from the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120199.png" />. If there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120200.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120201.png" />, then the original variational problem is equivalent to that of minimizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120202.png" />, which can be solved by various methods on a computer (for example, by solving the corresponding Euler equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120203.png" />). The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120204.png" /> minimizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120205.png" /> can be regarded as the result of applying to the right-hand side of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120206.png" /> a certain operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120207.png" /> depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120208.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120209.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120210.png" /> is determined by the discrepancy relation <img align="absmiddle" border="0" src="/legacyima
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| − | ges/i/i050/i050120/i050120211.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120212.png" /> is a regularizing operator for (1). Equivalence of the original variational problem with that of finding the minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120213.png" /> holds, for example, for linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120214.png" />. For non-linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120215.png" /> this need not be the case (see {{Cite|GoLeYa}}).
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| − | The so-called smoothing functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120216.png" /> can be introduced formally, without connecting it with a conditional extremum problem for the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120217.png" />, and for an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120218.png" /> minimizing it sought on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120219.png" />. This poses the problem of finding the regularization parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120220.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120221.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120222.png" />, such that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120223.png" /> determining the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120224.png" /> is regularizing for (1). Under certain conditions (for example, when it is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120225.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120226.png" /> is a linear operator) such a function exists and can be found from the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120227.png" />. There are also other methods for finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120228.png" />.
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| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120229.png" /> be a class of non-negative non-decreasing continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120230.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120231.png" /> a solution of (1) with right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120232.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120233.png" /> a continuous operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120234.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120235.png" />. For any positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120236.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120237.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120238.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120239.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120240.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120241.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120242.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120243.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120244.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120245.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120246.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120247.png" /> for all <img align="absmiddle" border=
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| − | "0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120248.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120249.png" />.
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| − | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120250.png" /> is known, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120251.png" />, then in accordance with the preceding it is natural to determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120252.png" /> by the discrepancy, that is, from the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120253.png" />.
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| − | | |
| − | The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120254.png" /> is monotone and semi-continuous for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120255.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120256.png" /> is a linear operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120257.png" /> a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120258.png" /> a strictly-convex functional (for example, quadratic), then the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120259.png" /> is unique and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120260.png" /> is a single-valued function. Under these conditions, for every positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120261.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120262.png" />, there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120263.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120264.png" /> (see {{Cite|TiArAr}}).
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| − | | |
| − | However, for a non-linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120265.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120266.png" /> may have no solution (see {{Cite|GoLeYa}}).
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| − | | |
| − | The regularization method is closely connected with the construction of splines (cf. [[Spline|Spline]]). For example, the problem of finding a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120267.png" /> with piecewise-continuous second-order derivative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120268.png" /> that minimizes the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120269.png" /> and takes given values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120270.png" /> on a grid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120271.png" />, is equivalent to the construction of a spline of the second degree.
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| − | | |
| − | A regularizing operator can be constructed by spectral methods (see {{Cite|TiArAr}}, {{Cite|GoLeYa}}), by means of the classical integral transforms in the case of equations of convolution type (see {{Cite|Ar}}, {{Cite|TiArAr}}), 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}}).
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| − | | |
| − | Next, suppose that not only the right-hand side of (1) but also the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120272.png" /> is given approximately, so that instead of the exact initial data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120273.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120274.png" />, where
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| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120275.png" /></td> </tr></table>
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| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120276.png" /></td> </tr></table>
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| − | | |
| − | Under these conditions the procedure for obtaining an approximate solution is the same, only instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120277.png" /> one has to consider the functional
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| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120278.png" /></td> </tr></table>
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| − | | |
| − | and the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120279.png" /> can be determined, for example, from the relation (see {{Cite|TiArAr}})
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| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120280.png" /></td> </tr></table>
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| − | | |
| − | If (1) has an infinite set of solutions, one introduces the concept of a normal solution. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120281.png" /> is a normed space. Then one can take, for example, a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120282.png" /> for which the deviation in norm from a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120283.png" /> is minimal, that is,
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| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120284.png" /></td> </tr></table>
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| − | | |
| − | An approximation to a normal solution that is stable under small changes in the right-hand side of (1) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120285.png" /> is replaced by its approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120286.png" />. As a normal solution of a corresponding degenerate system one can take a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120287.png" /> of minimal norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120288.png" />. In the smoothing functional one can take for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120289.png" /> the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120290.png" />. Approximate solutions of badly-conditioned systems can also be found by the regularization method with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120291.png" /> (see {{Cite|TiArAr}}).
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| − | | |
| − | Similar methods can be used to solve a Fredholm integral equation of the second kind in the spectrum, that is, when the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120292.png" /> of the equation is equal to one of the eigen values of the kernel.
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| − | | |
| − | ==Instability problems in the minimization of functionals.==
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| − | A number of problems important in practice leads to the minimization of functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120293.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120294.png" /> on any minimizing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120295.png" />.
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| − | | |
| − | In the second type of problems one has to find elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120296.png" /> on which the minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120297.png" /> 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.
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| − | | |
| − | Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120298.png" /> is a continuous functional on a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120299.png" /> and that there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120300.png" /> minimizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120301.png" />. A minimizing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120302.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120303.png" /> is called regularizing if there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120304.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120305.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120306.png" />. If the minimization problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120307.png" /> has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120308.png" />, then a regularizing minimizing sequence converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120309.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120310.png" />.
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| − | | |
| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120311.png" /> be a stabilizing functional defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120312.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120313.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120314.png" />. Frequently, instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120315.png" /> one takes its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120317.png" />-approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120318.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120319.png" />, that is, a functional such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120320.png" />,
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| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120321.png" /></td> </tr></table>
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| − | | |
| − | Then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120322.png" /> the problem of minimizing the functional
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| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120323.png" /></td> </tr></table>
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| − | | |
| − | over the argument is stable.
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| − | | |
| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120324.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120325.png" /> be null-sequences such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120326.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120327.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120328.png" /> be a sequence of elements minimizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120329.png" />. This is a regularizing minimizing sequence for the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120330.png" /> (see {{Cite|TiArAr}}), consequently, it converges as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120331.png" /> to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120332.png" />. As approximate solutions of the problems one can then take the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120333.png" />.
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| − | | |
| − | Similarly approximate solutions of ill-posed problems in optimal control can be constructed.
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| − | | |
| − | 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|TiArAr}}, {{Cite|LaVa}}).
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| − | | |
| − | ====References====
| |
| − | {|
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| − | |-
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| − | | 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
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| − | |-
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| − | | 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
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| − | |-
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| − | | 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
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| − | |-
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| − | | valign="top"|{{Ref|Iv}}||valign="top"| V.K. Ivanov, "On ill-posed problems" ''Mat. Sb.'', '''61''' : 2 (1963) pp. 211–223 (In Russian)
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| − | |-
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| − | | 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
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| − | |-
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| − | | 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
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| − | |-
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| − | | valign="top"|{{Ref|LaLa}}||valign="top"| M.M. [M.A. Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics", Springer (1967) (Translated from Russian)
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| − | |-
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| − | | valign="top"|{{Ref|LaLi}}||valign="top"| R. Lattes, J.L. Lions, "Méthode de quasi-réversibilité et applications", Dunod (1967)
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| − | |-
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| − | | 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
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| − | |-
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| − | | 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
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| − | |-
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| − | | 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
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| − | |-
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| − | | 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)
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| − | |-
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| − | | 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
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| − | |-
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| − | | valign="top"|{{Ref|TiArAr}}||valign="top"| A.N. Tikhonov, V.I. [V.I. Arsenin] Arsenine, "Solution of ill-posed problems", Winston (1977) (Translated from Russian)
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| − | |-
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| − | | 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
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| − | |-
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| − | |}
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| − | | |
| − | | |
| − | | |
| − | ====Comments====
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| − | The idea of conditional well-posedness was also found by B.L. Phillips {{Cite|Ph}}; the expression "Tikhonov well-posed" is not used in the West.
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| − | | |
| − | 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}}).
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120334.png" /> is a bounded linear operator between Hilbert spaces, then, as also mentioned above, regularization operators can be constructed via [[Spectral theory|spectral theory]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120335.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120336.png" />, then under mild assumptions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120337.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120338.png" />, the resulting method is called Tikhonov regularization: The regularized solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120339.png" /> is defined via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120340.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050120/i050120341.png" /> is called the discrepancy principle ({{Cite|Mo}}).
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| − | ====References====
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| − | | valign="top"|{{Ref|Ba4}}||valign="top"| H.P. Baltes (ed.), ''Inverse scattering problems in optics'', Springer (1980)
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| − | | valign="top"|{{Ref|CoKr}}||valign="top"| D.L. Colton, R. Kress, "Integral equation methods in scattering theory", Wiley (1983)
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| − | | 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
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| − | | valign="top"|{{Ref|EnGr}}||valign="top"| H.W. Engl (ed.) C.W. Groetsch (ed.), ''Inverse and ill-posed problems'', Acad. Press (1987)
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| − | | valign="top"|{{Ref|Gr}}||valign="top"| C.W. Groetsch, "The theory of Tikhonov regularization for Fredholm equations of the first kind", Pitman (1984)
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| − | | valign="top"|{{Ref|Gr2}}||valign="top"| C.W. Groetsch, "The theory of Tikhonov regularization for Fredholm equations of the first kind", Pitman (1984)
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| − | | valign="top"|{{Ref|He}}||valign="top"| G.T. Herman (ed.), ''Image reconstruction from projections'', Springer (1979)
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| − | | 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)
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| − | | valign="top"|{{Ref|Mo}}||valign="top"| V.A. Morozov, "Methods for solving incorrectly posed problems", Springer (1984) (Translated from Russian)
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| − | | valign="top"|{{Ref|Na}}||valign="top"| F. Natterer, "Error bounds for Tikhonov regularization in Hilbert scales" ''Applic. Analysis'', '''18''' (1984) pp. 29–37
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| − | | valign="top"|{{Ref|Na2}}||valign="top"| F. Natterer, "The mathematics of computerized tomography", Wiley (1986)
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| − | | 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
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| − | | valign="top"|{{Ref|Pa}}||valign="top"| L.E. Payne, "Improperly posed problems in partial differential equations", SIAM (1975)
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| − | | 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
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| − | |}
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