On truncated spectral regularization for an ill-posed evolution equation
M. Thamban Nair

TL;DR
This paper investigates spectral truncation as a regularization technique for ill-posed parabolic final value problems, providing error estimates under noise and comparing it to Lavrentieve's method, highlighting its lack of saturation.
Contribution
It introduces spectral truncation as a regularization method for ill-posed evolution equations and derives error estimates under general source conditions.
Findings
Spectral truncation yields error estimates without saturation.
Comparison shows spectral truncation performs favorably against Lavrentieve's method.
Error bounds are established under noisy data conditions.
Abstract
In this note we consider the {\it spectral truncation} as the regularization for an ill-posed non-homogeneous parabolic final value problem, and obtain error estimates under a genral source condition when the data, which consist of the non-homogeneous term as well as the final value, are noisy. The resulting error estimate is compared with the corresponding estimate under the Lavrentieve method, and showed that the truncation method has no index of saturation.
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On truncated spectral regularization for an
ill-posed evolution equation
M.Thamban Nair
Department of Matheamtics, IIT Madras, Chennai, INDIA
Abstract.
In this note we consider the spectral truncation as the regularization for an ill-posed non-homogeneous parabolic final value problem, and obtain error estimates under a genral source condition when the data, which consist of the non-homogeneous term as well as the final value, are noisy. The resulting error estimate is compared with the corresponding estimate under the Lavrentieve method, and showed that the truncation method has no index of saturation.
AMS Subject Classification: 35K05, 35K99, 47J06, 47H10
Keywords: Ill-posed problems, Evolution equations, Semigroup Regularization, Parameter choice.
1. Introduction
Let be a Hilbert space and be a densely defined positive self adjoint (unbounded) operator. Given and , consider the initial value problem (IVP):
[TABLE]
It is known [5] that if is a solution for (1), then
[TABLE]
Here, for , the operator is defined by
[TABLE]
We may also recall (see, e.g., [7]) that, corresponding to any , the operator is defined by
[TABLE]
where
[TABLE]
In fact, by spectral theorem, for any continuous function ,
[TABLE]
where
[TABLE]
and is a positive self adjoint operator. we have
[TABLE]
In particular,
[TABLE]
where
[TABLE]
and
[TABLE]
Thus, is a self adjoint operator which is also bounded below, so that it is onto, and hence
- (1)
is one-one, onto, and has bounded inverse, namely, ; 2. (2)
; 3. (3)
is a strongly continuous semigroup on with
[TABLE]
Further (see [5]), is the infinitesimal generator of , i.e.,
[TABLE]
and
In this paper, we are inerested in the final value problem (FVP), that is the problem of solving
[TABLE]
for a known for some .
Suppose is a solution of (3)-(4). Then, from (2), we have
[TABLE]
Thus,
[TABLE]
Since the operator is unbounded, the above representation of shows that
small error in the data can lead to large error in the solution .
In other words, the problem of solving the FVP (3)-(4) is ill-posed.
Definition 1**.**
If and are such that
[TABLE]
belongs to , then defined by
[TABLE]
is called the mild solution of the FVP (3).
It is to be observed that a mild solution of the FVP need not be a solution of the FVP. In fact, we have the following characterization of the solution of the FVP.
Theorem 2**.**
[1]** Let and let be defined by , . Then is a solution of the FVP
[TABLE]
if and only if .
Proof.
Note that, for and ,
[TABLE]
Since is the infinitesimal generator of the semigroup , it follows that
[TABLE]
and in that case . ∎
In view of the representation (5) of the mild solution of the FVP, the problem of finding with can be posed as a problem of solving the ill-posed operator equation
[TABLE]
where is the bounded operator defined by
[TABLE]
and Note that
- •
is an injective bounded self adjoint operator,
- •
is dense in , and
- •
is not continuous.
In order to obtain stable approximation for the mild solution given in (5) for the FVP, we shall apply the so called truncated spectral regularization, and obtain error estimate under a general source condition. The obtained rate will be compared with the rate resulting from the Lavrentieve method [4].
2. Truncated Spectral Regularization (TRS)
For and , the mild solution, as in (5), of the FVP has the spectral representation
[TABLE]
whenever
[TABLE]
belongs to
[TABLE]
The above representation (7) involving integral over the whole of suggests that a truncation of the inteegral would give a reasonable approximation for . That is the idea in truncated spectral regularization [6, 1, 2].
Definition 3**.**
The truncated spectral regularized solution for the mild solution is defined by
[TABLE]
for each .
The following theorem shows that is an approximation of for large .
Theorem 4**.**
Under the assumption ,
[TABLE]
Proof.
Since
[TABLE]
we obtain
[TABLE]
∎
Next, we show that is, in fact, stable under perturbations in the data .
Suppose and are the noisy data, in place of the actual data and , respectively. Let
[TABLE]
where
[TABLE]
Theorem 5**.**
Let and . The for each and ,
[TABLE]
Proof.
We observe that
[TABLE]
so that
[TABLE]
Note that
[TABLE]
so that
[TABLE]
Thus, we obtain the required result. ∎
We see that the map
[TABLE]
define a norm on . Thus, Theorem 5 shows that the truncated spectral regularized solution is stable under perturbations in the data with respect to the above norm .
3. Convergence and error estimates
3.1. Convergence
From Theorem 5, the following theorem is immediate.
Theorem 6**.**
Let and be such that
[TABLE]
for some . Then for each and ,
[TABLE]
Further, if
[TABLE]
for , then and
[TABLE]
3.2. Estimates under general source condition
For obtaining error estimates it is required to assume certain smoothness assumptions on the solution, the so called source conditions. For this purpose, we consider a general condition of the form
[TABLE]
where the function is continuous and for each ,
[TABLE]
Note that the condition (16) is equivalent to
[TABLE]
At this point one may recall that in [6], Tuan has considered the source conditions of the forms,
[TABLE]
for respectively. Note that the source conditions in (11) are special cases of (10) obtained by the choices
[TABLE]
respectively.
Theorem 7**.**
Suppose and are such that for each and there exists a monotonically increasing continuous function such that
- (i)
* as ,* 2. (ii)
,
Let be such that . Then
[TABLE]
where and as .
Proof.
Recall from Definition 1 that Hence,
[TABLE]
Since is monotonically increasing, from the above, we obtain
[TABLE]
By the assumption,
[TABLE]
Hence, taking such that
[TABLE]
the inequality (12) leads to
[TABLE]
where and as . ∎
Remark 8**.**
Recently, Jana [1] and Jana and Nair [2] used similar general source condition, but based on the data instead of the mild solution .
Combining the last two theorems, we obtain the following.
Theorem 9**.**
Suppose and are noisy data such that
[TABLE]
for some noise level . Then
[TABLE]
If and are as in Theorem 7, then we have
[TABLE]
4. Parameter Choice Strategy
In Theorem 9, we obtained an estimate for the error a smoothness assumption on . Now, we choose depending on and such that the obtained estimate converges to [math] as .
Theorem 10**.**
or , let , and let
[TABLE]
Then
[TABLE]
In particular,
[TABLE]
Proof.
Note that
[TABLE]
Thus, for the choice of , Theorem 9 implies
[TABLE]
Since as .
[TABLE]
∎
4.1. A special choice of the source condition
Suppose for some . Then
[TABLE]
for some . Thus,
[TABLE]
and
[TABLE]
Note that the funtion
[TABLE]
satisfies the properties (1)-(3) in Theorem 7. Thus, by Theorem 9,
[TABLE]
Theorem 11**.**
Suppose for some and
[TABLE]
Then
[TABLE]
Further, taking
[TABLE]
we have
[TABLE]
5. Comparison with Lavrentiev regularization
Recall that the operator defined by
[TABLE]
is injective, continuous, self adjoint, with dense in . Let be the solution of (LABEL:op-eq-1), that is,
[TABLE]
Let be the Lavrentive regularized solution, i.e.,
[TABLE]
Then, from the standard theory [3, 4], we know that
[TABLE]
and
[TABLE]
Next, suppose
[TABLE]
for some . Equivalently,
[TABLE]
Then we have the estimate
[TABLE]
and consequently,
[TABLE]
Note that
[TABLE]
Thus, the estimate in (17) is same as (13) for the choice of as in (18). However, the estimate in (13) is valid for all , whereas (17) is valid only for .
Acknowledgements: This work is completed while the author was a visiting mathematician at Sun Yat-sen University, Guanzhou, China, during the period June 13 to July 8, 2019. The support and the warm hospitality received from Prof. Hongqi Yang are gratefully acknowledged.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] A. Jana, Regularization of Ill-Posed Nonhomogeneous Parabolic Problems , Ph.D. Thesis, IIT Madras, August, 2018.
- 2[2] A. Jana and M.T. Nair, Truncated spectral regularization for an ill-posed nonhomogeneous parabolic problem, J. Math. Anal. Appl. Vol. 438, Issue 1, 1 June 2016, Pages 351-372
- 3[3] H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems , Dordrecht, Kluwer, 1993.
- 4[4] M. T. Nair and U. Tautenhahn, Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions, Zeitschrift für Analysis und ihre Anwendungen , Journal for Analysis and its Applications Volume 23 (2004), No. 1, 167-185.
- 5[5] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations , Springer - Verlag, New York, 1983.
- 6[6] N.H. Tuan, D.D. Trong, and H.O. Minh City, A simple regularization method for the ill-posed evolution equation, Czechoslovak Mathematical Journal , Vol. 61 (2011), No. 1, 85-95
- 7[7] K. Yosida, Functional Analysis , Springer,
