On truncated spectral regularization for an ill-posed evolution equation

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.

Equations

$$\frac{d}{dt}u(t)+Au(t)=f(t),u(0)={\phi }_0.$$

$$u(t)=e^{-tA}{\phi }_0+{\int }_0^t{e}^{-(t-s)A}f(s)ds.$$

$$e^{-rA}\phi :={\int }_0^{\infty }{e}^{r\lambda }d{E}_{\lambda }\phi ,\phi \in H,\phi \in H.$$

$$e^{rA}\phi :={\int }_0^{\infty }{e}^{r\lambda }d{E}_{\lambda }\phi ,\phi \in H,\phi \in D(e^{rA}),$$

$$D(e^{rA}):=\{\phi \in H:{\int }_0^{\infty }{e}^{2r\lambda }d\Vert E_{\lambda }\phi {\Vert }^2<\infty \}.$$

$$g(A)\phi :={\int }_0^{\infty }g(\lambda )d{E}_{\lambda }\phi ,\phi \in D(g(A)),$$

$$D(g(A)):=\{\phi \in H:{\int }_0^{\infty }|g(\lambda ){|}^{2}d\Vert E_{\lambda }\phi {\Vert }^2<\infty \},$$

$$\Vert g(A)\phi {\Vert }^2:={\int }_0^{\infty }|g(\lambda ){|}^{2}d\Vert E_{\lambda }\phi {\Vert }^2.$$

$$e^{tA}\phi :={\int }_0^{\infty }{e}^{t\lambda }d{E}_{\lambda }\phi ,\phi \in D(e^{tA}),$$

$$D(e^{tA}):=\{\phi \in H:{\int }_0^{\infty }{e}^{2t\lambda }d\Vert E_{\lambda }\phi {\Vert }^2<\infty \}$$

$$\Vert e^{tA}\phi {\Vert }^2={\int }_0^{\infty }{e}^{2t\lambda }d\Vert E_{\lambda }\phi {\Vert }^2\}\ge \Vert \phi {\Vert }^2.$$

$$\Big{\|}\frac{S(t)\varphi-\varphi}{t}+A\varphi\Big{\|}\to 0\quad\hbox{as}\quad t\to 0.$$

$$-A\phi :=\underset{t\to 0}{\lim }\frac{S(t)\phi -\phi }{t},\phi \in D(-A)$$

$$\frac{d}{dt}u(t)+Au(t)$$

$$u(\tau )$$

$${\phi }_{\tau }$$

$$u(t)=e^{(\tau-t)A}\Big{[}\varphi(\tau)-\int_{t}^{\tau}e^{-(\tau-s)A}f(s)ds\Big{]}.$$

$$\psi (t):={\phi }_{\tau }-{\int }_t^{\tau }{e}^{-(\tau -s)A}f(s)ds\forall t\in [0,\tau )$$

$$u(t)=e^{(\tau -t)A}\psi (t)$$

$$u_t+Au(t)=f(t),u(\tau )=\varphi$$

$$\frac{u(t+h)-u(t)}{h}=\frac{e^{(\tau -t-h)A}\varphi -e^{(\tau -t)A}\varphi }{h}=\frac{e^{-hA}u(t)-u(t)}{h}.$$

$$\underset{h\to 0}{\lim }\frac{u(t+h)-u(t)}{h}exists\; iffu(t)\in D(-A)iff\varphi \in D(A{e}^{\tau A})\forall t\ge 0,$$

$${\mathcal{A}}_{t}u(t)=\psi (t),$$

$${\mathcal{A}}_t\phi :=e^{-(\tau -t)A}\phi ,\phi \in H,$$

$$u(t)={\int }_0^{\infty }{e}^{(\tau -t)\lambda }d{E}_{\lambda }(\psi (t))$$

$$\psi (t):={\phi }_{\tau }-{\int }_t^{\tau }{e}^{-(\tau -s)A}f(s)ds$$

$$D(e^{(\tau -t)A}):=\{\phi \in H:{\int }_0^{\infty }{e}^{2(\tau -t)\lambda }d\Vert E_{\lambda }\phi {\Vert }^2<\infty \}.$$

$$u_{\beta }(t)={\int }_0^{\beta }{e}^{(\tau -t)\lambda }d{E}_{\lambda }(\psi (t))$$

$$\Vert u(t)-u_{\beta }(t)\Vert \to 0as\beta \to \infty .$$

$$\Vert u(t){\Vert }^2={\int }_0^{\infty }{e}^{2(\tau -t)\lambda }d\Vert E_{\lambda }(\psi (t)){\Vert }^2<\infty ,$$

$$\Vert u(t)-u_{\beta }(t){\Vert }^2={\int }_{\beta }^{\infty }{e}^{2(\tau -t)\lambda }d\Vert E_{\lambda }(\psi (t)){\Vert }^2\to 0as\beta \to \infty .$$

Full text

AMS Subject Classification: 35K05, 35K99, 47J06, 47H10

Keywords: Ill-posed problems, Evolution equations, Semigroup Regularization, Parameter choice.

1. Introduction

Let $H$ be a Hilbert space and $A:D(A)\subseteq H\to H$ be a densely defined positive self adjoint (unbounded) operator. Given $\varphi_{0}\in H$ and $f\in L^{1}([0,\infty),H)$, consider the initial value problem (IVP):

[TABLE]

It is known [5] that if $u(\cdot)$ is a solution for (1), then

[TABLE]

Here, for $r>0$, the operator $e^{-rA}$ is defined by

[TABLE]

We may also recall (see, e.g., [7]) that, corresponding to any $r\in\mathbb{R}$, the operator $e^{rA}$ is defined by

[TABLE]

where

[TABLE]

In fact, by spectral theorem, for any continuous function $g:[0,\infty)\to\mathbb{R}$,

[TABLE]

where

[TABLE]

and $g(A)$ is a positive self adjoint operator. we have

[TABLE]

In particular, $t\geq 0$

[TABLE]

where

[TABLE]

and

[TABLE]

Thus, $e^{tA}$ is a self adjoint operator which is also bounded below, so that it is onto, and hence

1. (1)

$e^{tA}$ is one-one, onto, and has bounded inverse, namely, $e^{-tA}$; 2. (2)

$R(e^{-tA})=D(e^{tA})\quad\forall\,t\geq 0$; 3. (3)

${\mathcal{S}}:=\{S(t):=e^{-tA}:t\geq 0\}$ is a strongly continuous semigroup on $H$ with

[TABLE]

Further (see [5]), $-A$ is the infinitesimal generator of ${\mathcal{S}}$, i.e.,

[TABLE]

and $\displaystyle D(-A):=\{\varphi\in H:\lim_{t\to 0}\frac{S(t)\varphi-\varphi}{t}\hbox{ exists}\}.$

In this paper, we are inerested in the final value problem (FVP), that is the problem of solving

[TABLE]

for a known $\varphi_{\tau}$ for some $\tau>0$.

Suppose $u$ is a solution of (3)-(4). Then, from (2), we have

[TABLE]

Thus,

[TABLE]

Since the operator $e^{(\tau-t)A}$ is unbounded, the above representation of $u(t)$ shows that

small error in the data $(\varphi_{\tau},f)$ can lead to large error in the solution $u(t)$.

In other words, the problem of solving the FVP (3)-(4) is ill-posed.

Definition 1**.**

If $\varphi_{\tau}\in H$ and $f\in L^{1}([0,\tau],H)$ are such that

[TABLE]

belongs to $D(e^{(\tau-t)A})$, then $u(\cdot)$ defined by

[TABLE]

is called the mild solution of the FVP (3). $\lhd$

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 $\phi\in D(e^{\tau A})$ and let $u:[0,\tau]\to H$ be defined by $u(t)=e^{(\tau-t)A}\phi$, $t\geq 0$. Then $u$ is a solution of the FVP

[TABLE]

if and only if $\phi\in D(Ae^{\tau A})$.

Proof.

Note that, for $t\geq o$ and $h>0$,

[TABLE]

Since $-A$ is the infinitesimal generator of the semigroup $\{e^{-hA}:h\geq 0\}$, it follows that

[TABLE]

and in that case $u^{\prime}(t)=-Au(t)$ $u(\tau)=\phi$. ∎

In view of the representation (5) of the mild solution $u(t)$ of the FVP, the problem of finding $u(t)$ with $u(\tau)=\varphi_{\tau}$ can be posed as a problem of solving the ill-posed operator equation

[TABLE]

where ${\mathcal{A}}_{t}:H\to H$ is the bounded operator defined by

[TABLE]

and $\psi(t):=\varphi_{\tau}-\int_{t}^{\tau}e^{-(\tau-s)A}f(s)ds.$ Note that

• •

${\mathcal{A}}_{t}$ is an injective bounded self adjoint operator,

• •

$R({\mathcal{A}}_{t})=D(e^{(\tau-t)A})$ is dense in $H$, and

• •

${\mathcal{A}}_{t}^{-1}=e^{(\tau-t)A}:R({\mathcal{A}}_{t})\to H$ 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 $\varphi_{\tau}$ and $f\in L^{1}([0,\tau],H)$, 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 $[0,\infty)$ suggests that a truncation of the inteegral would give a reasonable approximation for $u(t)$. 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 $\beta>0$. $\lhd$

The following theorem shows that $u_{\beta}(\cdot)$ is an approximation of $u(\cdot)$ for large $\beta$.

Theorem 4**.**

Under the assumption $\psi(t)\in D(e^{\tau A})$,

[TABLE]

Proof.

Since

[TABLE]

we obtain

[TABLE]

∎

Next, we show that $u_{\beta}(\cdot)$ is, in fact, stable under perturbations in the data $(\varphi_{\tau},f)$.

Suppose $\tilde{\varphi}_{\tau}\in H$ and $\tilde{f}\in L^{1}([0,\tau],H)$ are the noisy data, in place of the actual data $\varphi_{\tau}$ and $f$, respectively. Let

[TABLE]

where

[TABLE]

Theorem 5**.**

Let $\varphi_{\tau},\tilde{\varphi}_{\tau}\in H$ and $f,\tilde{f}\in L^{1}([0,\tau],H)$. The for each $t\in[0,\tau]$ and $\beta>0$,

[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 $H\times L^{1}([0,\tau],H)$. Thus, Theorem 5 shows that the truncated spectral regularized solution $u_{\beta}(t)$ is stable under perturbations in the data $(\varphi_{\tau},f)$ with respect to the above norm $\|\cdot\|_{*}$.

3. Convergence and error estimates

3.1. Convergence

From Theorem 5, the following theorem is immediate.

Theorem 6**.**

Let $\varphi_{\tau},\tilde{\varphi}_{\tau}\in H$ and $f,\tilde{f}\in L^{1}([0,\tau],H)$ be such that

[TABLE]

for some $\delta>0$. Then for each $t\in[0,\tau]$ and $\beta>0$,

[TABLE]

Further, if

[TABLE]

for $0<p<1$, then $e^{(\tau-t)\beta}\delta=\delta^{p}$ 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 $h_{t}:[0,\infty)\to(0,\infty)$ is continuous and for each $t\in[0,\tau)$,

[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 $p,q>0$ respectively. Note that the source conditions in (11) are special cases of (10) obtained by the choices

[TABLE]

respectively.

Theorem 7**.**

Suppose $\varphi_{\tau}\in H$ and $f\in L^{1}([0,\tau],H)$ are such that $\psi(t)\in D(e^{\tau A})$ for each $t\in[0,\tau)$ and there exists a monotonically increasing continuous function $h_{t}:[0,\tau]\to(0,\infty)$ such that

1. (i)

$h_{t}(\lambda)\to\infty$* as $\lambda\to\infty$,* 2. (ii)

$u(t)\in D(h_{t}(A))$,

Let $\rho_{t}>0$ be such that $\|h_{t}(A)u(t)\|\leq\rho_{t}$. Then

[TABLE]

where $\rho_{t,\beta}\leq\rho_{t}$ and $\rho_{t,\beta}\to 0$ as $\beta\to\infty$.

Proof.

Recall from Definition 1 that $u(t)=e^{(\tau-t)A}\psi(t).$ Hence,

[TABLE]

Since $h_{t}$ is monotonically increasing, from the above, we obtain

[TABLE]

By the assumption,

[TABLE]

Hence, taking $\rho_{t,\beta}$ such that

[TABLE]

the inequality (12) leads to

[TABLE]

where $\rho_{t,\beta}\leq\rho_{t}$ and $\rho_{t,\beta}\to 0$ as $\beta\to\infty$. ∎

Remark 8**.**

Recently, Jana [1] and Jana and Nair [2] used similar general source condition, but based on the data $\varphi_{\tau},f$ instead of the mild solution $u(t)$. $\lhd$

Combining the last two theorems, we obtain the following.

Theorem 9**.**

Suppose $\tilde{\varphi}_{\tau}$ and $\tilde{f}$ are noisy data such that

[TABLE]

for some noise level $\delta>0$. Then

[TABLE]

If $\rho_{t,\beta}$ and $h_{t}(\cdot)$ are as in Theorem 7, then we have

[TABLE]

4. Parameter Choice Strategy

In Theorem 9, we obtained an estimate for the error $\|u(t)-\tilde{u}_{\beta}(t)\|$ a smoothness assumption on $u(\cdot)$. Now, we choose $\beta:=\beta_{t}^{\delta}$ depending on $t$ and $\delta$ such that the obtained estimate converges to [math] as $\delta\to 0$.

Theorem 10**.**

or $\lambda>0$, let $\xi_{t}(\lambda):=h_{t}(\lambda)e^{(\tau-t)\lambda}$, and let

[TABLE]

Then

[TABLE]

In particular,

[TABLE]

Proof.

Note that

[TABLE]

Thus, for the choice of $\beta=\xi_{t}^{-1}(\rho_{t}/\delta)$, Theorem 9 implies

[TABLE]

Since $h(\xi_{t}^{-1}(\rho_{t}/\delta))\to\infty$ as $\delta\to 0$.

[TABLE]

∎

4.1. A special choice of the source condition

Suppose $u(t)\in R(e^{-\gamma(\tau-t)A})$ for some $\gamma>0$. Then

[TABLE]

for some $v_{\gamma}(t)\in H$. 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 $u(t)\in R(e^{-\gamma(\tau-t)A})$ for some $\gamma>0$ and

[TABLE]

Then

[TABLE]

Further, taking

[TABLE]

we have

[TABLE]

5. Comparison with Lavrentiev regularization

Recall that the operator ${\mathcal{A}}_{t}:H\to H$ defined by

[TABLE]

is injective, continuous, self adjoint, with $R({\mathcal{A}}_{t})$ dense in $H$. Let $u(t)$ be the solution of (LABEL:op-eq-1), that is,

[TABLE]

Let $u^{L}_{\alpha}(\cdot)$ 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 $\gamma>0$. 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 $\beta$ as in (18). However, the estimate in (13) is valid for all $\gamma>0$, whereas (17) is valid only for $0<\gamma\leq 1$.

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.