Convergent numerical methods for parabolic equations with reversed time   via a new Carleman estimate

Michael V. Klibanov; Anatoly G. Yagola

arXiv:1903.01238·math.AP·January 8, 2020

Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

Michael V. Klibanov, Anatoly G. Yagola

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TL;DR

This paper introduces a new Carleman estimate for second-order parabolic equations with reversed time, enabling stable numerical methods over arbitrary time intervals and extending to quasilinear cases with proven convergence.

Contribution

It develops a novel Carleman estimate applicable to any time interval, facilitating stable quasi-reversibility methods and global convergence results for reversed time parabolic equations.

Findings

01

Established a stability estimate for reversed time parabolic equations.

02

Proposed a quasi-reversibility numerical method with proven convergence.

03

Constructed a globally convex Tikhonov-like functional ensuring convergence.

Abstract

The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability estimate is proven. Next, the quasi-reversibility numerical method is proposed for an arbitrary time interval for the linear case. This is unlike a sufficiently small time interval in the previous work. The convergence rate for the quasi-reversibility method is established. Finally, the quasilinear parabolic equation with reversed time is considered. A weighted globally strictly convex Tikhonov-like functional is constructed. The weight is the Carleman Weight Function which is involved in that Carleman estimate. The global convergence of the…

Equations486

Q_{T} = Ω \times (0, T), S_{T} = \partial Ω \times (0, T), Q_{T τ} = Ω \times (τ, T) .

Q_{T} = Ω \times (0, T), S_{T} = \partial Ω \times (0, T), Q_{T τ} = Ω \times (τ, T) .

a_{ij} (x, t) = a_{ij} (x, t) \in C^{1} (\overline{Q}_{T}),

a_{ij} (x, t) = a_{ij} (x, t) \in C^{1} (\overline{Q}_{T}),

μ_{1} ∣ ξ ∣^{2} \leq i, j = 1 \sum n a_{ij} (x, t) ξ_{i} ξ_{j} \leq μ_{2} ∣ ξ ∣^{2}, \forall (x, t) \in \overline{Q}_{T}, \forall ξ \in R^{n},

μ_{1} ∣ ξ ∣^{2} \leq i, j = 1 \sum n a_{ij} (x, t) ξ_{i} ξ_{j} \leq μ_{2} ∣ ξ ∣^{2}, \forall (x, t) \in \overline{Q}_{T}, \forall ξ \in R^{n},

μ_{1}, μ_{2} = co n s t . > 0, μ_{1} \leq μ_{2} .

μ_{1}, μ_{2} = co n s t . > 0, μ_{1} \leq μ_{2} .

Lu = i, j = 1 \sum n a_{ij} (x, t) u_{x_{i} x_{j}}, (x, t) \in Q_{T} .

Lu = i, j = 1 \sum n a_{ij} (x, t) u_{x_{i} x_{j}}, (x, t) \in Q_{T} .

∣ F (y_{1}) - F (y_{2}) ∣ \leq \overline{C} ∣ y_{1} - y_{2} ∣, \forall y_{1}, y_{2} \in R^{2 n + 2} .

∣ F (y_{1}) - F (y_{2}) ∣ \leq \overline{C} ∣ y_{1} - y_{2} ∣, \forall y_{1}, y_{2} \in R^{2 n + 2} .

u_{t} = Lu + F (\nabla u, u, x, t), (x, t) \in Q_{T} .

u_{t} = Lu + F (\nabla u, u, x, t), (x, t) \in Q_{T} .

u ∣_{S_{T}} = 0.

u ∣_{S_{T}} = 0.

u (x, T) = g (x), x \in Ω.

u (x, T) = g (x), x \in Ω.

Q_{T} \int (u_{t} - Lu)^{2} exp (2 (t + 1)^{ν}) d x d t

Q_{T} \int (u_{t} - Lu)^{2} exp (2 (t + 1)^{ν}) d x d t

\geq C ν Q_{T} \int (\nabla u)^{2} exp (2 (t + 1)^{ν}) d x d t + C ν^{2} Q_{T} \int u^{2} exp (2 (t + 1)^{ν}) d x d t

\geq C ν Q_{T} \int (\nabla u)^{2} exp (2 (t + 1)^{ν}) d x d t + C ν^{2} Q_{T} \int u^{2} exp (2 (t + 1)^{ν}) d x d t

- C exp (3 (T + 1)^{ν}) ∥ u (x, T) ∥_{L_{2} (Ω)}^{2} - C ∥ \nabla u (x, 0) ∥_{L_{2} (Ω)}^{2}, \forall ν \geq ν_{0} .

- C exp (3 (T + 1)^{ν}) ∥ u (x, T) ∥_{L_{2} (Ω)}^{2} - C ∥ \nabla u (x, 0) ∥_{L_{2} (Ω)}^{2}, \forall ν \geq ν_{0} .

u \in C^{2} (\overline{Q}_{T}), u ∣_{S_{T}} = 0.

u \in C^{2} (\overline{Q}_{T}), u ∣_{S_{T}} = 0.

v (x, t) = u (x, t) exp ((t + 1)^{ν}) .

v (x, t) = u (x, t) exp ((t + 1)^{ν}) .

u = v exp (- (t + 1)^{ν}),

u = v exp (- (t + 1)^{ν}),

u_{t} = [v_{t} - ν (t + 1)^{ν - 1} v] exp (- (t + 1)^{ν}),

u_{t} = [v_{t} - ν (t + 1)^{ν - 1} v] exp (- (t + 1)^{ν}),

u_{x_{i} x_{j}} = v_{x_{i} x_{j}} exp (- (t + 1)^{ν}) .

u_{x_{i} x_{j}} = v_{x_{i} x_{j}} exp (- (t + 1)^{ν}) .

(u_{t} - Lu)^{2} exp (2 (t + 1)^{ν}) = [v_{t} - (Lv + ν (t + 1)^{ν - 1} v)]^{2}

(u_{t} - Lu)^{2} exp (2 (t + 1)^{ν}) = [v_{t} - (Lv + ν (t + 1)^{ν - 1} v)]^{2}

\geq v_{t}^{2} - 2 v_{t} (Lv + ν (t + 1)^{ν - 1} v)

\geq v_{t}^{2} - 2 v_{t} (Lv + ν (t + 1)^{ν - 1} v)

= v_{t}^{2} - 2 v_{t} Lv - 2 ν (t + 1)^{ν - 1} v_{t} v .

= v_{t}^{2} - 2 v_{t} Lv - 2 ν (t + 1)^{ν - 1} v_{t} v .

- 2 v_{t} Lv = - i, j = 1 \sum n (a_{i, j} v_{x_{i} x_{j}} v_{t} + a_{i, j} v_{x_{j} x_{i}} v_{t}) .

- 2 v_{t} Lv = - i, j = 1 \sum n (a_{i, j} v_{x_{i} x_{j}} v_{t} + a_{i, j} v_{x_{j} x_{i}} v_{t}) .

- (a_{i, j} v_{x_{i} x_{j}} v_{t} + a_{i, j} v_{x_{j} x_{i}} v_{t}) = - (a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + a_{i, j} v_{x_{i}} v_{t x_{j}} + (a_{i, j})_{x_{j}} v_{x_{i}} v_{t}

- (a_{i, j} v_{x_{i} x_{j}} v_{t} + a_{i, j} v_{x_{j} x_{i}} v_{t}) = - (a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + a_{i, j} v_{x_{i}} v_{t x_{j}} + (a_{i, j})_{x_{j}} v_{x_{i}} v_{t}

- (a_{i, j} v_{x_{j}} v_{t})_{x_{i}} + a_{i, j} v_{t x_{i}} v_{x_{j}} + (a_{i, j})_{x_{i}} v_{x_{j}} v_{t}

- (a_{i, j} v_{x_{j}} v_{t})_{x_{i}} + a_{i, j} v_{t x_{i}} v_{x_{j}} + (a_{i, j})_{x_{i}} v_{x_{j}} v_{t}

= (a_{i, j} v_{x_{i}} v_{x_{j}})_{t} - (a_{i, j})_{t} v_{x_{i}} v_{x_{j}} + (a_{i, j})_{x_{j}} v_{x_{i}} v_{t} + (a_{i, j})_{x_{i}} v_{x_{j}} v_{t}

= (a_{i, j} v_{x_{i}} v_{x_{j}})_{t} - (a_{i, j})_{t} v_{x_{i}} v_{x_{j}} + (a_{i, j})_{x_{j}} v_{x_{i}} v_{t} + (a_{i, j})_{x_{i}} v_{x_{j}} v_{t}

- [(a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + (a_{i, j} v_{x_{j}} v_{t})_{x_{i}}]

- [(a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + (a_{i, j} v_{x_{j}} v_{t})_{x_{i}}]

= (a_{i, j} v_{x_{i}} v_{x_{j}})_{t} - (a_{i, j})_{t} v_{x_{i}} v_{x_{j}} + (a_{i, j})_{x_{j}} v_{x_{i}} v_{t} + (a_{i, j})_{x_{i}} v_{x_{j}} v_{t}

= (a_{i, j} v_{x_{i}} v_{x_{j}})_{t} - (a_{i, j})_{t} v_{x_{i}} v_{x_{j}} + (a_{i, j})_{x_{j}} v_{x_{i}} v_{t} + (a_{i, j})_{x_{i}} v_{x_{j}} v_{t}

- [(a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + (a_{i, j} v_{x_{j}} v_{t})_{x_{i}}] .

- [(a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + (a_{i, j} v_{x_{j}} v_{t})_{x_{i}}] .

2 ab \geq - ε a^{2} - \frac{1}{ε} b^{2}, \forall a, b \in R, \forall ε > 0,

2 ab \geq - ε a^{2} - \frac{1}{ε} b^{2}, \forall a, b \in R, \forall ε > 0,

- 2 v_{t} Lv \geq - C (\nabla v)^{2} - \frac{1}{2} v_{t}^{2} + i, j = 1 \sum n (a_{i, j} v_{x_{i}} v_{x_{j}})_{t}

- 2 v_{t} Lv \geq - C (\nabla v)^{2} - \frac{1}{2} v_{t}^{2} + i, j = 1 \sum n (a_{i, j} v_{x_{i}} v_{x_{j}})_{t}

+ i, j = 1 \sum n [(a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + (a_{i, j} v_{x_{j}} v_{t})_{x_{i}}] .

+ i, j = 1 \sum n [(a_{i, j} v_{x_{i}} v_{t})_{x_{j}} + (a_{i, j} v_{x_{j}} v_{t})_{x_{i}}] .

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Full text

Convergent numerical methods for parabolic equations with reversed

time via a new Carleman estimate

Michael V. Klibanov and Anatoly G. Yagola Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC, 28223, [email protected] of Mathematics, Faculty of Physics, Moscow State University, Moscow, 119991, Russia, e-mail: [email protected]

Abstract

The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability estimate is proven. Next, the quasi-reversibility numerical method is proposed for an arbitrary time interval for the linear case. This is unlike a sufficiently small time interval in the previous work. The convergence rate for the quasi-reversibility method is established. Finally, the quasilinear parabolic equation with reversed time is considered. A weighted globally strictly convex Tikhonov-like functional is constructed. The weight is the Carleman Weight Function which is involved in that Carleman estimate. The global convergence of the gradient projection method to the exact solution is proved for this functional.

Key Words. linear and quasilinear parabolic equations, reversed time, Carleman estimate, stability estimate, convergent quasi-reversibility method for the linear case, globally convergent numerical method for the quasilinear case

AMS subject classification. 35R25, 35R30

1 Introduction

In this paper, we construct convergent numerical methods for linear and quasilinear parabolic equations with reversed time. The key tool of the convergence analysis is a new Carleman estimate. While the previously known Carleman estimate for these problems works only on a sufficiently small time interval, the one of this paper works on any finite time interval.

All functions below are real valued ones. Below $x=\left(x_{1},x_{2},...,x_{n}\right)$ denotes points in $\mathbb{R}^{n}$ and $\nabla f=\left(f_{x_{1}},f_{x_{2}},...,f_{x_{n}}\right)$ for any appropriate function $f\left(x\right).$ Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with a piecewise smooth boundary $\partial\Omega.$ Let $T>0$ and $\tau\in\left(0,T\right)$ be two numbers. Denote

[TABLE]

For $i,j=1,...,n$ , let functions $a_{ij}\left(x,t\right)$ be such that

[TABLE]

Introduce a uniform elliptic operator $L$ of the second order in the domain $Q_{T},$

[TABLE]

Let the function $F\left(y\right)\in C^{1}\left(\mathbb{R}^{2n+2}\right).$ For this function, we assume that there exists a constant $\overline{C}=\overline{C}\left(F\right)>0$ depending only on $F$ such that

[TABLE]

Consider the following quasilinear parabolic equation:

[TABLE]

We impose the zero Dirichlet boundary condition on the function $u,$

[TABLE]

Since we work with the time reversed case, we assume that the function $u\left(x,t\right)$ is known at the final time $T$ ,

[TABLE]

Thus, we have obtained the following problem:

Problem with Time Reversed Data. Suppose that conditions (1.1)-(1.5). Find a function $u\in C^{2,1}\left(\overline{Q}_{T}\right)$ satisfying conditions (1.7)-(1.9).

One of possible applications is in the case when a solid is heated and the initial temperature is unknown. However, one can measure the temperature of this solid at a final time. It is required to restore the temperature distribution inside of this solid at all preceding times. Another application of this problem, which was found recently, is in financial mathematics, more precisely in a the problem of forecast of prices of stock options using the Black-Scholes equation and real market data [17].

The problem of our interest is well known to be unstable, i.e. this is an ill-posed problem. Hölder stability estimates for this problem are known, see, e.g. [8, 14, 24]. To the best knowledge of the author, the strongest Hölder stability result, which is valid for an arbitrary large time interval $t\in\left(0,T\right),$ is obtained by Isakov, see Theorem 3.1.3 in [8]. In section 2 of chapter 4 of [24] and later in [14] a Carleman estimate was used to obtain the Hölder stability estimate. However, that Hölder stability estimate is valid only on a sufficiently small time interval $t\in\left(T-\varepsilon,T\right)$ for a sufficiently small $\varepsilon>0.$ The smallness of this interval is due to the Carleman Weight Function (CWF), which has been used in the Carleman estimates for that problem so far [14, 24]. This function is $\left(k+T-t\right)^{-2\lambda},$ where numbers $k,T>0$ are sufficiently small and the parameter $\lambda>0$ is sufficiently large. The same CWF was used in [7] for the proof of the uniqueness theorem.

Using this estimate, the first author has constructed in [14] the quasi-reversibility method (QRM) for the above problem in the linear case and has proven its convergence, again on a sufficiently small time interval. Below as well as in [14, 17] the QRM is realized via the minimization of a certain regularization functional. On the other hand, QRM is quite often realized via a proper perturbation of the underlying PDE operator [8, 25]. A surprising idea of the recent publication of Kaltenbacher and Rundell [9] it to use the non local operator of the fractional $t-$ derivative as the perturbing operator for the QRM. In the work of Tuan, Khoa and Au [27] another version of the QRM is constructed for the quasilinear case. Its convergence was proven for an arbitrary $T$ . However, the perturbation operator of [27] is a very complicated one. Another version of the QRM was proposed in [8]. This version works only in the case when the set of eigenfunctions of the underlying elliptic operator forms an orthonormal basis in $L_{2}\left(\Omega\right).$

The QRM was originally developed by Lattes and Lions in 1969 [25]. Their idea became quite popular since then with many publications treating a variety of ill-posed problems for PDEs. In this regard we refer to, e.g. [4, 5, 6, 8, 10, 17, 23, 27]. In particular, the first author has shown in the survey paper [14] that as soon as a proper Carleman estimate for an ill-posed problem for a linear PDE is available, then the QRM can be constructed for this problem and its convergence rate can be established. In the case of the time reversed data for the linear parabolic PDE the construction of the QRM in [14] is valid only on a sufficiently small time interval $\left(0,T\right).$

As to the ill-posed problems for quasilinear PDEs, it was shown in [15] that, again as soon as a proper Carleman estimate is available for the linear principal part of the PDE operator, a weighted globally strictly convex Tikhonov-like functional can be constructed, i.e. this problem can be “convexified”. The key element of this functional is the presence of the CWF in it, i.e. the function which is involved as the weight in the Carleman estimate for that linear principal part of the PDE operator. In the follow up paper [1] existence and uniqueness of the minimizer of that functional were established and global convergence to the exact solution of the gradient projection method of the minimization of this functional was proven. As to the quasilinear parabolic equations, they were considered in [1, 15, 18] only for the case of lateral Cauchy data with numerical results in [1, 18]. However, the case of time reversed data was not considered in [1, 15, 18].

The idea of convexifying coefficient inverse problems was first proposed in 1995-1997 in [11, 12], although without numerical studies, also see [13]. Recently the interest to the convexification approach was renewed. First, this was done only analytically [3, 16]. Next, a number of papers was published, which combined the theory with numerical studies, see, e.g. [1, 18] for the case of quasilinear PDEs and [19, 20, 21, 22] for coefficient inverse problems.

We call a numerical method for an ill-posed problem globally convergent if there is a theorem claiming that it converges to the exact solution of this problem without an a priori knowledge of a sufficiently small neighborhood of this solution. On the other hand, we call a numerical method for that problem locally convergent if its convergence is rigorously guaranteed only if iterations start in a sufficiently small neighborhood of the exact solution. Thus, the convexification is a globally convergent method (see Theorem 5.4 in section 5). On the other hand, a gradient-like method being applied to a non-convex Tikhonov-like functional, might have guaranteed convergence to the exact solution only if its starting point is located in a sufficiently small neighborhood of that solution. The latter is called the local convergence.

New elements of this paper are:

A new Carleman estimate for a general parabolic operator of the second order with time reversed data is proven. This estimate works on an arbitrary time interval $t\in\left(0,T\right)$ , unlike a sufficiently small interval of previous publications [7, 14, 24]. Results listed in items 2-4 below are based on this estimate. 2. 2.

A stability estimate is proven for the above Problem with Time Reversed Data. This estimate is somewhat “between” Hölder and logarithmic stability estimates. In other words, although it is weaker than the Hölder stability estimate of [8, 14, 24], it is stronger than the logarithmic stability estimate. Still, the main advantage of our stability estimate over ones in [14, 24] is that it works without a smallness assumption imposed on the time interval. 3. 3.

In the linear case, the QRM is constructed, existence and uniqueness of the minimizer as well as convergence of minimizers to the exact solution are proven. Unlike previous works [14, 17], a smallness assumption is not imposed on the time interval. 4. 4.

In the quasilinear case, this problem is convexified for the first time. In other words, a weighted globally strictly convex Tikhonov-like functional is constructed with the CWF in it. Both the existence and uniqueness of its minimizer are proved. In addition, the global convergence of the gradient projection method to the exact solution is established. This way we avoid the use of a complicated perturbation operator of [27].

In section 2 we prove the new Carleman estimate. In section 3 we present stability estimate. In section 4 we describe the quasi-reversibility method for the linear case, prove existence and uniqueness of the minimizer as well as convergence of minimizers to the exact solution when the level of the noise in the data tends to zero. In section 5 we construct the above mentioned weighted globally strictly convex Tikhonov-like functional and formulate corresponding theorems. These theorems are proved in section 6.

2 Carleman Estimate

Theorem 2.1. (Carleman estimate). Assume that conditions (1.1)-(1.5) are in place. Then there exists a sufficiently large number $\nu_{0}=\emph{\ }\nu_{0}\left(\mu_{1},\mu_{2},\max_{i,j}\left\|a_{ij}\right\|_{C^{1}\left(\overline{Q}_{T}\right)},Q_{T}\right)>1$ depending only on listed parameters and the number $C>0$ depending on the same parameters as ones of $\nu_{0}$ , such that for all functions $u\in H^{2}\left(Q_{T}\right)$ satisfying the zero Dirichlet boundary condition (1.8) the following Carleman estimate is valid

[TABLE]

Proof. In this proof,

[TABLE]

The case $u\in H^{2}\left(Q_{T}\right),$ $u\mid_{S_{T}}=0$ can be obtained from (2.2) via density arguments. Everywhere below in this paper $C=\emph{\ }C\left(\mu_{1},\mu_{2},\max_{i,j}\left\|a_{ij}\right\|_{C^{1}\left(\overline{Q}_{T}\right)},Q_{T}\right)>0$ denotes different constants depending only on listed parameters.

We prove this theorem in six steps. Introduce a new function $v\left(x,t\right),$

[TABLE]

Hence,

[TABLE]

Hence,

[TABLE]

Step 1. First, we estimate from the below $-2v_{t}Lv,$

[TABLE]

Next,

[TABLE]

Applying the Cauchy-Schwarz inequality “with $\varepsilon"$ to (2.5),

[TABLE]

and using (2.4), we obtain

[TABLE]

Hence,

[TABLE]

Step 2. Estimate the term $-2\nu\left(t+1\right)^{\nu-1}v_{t}v$ in the third line of (2.3). We have:

[TABLE]

Step 3. Sum up (2.7), (2.8). Then, taking into account (2.3), we obtain

[TABLE]

Replacing here $v$ with $u=v\exp\left(-\left(t+1\right)^{\nu}\right)$ and using $v_{t}^{2}/2\geq 0,$ we obtain

[TABLE]

What is not good in estimate (2.9) is that the first line in its right hand side contains both positive and negative terms. However, only positive terms must be in such cases in any Carleman estimate. Hence, we continue with further steps.

Step 4. Estimate from the below the expression $\left(u_{t}-Lu\right)u\exp\left(2\left(t+1\right)^{\nu}\right).$ We have

[TABLE]

By (1.3)

[TABLE]

Hence, using (2.6) and (2.10), we obtain for all $\nu\geq\nu_{0}$

[TABLE]

Step 5. Multiply (2.11) by $\sqrt{\nu}$ and sum up with (2.9). Since $\sqrt{\nu}>>1$ and $\nu^{2}>>\nu^{3/2}$ for sufficiently large $\nu,$ then we obtain for $\nu\geq\nu_{0}$

[TABLE]

Next, we estimate from the above the left hand side of inequality (2.12) as

[TABLE]

Comparing this with (2.12), we obtain

[TABLE]

where the vector function $U$ is such that

[TABLE]

Condition (2.14) follows from the boundary condition in (2.2) and the lines 5 and 6 of (2.12).

Step 6. Integrate the pointwise Carleman estimate (2.13) over the domain $Q_{T}.$ Using Gauss’ formula and (2.14), we obtain (2.1). $\square$

3 Stability Estimate

For an arbitrary $\tau\in\left(0,T\right),$ denote

[TABLE]

Prior establishing our stability estimate, we prove Lemma 3.1.

Lemma 3.1. Let $\delta\in\left(0,1\right)$ be a sufficiently small number and let the number $k>0$ . Choose a sufficiently large number $\nu=\nu\left(\delta\right)$ such that

[TABLE]

*Then for any $\tau\in\left(0,T\right)$ and for any number $y>0$ *

[TABLE]

Remark 3.1. It follows (3.2) and (3.3) that any stability estimate via $\exp\left(-2\left(\tau+1\right)^{\nu}\right)$ is weaker than Hölder and stronger than logarithmic stability estimate.

Proof of Lemma 3.1. By (3.1)* * $k\left(T+1\right)^{\nu\left(\delta\right)}=\ln\left(\delta^{-1}\right).$ Hence, $\left(T+1\right)^{\nu\left(\delta\right)}=\ln\left(\delta^{-1/k}\right).$ Hence,

[TABLE]

Next,

[TABLE]

By (3.4)

[TABLE]

Using (3.5)-(3.7), we obtain

[TABLE]

Hence,

[TABLE]

Using (3.8), we now prove (3.2). Indeed,

[TABLE]

Since by (3.7) $c\in\left(0,1\right),$ then

[TABLE]

Hence,

[TABLE]

This and (3.9) prove (3.2).

We now prove (3.3), which is equivalent with

[TABLE]

Next,

[TABLE]

Obviously

[TABLE]

Thus, (3.10) follows from (3.11) and (3.12). $\square$

Theorem 3.1 (stability estimate). Assume that conditions (1.1)-(1.5) are in place. Suppose that two functions $u_{1},u_{2}\in H^{2}\left(Q_{T}\right)$ are solutions of problem (1.7), (1.8) with different data at $\left\{t=T\right\}$ ,

[TABLE]

Suppose that

[TABLE]

**

[TABLE]

*where $M>0$ is a number and the parameter $\delta$ characterizes the level of noise in the data $u\left(x,T\right)$ . Denote * $w=u_{1}-u_{2}.$ Let $\overline{C}>0$ be the number in (1.6) and $\nu_{0}$ be the parameter of Theorem 2.1. Then there exist constants

[TABLE]

* depending only on listed parameters such that if the number $\delta_{0}$ is so small that*

[TABLE]

*then the following estimate holds for any $\tau\in\left(0,T\right)$ and for all * $\delta\in\left(0,\delta_{0}\right)$ **

[TABLE]

where *the constant $C_{1}$ is independent on $M$ and the number * $c=c\left(\tau,T\right)\in\left(0,1\right)$ is defined in (3.7).

Estimate (3.17) is between Hölder and logarithmic stability estimates, see Lemma 3.1 and Remark 3.1. Everywhere below $C_{1}>0$ denotes different numbers depending on the same parameters as ones listed in this theorem.

Proof. It follows from (1.6)-(1.9) and (3.13) that

[TABLE]

Square both sides of inequality (3.18), then multiply by $\exp\left(2\left(t+1\right)^{\nu}\right)$ integrate over $Q_{T}$ and then apply Theorem 2.1 taking into account (3.19) and (3.20). We obtain

[TABLE]

Choose $\nu_{1}=\nu_{1}\left(\mu_{1},\mu_{2},\max_{i,j}\left\|a_{ij}\right\|_{C^{1}\left(\overline{Q}_{T}\right)},Q_{T},\overline{C}\right)\geq\nu_{0}>1$ so that $C\sqrt{\nu_{1}}/4\geq\overline{C}^{2}.$ Then (3.14) and (3.21) imply that for all $\nu\geq\nu_{1}$

[TABLE]

Hence,

[TABLE]

Choose $\nu=\nu\left(\delta\right)$ such that (3.1) would be satisfied with $k=3$ , i.e.

[TABLE]

Hence,

[TABLE]

The choice (3.25) is possible since (3.16) holds and $\delta\in\left(0,\delta_{0}\right).$ Hence, by (3.8)

[TABLE]

Hence, (3.15), (3.23), (3.24) and (3.26) imply that

[TABLE]

The target estimate (3.17) of this theorem obviously follows from (3.2) and (3.27). $\square$

4 The Quasi-Reversibility Method for the Linear Case

We assume in this section that the function $F\left(\nabla u,u,x,t\right)$ in (1.7) is linear with respect to the function $u$ and its first derivatives,

[TABLE]

where functions $b_{j},c,p\in C\left(\overline{Q}_{T}\right).$ Then (1.7)-(1.9) become

[TABLE]

Assuming that $g\left(x\right)=0$ for $x\in\partial\Omega$ and

[TABLE]

consider the function $v\left(x,t\right)=u\left(x,t\right)-g\left(x\right).$ Then (4.2)-(4.5) lead to

[TABLE]

We introduce the subspace $H_{0}^{2}\left(Q_{T}\right)$ of the space $H^{2}\left(Q_{T}\right)$ as

[TABLE]

The QRM for problem (4.6)-(4.9) amounts to the minimization of the following functional $J_{\alpha},$

[TABLE]

where $\alpha\in\left(0,1\right)$ is the regularization parameter. We arrive at the following problem:

Minimization Problem 1. Minimize the functional $J_{\alpha}\left(v\right)$ on the space $H_{0}^{2}\left(Q_{T}\right).$

Theorem 4.1. Assume that conditions (1.1)-(1.5), (4.9) hold. Then there exists unique minimizer $v_{\min}\in H_{0}^{2}\left(Q_{T}\right)$ of the functional $J_{\alpha}\left(v\right).$

Proof. Let $\left[.,.\right]$ denotes the scalar product in $H^{2}\left(Q_{T}\right).$ By the variational principle, any minimizer $v_{\min}\in H_{0}^{2}\left(Q_{T}\right),$ if it exists, satisfies the following integral identity

[TABLE]

Since

[TABLE]

then the equality

[TABLE]

defines a new scalar product in the space $H_{0}^{2}\left(Q_{T}\right).$ We rewrite integral identity (4.11) as

[TABLE]

Next, by the Cauchy-Schwarz inequality

[TABLE]

Hence, by Riesz theorem, there exists a unique function $p\in H_{0}^{2}\left(Q_{T}\right)$ such that

[TABLE]

Comparing this with (4.12), we obtain

[TABLE]

Therefore, there exists unique minimizer $v_{\min}=p\in H_{0}^{2}\left(Q_{T}\right)$ of the functional $J_{\alpha}\left(v\right).$ $\square$

In the regularization theory, the function $v_{\min}$ is called the regularized solution of problem (4.6)-(4.9) [2, 26]. The next step after Theorem 4.1 is to prove convergence of regularized solutions to the exact one when the noise in the data tends to zero. While we have used only Riesz theorem to prove existence and uniqueness of the minimizer, the convergence result requires the Carleman estimate of Theorem 2.1.

Let the function $q^{\ast}\in L_{2}\left(Q_{T}\right)$ be the exact data in problem (4.6)-(4.9), i.e. the data without a noise in it. Suppose that there exists the exact solution $v^{\ast}\in H_{0}^{2}\left(Q_{T}\right)$ of problem (4.6)-(4.9) with $q=q^{\ast}.$ Theorem 1.2 implies that this solution is unique. Let $q\in L_{2}\left(Q_{T}\right)$ be the noisy data and let $v_{\min}\in H_{0}^{2}\left(Q_{T}\right)$ be the corresponding minimizer of functional (4.10) (Theorem 4.1). We assume that

[TABLE]

where $\delta>0$ is the noise level. Consider the difference

[TABLE]

Theorem 4.2 estimates the function $w$ via $\delta.$ It follows from the regularization theory that we need to assume a certain dependence $\alpha=\alpha\left(\delta\right)$ of the regularization parameter on the noise level $\delta.$

Theorem 4.2 (convergence rate). Assume that conditions (1.1)-(1.6) and (4.13) are in place. Let $\nu_{0}>1$ and $\nu_{1}\geq\nu_{0}$ be the numbers of Theorems 2.1 and 3.1 respectively and let $\alpha=\alpha\left(\delta\right)=\delta^{2}.$ Then there exists a number

[TABLE]

depending only on listed parameters *such that if * $\delta\in\left(0,\delta_{0}\right)$ and

[TABLE]

then the following convergence estimate of the QRM holds for every $\tau\in\left(0,T\right):$

[TABLE]

*where the function * $w_{\delta}$ *is defined in (4.14) and the number $c=c\left(\tau,T\right)\in\left(0,1\right)$ is defined in (3.7). *

As to estimate (4.16), also see Lemma 3.1 and Remark 3.1.

Proof of Theorem 4.2. The function $v^{\ast}\in H_{0}^{2}\left(Q_{T}\right)$ satisfies the following integral identity

[TABLE]

Subtracting (4.17) from (4.11) and using (4.14), we obtain

[TABLE]

Set in (4.18) $h=w_{\delta}.$ Using Cauchy-Schwarz inequality, we obtain

[TABLE]

Hence,

[TABLE]

Since $\alpha=\alpha\left(\delta\right)=\delta^{2},$ then (4.19) implies that $\left\|w_{\delta}\right\|_{H^{2}\left(Q_{T}\right)}^{2}\leq 1+\left\|v^{\ast}\right\|_{H^{2}\left(Q_{T}\right)}^{2}.$ Hence, trace theorem leads to

[TABLE]

To proceed further, we apply to (4.19) the Carleman estimate of Theorem 2.1. Let $\nu\geq\nu_{1}.$ We have

[TABLE]

Hence, using (4.19), we obtain

[TABLE]

We have

[TABLE]

Next, since $w_{\delta}\left(x,T\right)=0,$ then by (3.20) and (3.21)

[TABLE]

Choose $\nu_{2}\geq\nu_{1}>1$ such that $C\sqrt{\nu_{2}}/2\geq C_{1}.$ Then, using (4.20) and (4.23), we obtain

[TABLE]

Hence, (4.21)-(4.24) imply that

[TABLE]

Dividing this by $\exp\left(2\left(\tau+1\right)^{\nu}\right),$ we obtain for all $\forall\nu\geq\nu_{2}$

[TABLE]

Choose $\nu=\nu\left(\delta\right)$ such that (3.1) would be satisfied with $k=2$ , i.e.

[TABLE]

Hence,

[TABLE]

The choice (4.26), (4.27) is possible since (4.15) holds and $\delta\in\left(0,\delta_{0}\right).$ It follows from (3.1), (3.8) and (4.26) that

[TABLE]

Hence, using (4.25)-(4.28) and Lemma 3.1, we obtain

[TABLE]

which implies (4.16). $\square$

5 The Global Strict Convexity

5.1 The weighted Tikhonov-like functional

While the linear case (4.1) was studied in section 4, in this section we consider the quasilinear case. First, just like in section 4, we consider the function $v\left(x,t\right)=u\left(x,t\right)-g\left(x\right).$ Then, assuming (4.5), we obtain instead of (1.7)-(1.9):

[TABLE]

In our derivations below $v_{t},Lv$ and arguments of the function $F\left(\nabla v+\nabla g,v+g,x,t\right)$ must be uniformly bounded for all $\left(x,t\right)\in\overline{Q}_{T}.$ Hence, similarly with [1, 15], we now need to impose a higher smoothness than just $v\in H_{0}^{2}\left(Q_{T}\right)$ as in section 4. Consider an integer $k$ such that $k>\left[\left(n+1\right)/2\right]+2,$ where $\left[\left(n+1\right)/2\right]$ denotes the maximal integer which does not exceed $\left(n+1\right)/2.$ Then embedding theorem implies that $H^{k}\left(Q_{T}\right)\subset C^{2}\left(\overline{Q}_{T}\right)$ and

[TABLE]

where the constant $E=E\left(Q_{T}\right)>0$ depends only on the domain $Q_{T}.$ Define the subspace $H_{0}^{k}\left(Q_{T}\right)\subset H^{k}\left(Q_{T}\right)$ as

[TABLE]

In addition, since we need below the function $\left(Lg\right)\left(x,t\right)$ to be bounded in $\overline{Q}_{T},$ then we assume that

[TABLE]

Let $R>0$ be an arbitrary number. We consider the ball $B\left(R\right)$ in the space $H_{0}^{k}\left(Q_{T}\right),$

[TABLE]

Hence, by (5.5)

[TABLE]

where the number $E_{R}=E_{R}\left(Q_{T},R\right)=const.>0$ depends only on listed parameters.

We want to find an approximate solution of problem (5.1)-(5.4) in the closed ball $\overline{B\left(R\right)}$ . To do this, we select a number $\tau\in\left(0,T\right)$ and minimize the following weighted Tikhonov-like functional

[TABLE]

The multiplier $\exp\left(-2\left(\tau+1\right)^{\nu}\right)$ in (5.10) introduced to balance two terms in the right hand side of (5.10). Indeed, the regularization parameter $\alpha\in\left(0,1\right)$ and

[TABLE]

also, see (5.14).

Minimization Problem 2. *Minimize the functional $I_{\alpha,\lambda,\nu}\left(v\right)$ on the ball * $\overline{B\left(R\right)}.$

5.2 Theorems about the functional $I_{\alpha,\nu}\left(v\right)$

The central theorem of this section is Theorem 5.1.

Theorem 5.1 (global strict convexity). Assume that conditions (1.1)-(1.6) and (5.4) hold. Then the functional $I_{\alpha,\nu}\left(v\right)$ has the Fréchet derivative $I_{\alpha,\nu}^{\prime}\left(v\right)\in H_{0}^{2}\left(Q_{T}\right)$ at every point $v\in H_{0}^{2}\left(Q_{T}\right)$ and for all values of parameters $\alpha,\nu\geq 0.$ Let $\nu_{0}>1$ be the number of Theorem 2.1. Then there exists a number

[TABLE]

*depending only on listed parameters such that * $2C_{2}\exp\left(-2\left(\tau+1\right)^{\nu}\right)\in\left(0,1\right)$ for $\nu\geq\nu_{3}$ and if for these values of $\nu$ the regularization parameter $\alpha$ is such that

[TABLE]

* then the functional $I_{\alpha,\nu}\left(v\right)$ is strictly convex on $\overline{B\left(R\right)}.$ More precisely, for every $\tau\in\left(0,T\right)$ the following strict convexity estimate holds*

[TABLE]

*where the constant $C_{2}>0$ depends only on parameters listed in (5.12). *

Everywhere below $C_{2}>0$ denotes different positive constants depending on the same parameters as those listed in (5.12).

Remark 5.1. *The presence of the term $C_{2}\left\|h\right\|_{H^{1,0}\left(Q_{T\tau}\right)}^{2}$ in the right hand side of (5.14) indicates that the convergence of a gradient-like method of the minimization of functional $I_{\alpha,\nu}\left(v\right)$ is likely faster in the space $H^{1,0}\left(Q_{T\tau}\right)$ then in the space $H^{k}\left(Q_{T}\right).$ *

Theorem 5.2. The Fréchet derivative $I_{\alpha,\nu}^{\prime}\left(v\right)$ of the functional $I_{\alpha,\nu}\left(v\right)$ is Lipschitz continuous on $B\left(2R\right)$ for all values of parameters $\alpha,\lambda,\nu\geq 0.$ In other words, there exists a number

[TABLE]

depending only on listed parameters such that

[TABLE]

Furthermore, let $\nu_{3}$ be the number of Theorem 5.1. Then for every pair $\alpha>0,$ $\nu\geq\nu_{3}$ there exists unique minimizer $v_{\min}\in\overline{B\left(R\right)}$ of the functional $I_{\alpha,\nu}\left(v\right)$ on the closed ball $\overline{B\left(R\right)}$ and the following inequality holds

[TABLE]

Let $P_{\overline{B}}:H_{0}^{k}\left(Q_{T}\right)\rightarrow\overline{B\left(R\right)}$ be the orthogonal projection operator mapping the space $H_{0}^{k}\left(Q_{T}\right)$ onto the closed ball $\overline{B\left(R\right)}.$ Let $v_{0}\in B\left(R\right)$ be an arbitrary point of $B\left(R\right).$ Let the number $\gamma\in\left(0,1\right).$ Consider the sequence of the gradient projection method,

[TABLE]

Theorem 5.3. Let $\nu_{3}$ be the number of Theorem 5.1. Choose the number $\nu\geq\nu_{3}.$ Let $v_{\min}\in\overline{B\left(R\right)}$ be the unique minimizer of the functional $I_{\alpha,\nu}\left(v\right)$ on the set $\overline{B\left(R\right)}$ (Theorem 5.2). Then there exists a sufficiently small number

[TABLE]

such that for every $\gamma\in\left(0,\gamma_{0}\right)$ there exists a number $\theta=\theta\left(\gamma\right)$ such that

[TABLE]

Consider now the case of noise in the data. Following one of the Tikhonov’s concept of the regularization [2, 26], we assume that there exists exact noiseless data $g^{\ast}\in H^{2}\left(\Omega\right)$ in (5.4) and, respectively, there exists exact solution $v^{\ast}\in B\left(R\right)$ of problem (5.1)-(5.4). Let $\delta\in\left(0,1\right)$ be the level of noise in the data $g\left(x\right)$ , i.e.

[TABLE]

In Theorem 5.4 we estimate the accuracy of the minimizer, i.e. the norm $\left\|v_{\min}-v^{\ast}\right\|_{H^{1,0}\left(Q_{T\tau}\right)}$ for any $\tau\in\left(0,T\right).$ In turn, this estimate, combined with (5.18), provides the convergence rate of the sequence (5.17) to the exact solution. Note that since $\delta\in\left(0,1\right),$ then by (5.19), we replace below dependencies on $\left\|g\right\|_{C^{2}\left(\overline{\Omega}\right)}$ of the above numbers $\nu_{3},C_{2},D,\gamma_{0}$ with dependencies on $\left\|g^{\ast}\right\|_{C^{2}\left(\overline{\Omega}\right)}.$

**Theorem 5.4 **(estimates of the accuracy and the convergence rate). *Assume that the exact solution of problem (5.1)-(5.5) $v^{\ast}\in B\left(R\right)$ and that (5.19) holds. Let * $\nu_{3}>1$ *be the number of Theorem 5.1. Select an arbitrary number * $\tau\in\left(0,T\right).$ For any $\delta\in\left(0,1\right)$ set the number $\nu=\nu\left(\delta\right)$ be the same as in (4.27). Let the number $\delta_{0}=\delta_{0}\left(\mu_{1},\mu_{2},\max_{i,j}\left\|a_{ij}\right\|_{C^{1}\left(\overline{Q}_{T}\right)},\left\|g^{\ast}\right\|_{C^{2}\left(\overline{\Omega}\right)},Q_{T},R,\overline{C}\right)>0$ be so small that

[TABLE]

Let $\delta\in\left(0,\delta_{0}\right)$ *and let $v_{\min}\in\overline{B\left(R\right)}$ be the unique minimizer of the functional $I_{\alpha,\nu}\left(v\right)$ on the set $\overline{B\left(R\right)}$ (Theorem 5.2). Let * $\gamma_{0}$ *be the number defined in Theorem 5.3. Let $\gamma\in\left(0,\gamma_{0}\right)$ and $\theta=\theta\left(\gamma\right)\in\left(0,1\right)$ also be the numbers of Theorem 5.3. Choose the regularization parameter * $\alpha$ as

[TABLE]

* * Then the following accuracy and convergence estimates hold

[TABLE]

where the number $c=c\left(\tau,T\right)\in\left(0,1\right)$ is defined in (3.7).

Remark 5.2. *According to section 1, since $R>0$ is an arbitrary number and since the starting point $v_{0}\in B\left(R\right)$ of the gradient projection method is an arbitrary point of $B\left(R\right),$ then Theorem 5.4 implies the global convergence to the exact solution of the gradient projection method of the minimization of the functional $I_{\alpha,\nu}\left(v\right).$ *

In this paragraph, we temporary assume that Theorem 5.1 is proved. Then the proof of the Lipschitz continuity (5.15) of the Fréchet derivative $I_{\alpha,\nu}^{\prime}$ is very similar to the proof of Theorem 3.1 of [1]. The rest of Theorem 5.2 follows from (5.15) and Lemma 2.1 of [1]. Given Theorem 5.1, Theorem 5.3 follows from Theorem 3.3 of [1].

Therefore, we prove in section 6 only Theorems 5.1 and 5.4. Below, if we say that a vector function belongs to a certain Banah space, then this means that each of its components belongs to that space. The norm of that vector function in that space is defined as the square root of the sum of squared norms of its components.

6 Proofs of Theorems 5.1 and 5.4

6.1 Proof of Theorem 5.1

Consider two arbitrary functions $v_{1},v_{2}\in B\left(R\right).$ Denote $h=v_{2}-v_{1}.$ Then

[TABLE]

Hence, by (5.9)

[TABLE]

Consider the expression

[TABLE]

Using the multidimensional analog of Taylor formula [28], (1.6), (5.4), (5.9) and (6.2), we obtain

[TABLE]

where the $n-$ D vector function $G_{1}\left(x,t\right)\in C\left(\overline{Q}_{T}\right),$ the function $G_{2}\left(x,t\right)\in C\left(\overline{Q}_{T}\right)$ and the function $G_{3}\left(x,t,\nabla h,h\right)$ is such that

[TABLE]

In addition,

[TABLE]

Hence, (6.3) and (6.4) imply

[TABLE]

where $Lin\left(h\right)$ and $Nonlin\left(h\right)$ denote linear and nonlinear expressions with respect to $h$ respectively,

[TABLE]

We represent the term $Nonlin\left(h\right)\left(x,t\right)$ in (6.7) as

[TABLE]

Hence, using Cauchy-Schwarz inequality, (6.5) and (6.6), we obtain

[TABLE]

Also, by (5.5) and (6.5)-(6.7)

[TABLE]

Thus, (5.10), (6.7) and (6.8) imply that

[TABLE]

where $\left(.,.\right)_{k}$ is the scalar product in $H^{k}\left(Q_{T}\right).$ Assuming temporary that $h$ is an arbitrary function of $H^{k}\left(Q_{T}\right),$ consider the expression $X\left(h\right),$

[TABLE]

It follows from (5.9), (6.6) and (6.8) that $X:H^{k}\left(Q_{T}\right)\rightarrow\mathbb{R}$ is a bounded linear functional. Hence, by Riesz theorem there exists a unique function $\widetilde{X}\in H^{k}\left(Q_{T}\right)$ such that

[TABLE]

At the same time, it follows from (6.10)-(6.13) that

[TABLE]

Hence, $\widetilde{X}$ is the Fréchet derivative $I_{\alpha,\nu}^{\prime}\left(v_{1}\right)\in H^{k}\left(Q_{T}\right)$ of the functional $I_{\alpha,\nu}\left(v\right)$ at the point $v_{1},$

[TABLE]

We now come back again to the case when $h\in B\left(2R\right)$ as in (6.1). Using (6.9), (6.11) and (6.13)-(6.15), we obtain

[TABLE]

We now use the Carleman estimate (2.1). Recalling that $h\left(x,T\right)=0$ and using (6.16), we obtain

[TABLE]

By (5.5)

[TABLE]

Choose the number $\nu_{3}\geq\nu_{0}>1$ depending on the same parameters as those listed in (5.12) and such that $C\sqrt{\nu_{3}}/2>C_{2}.$ Hence, (5.11), (6.16)-(6.18) and (5.13) imply that for all $\nu\geq\nu_{3}$

[TABLE]

6.2 Proof of Theorem 5.4

Denote

[TABLE]

By (5.10) and (6.19)

[TABLE]

By (5.4)

[TABLE]

Hence, using the multidimensional analog of Taylor formula [28] and (5.19), we obtain similarly with (6.4)-(6.6)

[TABLE]

where the function $P\left(x,t\right)$ is such that $\left\|P\right\|_{L_{2}\left(Q_{T}\right)}\leq C_{2}\delta.$ Since $v_{t}^{\ast}-Lv^{\ast}-G\left(\nabla v^{\ast},v^{\ast},g^{\ast},x,t\right)=0,$ then

[TABLE]

Hence, using (6.20), we obtain

[TABLE]

Next, by (5.14)

[TABLE]

By (5.16) $-I_{\alpha,\lambda,\nu}^{\prime}\left(v_{\min}\right)\left(v^{\ast}-v_{\min}\right)\leq 0.$ Hence, (6.22) and (6.23) imply that

[TABLE]

Recall that the numbers $\nu=\nu\left(\delta\right)$ and $\alpha=\alpha\left(\delta\right)$ are defined in (4.27) and (5.21) respectively. These choices of $\nu\left(\delta\right)$ and $\alpha\left(\delta\right)$ are possible since (5.20) holds and $\delta\in\left(0,\delta_{0}\right).$ Thus, condition (5.13) of Theorem 5.1 imposed on $\alpha$ is in place. Hence, (4.26) and (4.28) hold. Hence, using (3.2), (4.26)-(4.28), and (6.24), we obtain

[TABLE]

which implies (5.22). To prove (5.23), we use the triangle inequality,

[TABLE]

Using (5.18), (5.22) and (6.25), we obtain (5.23). $\square$

Acknowledgment

The work of M.V. Klibanov was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044.

Bibliography28

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Convergent numerical methods for parabolic equations with reversed

Abstract

1 Introduction

2 Carleman Estimate

3 Stability Estimate

4 The Quasi-Reversibility Method for the Linear Case

5 The Global Strict Convexity

5.1 The weighted Tikhonov-like functional

5.2 Theorems about the functional Iα,ν(v)I_{\alpha,\nu}\left(v\right)Iα,ν​(v)

6 Proofs of Theorems 5.1 and 5.4

6.1 Proof of Theorem 5.1

6.2 Proof of Theorem 5.4

5.2 Theorems about the functional $I_{\alpha,\nu}\left(v\right)$