Convergence Rate Analysis of Galerkin Approximation of Inverse Potential Problem

1 Introduction

In this work, we study the inverse problem of recovering a space-dependent potential coefficient in elliptic and parabolic equations. Let $Ω \subset R^{d} (d = 1, 2, 3)$ be a simply connected convex polyhedral domain with a boundary $\partial Ω$ . Then the governing equation in the elliptic and parabolic cases are given respectively by

{\begin{matrix} - Δ u + q u & = f, & in & Ω, u & = 0, & on & \partial Ω, \end{matrix}

(1.1)

and

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \partial_{t} u - Δ u + q u & = f, & in & Ω \times (0, T), u & = 0, & on & \partial Ω \times (0, T), u (\cdot, 0) & = u_{0}, & in & Ω, \end{matrix}

(1.2)

where $T > 0$ is the final time. The functions $f$ and $u_{0}$ in (1.1) and (1.2) are the source and initial data, respectively. The space-dependent potential $q$ belongs to the admissible set $K$ such that

K = {q \in L^{\infty} (Ω) : c_{0} \leq q (x) \leq c_{1} a.e. in Ω},

with $0 \leq c_{0} < c_{1} < \infty$ . To explicitly indicate the dependence of the solution $u$ to problems (1.1) and (1.2) on the potential $q$ , we write $u (q)$ . Further, we are given the observational data $z^{δ}$ on $Ω$ or $Ω \times (T_{0}, T)$ :

{\begin{matrix} z^{δ} (x) & = u (q^{†}) (x) + ξ (x), x \in Ω, & elliptic z^{δ} (x, t) & = u (q^{†}) (x, t) + ξ (x, t), (x, t) \in Ω \times (T_{0}, T), & parabolic \end{matrix}

where $u (q^{†})$ denotes the exact data (corresponding to the exact potential $q^{†}$ ), $0 \leq T_{0} < T$ , and $ξ$ denotes the measurement noise. The accuracy of the data $z^{δ}$ is measured by the noise level $δ = ∥ u (q^{†}) - z^{δ} ∥_{L^{2} (Ω)}$ or $δ = ∥ u (q^{†}) - z^{δ} ∥_{L^{2} (T_{0}, T; L^{2} (Ω))}$ , in the elliptic and parabolic cases, respectively. The inverse potential problem is to recover the potential $q$ from the noisy observation $z^{δ}$ . It arises in several practical applications, where $q$ represents the radiativity coefficient in heat conduction [37] and perfusion coefficient in Pennes’ bio-heat equation in human physiology [29, 33] (see [31, 39] for experimental studies) and the elliptic case also in quantitative dynamic elastography [12].

The inverse potential problem is ill-posed, which poses challenges to construct accurate and stable numerical approximations. A number of reconstruction methods have been designed to overcome the ill-posed nature, with the most prominent one being Tikhonov regularization [16, 23]. In practical computation, one still needs to discretize the continuous regularized formulation. This is often achieved by the Galerkin finite element method (FEM) when the domain $Ω$ is irregular and the problem data ( $u_{0}$ and $f$ ) have only limited regularity. This strategy has been widely used [37, 14, 38]. Yamamoto and Zou [37] proved the convergence of the discrete approximations in the parabolic case. However, the convergence rates of discrete approximations are generally very challenging to obtain, due to the strong nonconvexity of the regularized functional, which itself stems from the high degree nonlinearity of the parameter-to-state map. Indeed, this has been a long standing issue for the numerical analysis of many nonlinear inverse problems, e.g., parameter identifications for PDEs. So far there have been only very few error bounds on discrete approximations, despite the fact that such an analysis would provide useful guidelines for choosing suitable discretization parameters. For the related inverse conductivity problem, the works [35, 24] derived error bounds in a weighted $L^{2} (Ω)$ norm by employing a special test function for elliptic and parabolic cases, and the latter work [24] also gives the standard $L^{2} (Ω)$ error estimates with the help of a positivity condition on the weighted function.

In this work we study the concerned elliptic and parabolic inverse potential problems, and contribute in the following two aspects. First, we establish novel conditional stability estimates for the concerned inverse problem, including both weighted $L^{2} (Ω)$ and standard $L^{2} (Ω)$ stability. The latter is obtained under a certain positivity condition, which can be verified for a class of problem data. The derivation is purely variational, using only a nonstandard test function, and extends directly to the error analysis. Second, we derive novel weighted $L^{2} (Ω)$ error bounds for the discrete approximations under very mild regularity conditions on the problem data and unknown coefficient $q$ as well as the standard $L^{2} (Ω)$ error bounds under some positivity condition. Note that the analysis does not employ standard source type conditions. Instead, it is achieved by a novel choice of the test function in the weak formulation, as in the conditional stability analysis, adapting the stability argument to the discrete setting, which allows us to bypass the standard source condition. To the best of our knowledge, these results represent the first error bounds for the discrete approximations for the inverse potential problem. Further, we provide several numerical experiments to complement the theoretical analysis.

Now we review existing works on the analysis and numerics of the inverse potential problem. Several uniqueness and stability results have been obtained [30, 11, 25, 3, 12]. Choulli and Yamamoto [11] proved the uniqueness of recovering the potential $q^{†}$ , initial condition and boundary coefficient from terminal measurement, and also gave a stability result under a smallness condition. In the parabolic case, Beretta and Cavaterra [3] proved the unique recovery of the potential $q (x)$ from the time-averaged observation. More recently, Choulli [12] derived a new stability estimate in the elliptic case. The well-posedness of the continuous regularized formulation has been analyzed for both elliptic / parabolic cases [17, 37, 14, 38], and convergence rates with respect to the noise level $δ$ were obtained under various conditions. In the 1D elliptic case, Engl et al [17, Example 3.1] derived a convergence rate of the regularized approximation by Tikhonov regularization under the standard source condition with a small sourcewise representer. Hao and Quyen [22] presented a different approach without explicitly using the source condition. More recently, Chen et al [9] proved conditional stability of the inverse problem in negative Sobolev spaces, which allows deriving variational inequality type source conditions and also showing convergence rates of the regularized solutions, and the authors studied both elliptic and parabolic cases. Klibanov, Li, and Zhang [26] presented an interesting convexification method for the inverse problem which allows proving globle convergence despite the nonlinearity of the inverse problem (and actually the method allows recovering a time-independent source simultaneously). This work extends the current literature with new stability analysis and error analysis of discrete approximations Broadly speaking, this work is along the line of research which connects stability analysis with error analysis of discrete schemes (see, e.g., [7, 8] and references therein) and convergence rate with conditional stability (see, e.g., [10, 36]).

The rest of the paper is organized as follows. In Section 2 we present novel conditional stability estimates for the concerned inverse problems. In Sections 3 and 4, we describe the regularized formulation, their finite element discretizations, and derive novel error bounds on the discrete approximations, for the elliptic and parabolic cases, respectively. In Section 5, we present one- and two-dimensional numerical experiments to complement the theoretical analysis. We conclude with some useful notation. For any $m \geq 0$ and $p \geq 1$ , we denote by $W^{m, p} (Ω)$ the standard Sobolev spaces of order $m$ , equipped with the norm $∥ \cdot ∥_{W^{m, p} (Ω)}$ and also write $H^{m} (Ω)$ and $H_{0}^{m} (Ω)$ with the norm $∥ \cdot ∥_{H^{m} (Ω)}$ when $p = 2$ [1]. We denote the $L^{2} (Ω)$ inner product by $(\cdot, \cdot)$ . We also use Bochner spaces: For a Banach space $B$ , we define by

H^{m} (0, T; B) = {v : v (\cdot, t) \in B for a.e. t \in (0, T) and ∥ v ∥_{H^{m} (0, T; B)} < \infty} .

The space $L^{\infty} (0, T; B)$ is defined similarly. Throughout, we denote by $C$ a positive constant not necessarily the same at each occurrence but always independent of the discretization parameters $h$ and $τ$ , the noise level $δ$ and the regularization parameter $α$ .

2 Conditional stability estimates

In this section, we present novel conditional stability estimates for the concerned inverse problem. The analysis will also inspire the error analysis of the discrete approximations in Sections 3 and 4.

2.1 Elliptic inverse problem

We have the following conditional stability results in weighted and standard $L^{2} (Ω)$ norms.

Theorem 2.1.

Suppose that $q_{1}, q_{2} \in K \cap H^{1} (Ω)$ and $f \in L^{2} (Ω)$ . Let $u (q_{1})$ and $u (q_{2})$ be the corresponding weak solutions of problem (1.1). Then there holds

∥ (q_{1} - q_{2}) u (q_{1}) ∥_{L^{2} (Ω)} \leq C ∥ u (q_{1}) - u (q_{2}) ∥_{H^{1} (Ω)}^{\frac{1}{2}} .

Moreover, if there exists a $β \geq 0$ such that

u(q1)(x)≥C\rm dist(x,∂Ω)βa.e. in Ω,

(2.1)

then the following estimate holds

∥ q_{1} - q_{2} ∥_{L^{2} (Ω)} \leq C ∥ u (q_{1}) - u (q_{2}) ∥_{H^{1} (Ω)}^{\frac{1}{2 (1 + 2 β)}} .

Proof.

By the weak formulations of $u (q_{1})$ and $u (q_{2})$ , for any $φ \in H_{0}^{1} (Ω)$

((q_{1} - q_{2}) u (q_{1}), φ) = - (\nabla (u (q_{1}) - u (q_{2})), \nabla φ) - (q_{2} (u (q_{1}) - u (q_{2})), φ) =: I .

Let $φ = (q_{1} - q_{2}) u (q_{1})$ . Note that $\nabla φ = (\nabla q_{1} - \nabla q_{2}) u (q_{1}) + (q_{1} - q_{2}) \nabla u (q_{1})$ . Since $q_{1} \in K \cap H^{1} (Ω)$ , elliptic regularity theory implies $u (q_{1}) \in H^{2} (Ω)$ , and by Sobolev embedding, we have $u (q_{1}) \in L^{\infty} (Ω)$ for $d = 1, 2, 3$ . Then we have $∥ φ ∥_{L^{2} (Ω)} \leq C$ and $∥ \nabla φ ∥_{L^{2} (Ω)} \leq C$ , i.e., $φ \in H_{0}^{1} (Ω)$ . Now by the Cauchy-Schwarz inequality, we obtain the first assertion by

| I | \leq C (∥ \nabla (u (q_{1}) - u (q_{2})) ∥_{L^{2} (Ω)} + ∥ (u (q_{1}) - u (q_{2})) ∥_{L^{2} (Ω)}) \leq C ∥ (u (q_{1}) - u (q_{2})) ∥_{H^{1} (Ω)} .

(2.2)

Next we decompose the domain $Ω$ into two disjoint sets $Ω = Ω_{ρ} \cup Ω_{ρ}^{c}$ , with $Ω_{ρ} = {x \in Ω : d i s t (x, \partial Ω) \geq ρ}$ and $Ω_{ρ}^{c} = Ω ∖ Ω_{ρ}$ , with the constant $ρ > 0$ to be chosen. On the subdomain $Ω_{ρ}$ , we have

	$\int_{Ω_{ρ}} (q_{1} - q_{2})^{2} d x$	$= ρ^{- 2 β} \int_{Ω_{ρ}} (q_{1} - q_{2})^{2} ρ^{2 β} d x \leq ρ^{- 2 β} \int_{Ω_{ρ}} (q_{1} - q_{2})^{2} d i s t (x, \partial Ω)^{2 β} d x$
		$\leq C ρ^{- 2 β} \int_{Ω_{ρ}} (q_{1} - q_{2})^{2} u (q_{1})^{2} d x \leq C ρ^{- 2 β} ∥ u (q_{1}) - u (q_{2}) ∥_{H^{1} (Ω)} .$

Meanwhile, by the box constraint of $K$ , we have

\int_{Ω_{ρ}^{c}} (q_{1} - q_{2})^{2} d x \leq C | Ω_{ρ}^{c} | \leq C ρ .

Then the desired result follows by balancing the last two estimates with $ρ$ . ∎

The positivity condition (2.1) quantifies the decay rate of the solution $u (q_{1})$ to zero as $d i s t (x, \partial Ω) \to 0^{+}$ (due to the presence of a zero Dirichlet boundary condition). It can be verified under suitable conditions on the source $f$ . This requires the following property of Green’s function for the elliptic problem. The notation $B (x, r)$ denotes the ball centered at $x \in R^{d}$ with a radius $r$ .

Theorem 2.2.

Let the diffusion coefficient $a \in L^{\infty} (Ω)$ with a strictly positive lower bound over $Ω$ . For any $y \in Ω$ and $r > 0$ , let $G_{q} (x) := G_{q} (x, y) \in H^{1} (Ω ∖ B (y, r)) \cap W_{0}^{1, 1} (Ω)$ be Green’s function for the elliptic operator $- d i v (a \nabla \cdot) + q I$ (with a zero Dirichlet boundary condition). Then for $d \geq 2$ , the following estimate holds

G_{q} (x, y) \geq C | x - y |^{2 - d}, for | x - y | \leq \frac{1}{2} d i s t (x, \partial Ω) .

Proof.

When $q \equiv 0$ , the result is well known for $d \geq 3$ [27, 21]. We prove the slightly more general case for completeness. Let $ρ (x) = d i s t (x, \partial Ω)$ . Since the operator $- d i v (a \nabla \cdot) + q I$ is self-adjoint, there holds $G_{q} (x, y) = G_{q} (y, x)$ . It suffices to prove

G_{q} (x, y) \geq C | x - y |^{2 - d}, for | x - y | \leq \frac{1}{2} ρ (y) .

(2.3)

By definition, we have

\int_{Ω} a (z) \nabla G_{q} (z, y) \cdot \nabla φ (z) + q (z) G_{q} (z, y) φ (z) d z = φ (y), \forall φ \in C_{0}^{\infty} (Ω) .

(2.4)

Let $r := | x - y |$ . Consider a cut-off function $φ_{1} \in C_{0}^{\infty} (Ω)$ with the following properties: $φ_{1} \equiv 1$ on $B (y, r) \cap (Ω ∖ B (y, \frac{r}{2}))$ and $φ_{1} \equiv 0$ on $(Ω ∖ B (y, \frac{3 r}{2})) \cup B (y, \frac{r}{4})$ , meanwhile $0 \leq φ_{1} \leq 1$ and $| \nabla φ_{1} | \leq C r^{- 1}$ . By inserting a test function $(φ_{1} (z))^{2} G_{q} (z, y)$ into (2.4) and applying the boundedness and uniform ellipticity of the operator and the Cauchy-Schwarz inequality, since $G_{q} (z) := G_{q} (z, y) \in H^{1} (Ω ∖ B (y, \frac{r}{4}))$ , we derive

		$\int_{\frac{r}{4} \leq \| z - y \| \leq \frac{3 r}{2}} φ_{1} (z)^{2} \| \nabla G_{q} (z, y) \|^{2} d z \leq C \int_{\frac{r}{4} \| z - y \| \leq \frac{3 r}{2}} φ_{1} (z) G_{q} (z, y) \| \nabla φ_{1} (z) \| \| \nabla G_{q} (z, y) \| d z$
	$\leq$	$C (\int_{\frac{r}{4} \leq \| z - y \| \leq \frac{3 r}{2}} φ_{1} (z)^{2} \| \nabla G_{q} (z, y) \|^{2} d z)^{\frac{1}{2}} (\int_{\frac{r}{4} \leq \| z - y \| \leq \frac{3 r}{2}} G_{q} (z, y)^{2} \| \nabla φ_{1} (z) \|^{2} d z)^{\frac{1}{2}} .$

This and the construction of $φ_{1}$ lead to

\int_{\frac{r}{2} \leq | z - y | \leq r} | \nabla G_{q} (z) |^{2} d z \leq C r^{d - 2} sup \frac{r}{4} \leq | z - y | \leq \frac{3 r}{2} | G_{q} (z, y) |^{2} .

(2.5)

Since $G_{q} (x, y) \in W_{0}^{1, 1} (Ω)$ and $q \leq c_{1}$ , we can choose a sufficiently small radius $r_{0} := r_{0} (c_{1}) < \frac{r}{3}$ such that

\int_{z \in B (y, r_{0})} G_{q} (z, y) d z \leq \frac{1}{2 c_{1}} .

Replacing the radius $r$ by $r_{0}$ and repeating the argument of (2.5) yield

\int_{\frac{r_{0}}{2} \leq | z - y | \leq r_{0}} | \nabla G_{q} (z) |^{2} d z \leq C r_{0}^{d - 2} sup \frac{r_{0}}{4} \leq | z - y | \leq \frac{3 r_{0}}{2} | G_{q} (z, y) |^{2} .

Let $φ_{2} \in C_{0}^{\infty} (Ω)$ be a test function of (2.4) satisfying $φ_{2} \equiv 1$ on $B (y, \frac{r_{0}}{2})$ and $φ_{2} \equiv 0$ on $((Ω ∖ B (y, r_{0})) \cap B (y, \frac{r}{2})) \cup (Ω ∖ B (y, r))$ , with $0 \leq φ_{2} \leq 1$ on $Ω$ , $| \nabla φ_{2} | \leq C r_{0}^{- 1}$ on $B (y, r_{0})$ and $| \nabla φ_{2} | \leq C r^{- 1}$ on $(Ω ∖ B (y, r / 2)) \cap B (y, r)$ . It follows from the boundedness of the operator, and the last three estimates that

	$1$	$= \int_{Ω} a (z) \nabla G_{q} (z, y) \cdot \nabla φ_{2} (z) + q (z) G_{q} (z, y) φ_{2} (z) d z$
		$\leq \int_{\frac{r_{0}}{2} \leq \| z - y \| \leq r_{0}} a (z) \nabla G_{q} (z, y) \cdot \nabla φ_{2} (z) d z + \int_{\frac{r}{2} \leq \| z - y \| \leq r} a (z) \nabla G_{q} (z, y) \cdot \nabla φ_{2} (z) d z$
		$+ \frac{1}{2} + \int_{\frac{r}{2} \leq \| z - y \| \leq r} q (z) G_{q} (z, y) φ_{2} (z) d z$
		$\leq C r_{0}^{d - 2} sup \frac{r_{0}}{4} \leq \| z - y \| \leq \frac{3 r_{0}}{2} G_{q} (z, y) + C r^{d - 2} sup \frac{r}{4} \leq \| z - y \| \leq \frac{3 r}{2} G_{q} (z, y) + \frac{1}{2} + C r^{d} sup \frac{r}{2} \leq \| z - y \| \leq r G_{q} (z, y)$
		$\leq C r^{d - 2} sup \frac{r_{0}}{4} \leq \| z - y \| \leq \frac{3 r}{2} G_{q} (z, y) + \frac{1}{2} \leq C r^{d - 2} inf \frac{r_{0}}{4} \leq \| z - y \| \leq \frac{3 r}{2} G_{q} (z, y) + \frac{1}{2} \leq C \| x - y \|^{d - 2} G_{q} (x, y) + \frac{1}{2},$

where we have used Harnack’s inequality [20, p. 189] for Green’s function $G_{q} (z)$ on the compact subset ${z \in Ω : \frac{r_{0}}{4} \leq | z - y | \leq \frac{3 r}{2}} \subset\subset Ω$ , with the constant $C$ depending on the $d$ , $c_{1}$ and $Ω$ . This completes the proof of the theorem. ∎

Remark 2.1.

When $d = 1$ , i.e., $Ω = (a, b)$ with $- \infty < a < b < \infty$ , Green’s function $G_{c_{1}} (x, y)$ of the operator $- Δ + c_{1} I$ is explicitly given by

(2.6)

Now consider the asymptotics of the function $G_{c_{1}} (x, y)$ near the boundary. Let $y$ be close to the point $a$ and $| x - y | \leq \frac{1}{2} (y - a)$ . Since $e^{x} - 1 \geq x$ on $R$ , we have

G_{c_{1}} (x, y) {\begin{matrix} ≃ e^{\sqrt{c_{1}} (x - 2 a)} - e^{- \sqrt{c_{1}} x} \geq C (x - a) \geq C | x - y |, & a \leq x \leq y, ≃ e^{2 \sqrt{c_{1}} y} - e^{2 \sqrt{c_{1}} a} \geq C (y - a) \geq C | x - y |, & a \leq y \leq x, \end{matrix}

where the symbol $‘ ‘ ≃"$ denotes by $‘ ‘ ="$ up to a positive constant depending on $Ω$ and $c_{1}$ . A similar result holds when $y$ is close to the point $b$ . Since the operator $- Δ + c_{1} I$ is self-adjoint, we have

G_{c_{1}} (x, y) \geq C | x - y | for | x - y | \leq \frac{1}{2} ρ (x) := d i s t (x, \partial Ω) .

That is, the assertion in Theorem 2.2 holds also for $d = 1$ . For a general potential $q \in K \cap H^{1} (Ω)$ , by the weak maximum principle, for any fixed $y$ , we have $G_{q} (x, y) \geq G_{c_{1}} (x, y)$ a.e. $x \in Ω$ , and thus the desired assertion follows.

Now we can state a sufficient condition for the positivity condition (2.1).

Proposition 2.1.

Let $q \in H^{1} (Ω) \cap K$ and $f \geq c_{f} > 0$ a.e. in $Ω$ . Then condition (2.1) holds with $β = 2$ .

Proof.

Recall that for every $y \in Ω$ , there exists a unique Green’s function $G_{q} (\cdot, y) \in H^{1} (Ω ∖ B (y, r)) \cap W_{0}^{1, 1} (Ω)$ for the elliptic operator $- Δ + q I$ , such that

\int_{Ω} \nabla G_{q} (x, y) \cdot \nabla φ (x) + q G_{q} (x, y) φ (x) d x = φ (y), \forall φ \in C_{0}^{\infty} (Ω) .

By Theorem 2.2, we have

G_{q} (x, y) \geq C | x - y |^{- (d - 2)} for | x - y | \leq \frac{1}{2} ρ (x) := d i s t (x, \partial Ω), d \geq 2.

Now for any $x \in Ω$ , let $B (x, \frac{ρ (x)}{2}) \subset Ω$ be the ball centered at $x$ with a radius $\frac{ρ (x)}{2}$ . Since $G_{q} (x, y) \geq 0$ , $x, y \in Ω$ , we have

	$u (q) (x)$	$= \int_{Ω} G_{q} (x, y) f (y) d y \geq \int_{B (x, \frac{ρ (x)}{2})} f (y) G_{q} (x, y) d y$
		$\geq c_{f} C \int_{B (x, \frac{ρ (x)}{2})} \| x - y \|^{- (d - 2)} d y \geq C ρ^{2} (x) = C d i s t (x, \partial Ω)^{2},$

and thus the desired result follows directly. ∎

2.2 Parabolic inverse problem

The next result gives a conditional stability estimate for the parabolic inverse problem.

Theorem 2.3.

Suppose that $q_{1}, q_{2} \in K \cap H^{1} (Ω)$ , $u_{0} \in H^{2} (Ω) \cap H_{0}^{1} (Ω)$ and $f \in H^{1} (0, T; L^{2} (Ω))$ . Let $u (q_{1})$ and $u (q_{2})$ be the corresponding weak solutions of problem (1.2). Then with $T_{0} \leq s \leq t \leq T$ , there holds

\int_{T_{0}}^{T} \int_{T_{0}}^{t} \int_{s}^{t} ∥ (q_{1} - q_{2}) u (q_{1}) (ξ) ∥_{L^{2} (Ω)} d ξ d s d t \leq C ∥ u (q_{1}) - u (q_{2}) ∥_{L^{2} (T_{0}, T; H^{1} (Ω))}^{\frac{1}{2}} .

Moreover, suppose that there exists a $β \geq 0$ such that

u (q_{1}) (x, t) \geq C d i s t (x, \partial Ω)^{β} a.e. in Ω

(2.7)

for any $t \in (T_{0}, T)$ . Then there holds

∥ q_{1} - q_{2} ∥_{L^{2} (Ω)} \leq C ∥ u (q_{1}) - u (q_{2}) ∥_{L^{2} (T_{0}, T; H^{1} (Ω))}^{\frac{1}{2 (1 + 2 β)}} .

Proof.

By the weak formulations of $u (q_{1})$ and $u (q_{2})$ , for any $φ \in H_{0}^{1} (Ω)$

((q_{1} - q_{2}) u (q_{1}), φ) = - (\partial_{ξ} u (q_{1}) - \partial_{ξ} u (q_{2}), φ) - (\nabla (u (q_{1}) - u (q_{2})), \nabla φ) - (q_{2} (u (q_{1}) - u (q_{2})), φ) =: I_{1} + I_{2} + I_{3} .

Let $φ = (q_{1} - q_{2}) u (q_{1})$ . Then $\nabla φ = (\nabla q_{1} - \nabla q_{2}) u (q_{1}) + (q_{1} - q_{2}) \nabla u (q_{1})$ . By the standard parabolic regularity theory [18], problem (4.2) has a unique solution $u (q_{1}) \in H^{1} (0, T; H_{0}^{1} (Ω)) \cap L^{\infty} (0, T; H^{2} (Ω))$ , and then by Sobolev embedding theorem [1], $u (q_{1}) \in L^{\infty} (0, T; L^{\infty} (Ω))$ . Then there holds $∥ φ ∥_{L^{2} (Ω)} \leq C$ and $∥ \nabla φ ∥_{L^{2} (Ω)} \leq C$ . Thus we have $φ \in H_{0}^{1} (Ω)$ . Meanwhile, the Cauchy-Schwarz inequality and the box constraint yield

\int_{T_{0}}^{T} | I_{2} | d t \leq C ∥ \nabla (u (q_{1}) - u (q_{2})) ∥_{L^{2} (T_{0}, T; L^{2} (Ω))} and \int_{T_{0}}^{T} | I_{3} | d t \leq C ∥ u (q_{1}) - u (q_{2}) ∥_{L^{2} (T_{0}, T; L^{2} (Ω))} .

It remains to bound the term $I_{1}$ . By integration by parts, we have

	$\int_{s}^{t} (\partial_{ξ} u (q_{1}) (ξ) - \partial_{ξ} u (q_{2}) (ξ), φ (ξ)) d ξ$	$= (u (q_{1}) (t) - u (q_{2}) (t), φ (t)) - (u (q_{1}) (s) - u (q_{2}) (s), φ (s))$
		$- \int_{s}^{t} (u (q_{1}) (ξ) - u (q_{2}) (ξ), \partial_{ξ} φ (ξ)) d ξ .$

For the first two terms, by the Cauchy-Schwarz inequality, since $∥ φ (t) ∥_{L^{2} (Ω)} \leq C$ , we have

∣ ∣ \int_{T_{0}}^{T} \int_{T_{0}}^{t} (u (q_{1}) - u (q_{2}), φ) (t) - (u (q_{1}) - u (q_{2}), φ) (s) d s d t ∣ ∣ \leq C ∥ u (q_{1}) - u (q_{2}) ∥_{L^{2} (T_{0}, T; L^{2} (Ω))} .

Next by the regularity $u \in H^{1} (0, T; L^{2} (Ω))$ , and the box constraint in $K$ , we have $\partial_{ξ} φ = (q_{1} - q_{2}) \partial_{ξ} u (q_{1}) (ξ) \in L^{2} (0, T; L^{2} (Ω))$ . Using Cauchy-Schwarz inequality leads to

	$\| \int_{T_{0}}^{T} \int_{T_{0}}^{t} \int_{s}^{t} (u (q_{1}) (ξ) - u (q_{2}) (ξ), \partial_{ξ} φ (ξ)) d ξ d s d t \|$	$\leq C (\int_{T_{0}}^{T} ∥ u (q_{1}) - u (q_{2}) ∥_{L^{2} (Ω)}^{2} d t)^{\frac{1}{2}} (\int_{T_{0}}^{T} ∥ \partial_{t} φ ∥_{L^{2} (Ω)}^{2} d t)^{\frac{1}{2}}$
		$\leq C ∥ u (q_{1}) - u (q_{2}) ∥_{L^{2} (T_{0}, T; L^{2} (Ω))} ∥ \partial_{t} u ∥_{L^{2} (T_{0}, T; L^{2} (Ω))} .$

These estimates directly imply the first desired assertion. Under the positivity condition (2.7), the $L^{2} (Ω)$ estimate follows from the argument of Theorem 2.1. ∎

The next result gives a sufficient condition on the positivity condition (2.7).

Proposition 2.2.

Let $q \in H^{1} (Ω) \cap K$ , the source $f \geq c_{f} > 0$ and $\partial_{t} f \leq 0$ a.e. in $Ω \times (0, T)$ , $u_{0} \geq 0$ and $f (0) + Δ u_{0} - q u_{0} \leq 0$ a.e. in $Ω$ . Then there exists a positive constant $C$ , depending only on $c_{0}$ , $c_{1}$ , $c_{f}$ and $Ω$ , such that the positivity condition (2.7) holds with $β = 2$ .

Proof.

Since $f \geq 0$ and $u_{0} \geq 0$ , the standard parabolic maximum principle (see, e.g., [28, 19]) implies $u \geq 0$ , a.e. in $Ω \times [0, T] .$ Let $w := \partial_{t} u$ , which satisfies

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \partial_{t} w - Δ w + q w & = \partial_{t} f, & in & Ω \times (0, T), w & = 0, & on & \partial Ω \times (0, T), w (0) & = f (0) + Δ u_{0} - q u_{0}, & % in & Ω . \end{matrix}

Since $\partial_{t} f \leq 0$ a.e. in $Ω \times (0, T)$ and $f (0) + Δ u_{0} - q u_{0} \leq 0$ a.e. in $Ω$ , the parabolic maximum principle yields $w \leq 0$ a.e. in $Ω \times [0, T]$ . It suffices to prove that (2.7) with $β = 2$ holds for any $t \in (0, T]$ . By fixing $t \in [T_{0}, T]$ , we have $f (t) - w (t) \in L^{2} (Ω)$ . Then consider the following elliptic problem

{\begin{matrix} - Δ u (t) + q u (t) & = f (t) - w (t), & in & Ω, u (t) & = 0, & on & \partial Ω . \end{matrix}

By Green’s function $G_{q} (x, y)$ in Theorem 2.2, there holds

	$u (q) (x, t)$	$= \int_{Ω} G_{q} (x, y) (f (y, t) - w (y, t)) d y \geq \int_{B (x, \frac{ρ (x)}{2})} f (y, t) G_{q} (x, y) d y$
		$\geq c_{f} C \int_{B (x, \frac{ρ (x)}{2})} \| x - y \|^{- (d - 2)} d y \geq C ρ^{2} (x) = C d i s t (x, \partial Ω)^{2},$

for any $x \in Ω$ and $t \in [T_{0}, T]$ , i.e., the positivity condition (2.7) with $β = 2$ holds. ∎

3 Error analysis for the elliptic inverse problem

Now we formulate the regularized output least-squares formulation for the elliptic inverse problem, discretize the continuous formulation by the Galerkin FEM with continuous piecewise linear elements, and provide a complete error analysis.

3.1 Regularization problem and its FEM approximation

To reconstruct the coefficient $q$ , we employ the standard Tikhonov regularization with an $H^{1} (Ω)$ seminorm penalty [16, 23], which is equivalent to minimizing the following regularized functional:

min q \in K J_{α} (q) = \frac{1}{2} ∥ u (q) - z^{δ} ∥_{L^{2} (Ω)}^{2} + \frac{α}{2} ∥ \nabla q ∥_{L^{2} (Ω)}^{2},

(3.1)

where $u (q) \in H_{0}^{1} (Ω)$ satisfies

(\nabla u (q), \nabla φ) + (q u (q), φ) = (f, φ), \forall φ \in H_{0}^{1} (Ω) .

(3.2)

Recall that the given data $z^{δ} \in L^{2} (Ω)$ is noisy with a noise level $δ$ relative to the exact data $u (q^{†})$ (corresponding to the exact radiativity $q^{†}$ ), i.e., $∥ u (q^{†}) - z^{δ} ∥_{L^{2} (Ω)} = δ .$ The continuous problem (3.1)–(3.2) is well-posed in the sense that it has at least one globale minimizer, and the minimizer is continuous with respect to the perturbations in the data, and further as the noise level tends to zero, the sequence of minimizers converges to the exact solution in $H^{1} (Ω)$ (if $α$ is chosen properly) [17, 16, 23].

To discretize problem (3.1)–(3.2), we employ the standard Galerkin FEM [5]. Let $T_{h}$ be a shape regular quasi-uniform simplicial triangulation of the domain $Ω$ , with a grid size $h$ . On the triangulation $T_{h}$ , we define the conforming piecewise linear finite element spaces $V_{h}$ and $V_{h 0}$ by

V_{h} := {v_{h} \in H^{1} (Ω) : v_{h} |_{T} is a linear polynomial, \forall T \in T_{h}}

and $V_{h 0} := V_{h} \cap H_{0}^{1} (Ω)$ . We use the spaces $V_{h 0}$ and $V_{h}$ to approximate the state $u$ and the parameter $q$ , respectively. The following inverse inequality holds in the finite element space $V_{h 0}$ [5]

∥ v_{h} ∥_{H^{1} (Ω)} \leq C h^{- 1} ∥ v_{h} ∥_{L^{2} (Ω)}, \forall v_{h} \in V_{h 0} .

(3.3)

We denote by $P_{h}$ the standard $L^{2} (Ω)$ -projection operator associated with the finite element space $V_{h 0}$ . Then it is known that for $s =$

	$1$	$= \int_{Ω} a (z) \nabla G_{q} (z, y) \cdot \nabla φ_{2} (z) + q (z) G_{q} (z, y) φ_{2} (z) d z$
		$\leq \int_{\frac{r_{0}}{2} \leq \| z - y \| \leq r_{0}} a (z) \nabla G_{q} (z, y) \cdot \nabla φ_{2} (z) d z + \int_{\frac{r}{2} \leq \| z - y \| \leq r} a (z) \nabla G_{q} (z, y) \cdot \nabla φ_{2} (z) d z$
		$+ \frac{1}{2} + \int_{\frac{r}{2} \leq \| z - y \| \leq r} q (z) G_{q} (z, y) φ_{2} (z) d z$
		$\leq C r_{0}^{d - 2} sup \frac{r_{0}}{4} \leq \| z - y \| \leq \frac{3 r_{0}}{2} G_{q} (z, y) + C r^{d - 2} sup \frac{r}{4} \leq \| z - y \| \leq \frac{3 r}{2} G_{q} (z, y) + \frac{1}{2} + C r^{d} sup \frac{r}{2} \leq \| z - y \| \leq r G_{q} (z, y)$
		$\leq C r^{d - 2} sup \frac{r_{0}}{4} \leq \| z - y \| \leq \frac{3 r}{2} G_{q} (z, y) + \frac{1}{2} \leq C r^{d - 2} inf \frac{r_{0}}{4} \leq \| z - y \| \leq \frac{3 r}{2} G_{q} (z, y) + \frac{1}{2} \leq C \| x - y \|^{d - 2} G_{q} (x, y) + \frac{1}{2},$

Convergence Rate Analysis of Galerkin Approximation of Inverse Potential Problem

Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Incomplete Iterative Solution of the Subdiffusion Problem

Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic systems with boundary observations

Numerical Estimation of a Diffusion Coefficient in Subdiffusion

Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

Accuracy controlled data assimilation for parabolic problems

1 Introduction

2 Conditional stability estimates

2.1 Elliptic inverse problem

Theorem 2.1.

Proof.

Theorem 2.2.

Proof.

Remark 2.1.

Proposition 2.1.

Proof.

2.2 Parabolic inverse problem

Theorem 2.3.

Proof.

Proposition 2.2.

Proof.

3 Error analysis for the elliptic inverse problem

3.1 Regularization problem and its FEM approximation

Convergence Rate Analysis of Galerkin Approximation of Inverse Potential Problem

Related Research

Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Incomplete Iterative Solution of the Subdiffusion Problem

Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic systems with boundary observations

Numerical Estimation of a Diffusion Coefficient in Subdiffusion

Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

Accuracy controlled data assimilation for parabolic problems

1 Introduction

2 Conditional stability estimates

2.1 Elliptic inverse problem

Theorem 2.1.

Proof.

Theorem 2.2.

Proof.

Remark 2.1.

Proposition 2.1.

Proof.

2.2 Parabolic inverse problem

Theorem 2.3.

Proof.

Proposition 2.2.

Proof.

3 Error analysis for the elliptic inverse problem

3.1 Regularization problem and its FEM approximation