Convergence of a spectral regularization of a time-reversed reaction-diffusion problem with high-order Sobolev-Gevrey smoothness

Here we are interested in a time-reversed reaction-diffusion model, denoted by $\left(\mathcal{B}\right)$, with nonlinear source terms. Let $\Omega=\left[0,\ell\right]^{d}$ be a cube of $\mathbb{R}^{d}$ for $d\in\mathbb{N}$. In this context, we consider a population density $u=u\left(x,t\right)$ where $\left(x,t\right)\in Q_{T}:=\Omega\times\left(0,T\right)$ with $T>0$, obeying the following evolution equation:

associated with the periodic boundary conditions, i.e. $u\left(x+e_{i}\ell,\cdot\right)=u\left(x,\cdot\right)$ for $1\leq i\leq d$ where $e_{i}$ denotes the standard basis vector for $\mathbb{R}^{d}$.

Here, $\mathcal{A}:=I-\Delta$ involves the linear second-order differential operator and thus accounts for the anisotropic diffusion of the population. The nonlinearity $F$ indicates either the deterministic reaction rate or the proliferation rate for some mechanism processes. Eventually, we complete the time-reversed model by the final condition

Together with the periodic boundary condition, (1.1) and (1.2) structure our time-reversed reaction-diffusion model. In principle, the problem $\left(\mathcal{B}\right)$ is well known to be severely ill-posed and has been investigated in a wide range of real-world applications; see e.g. [17, 10, 5] and references cited therein for an overview of recent results and existing models. Taking into account tumour models (see e.g. [10]), the physical meaning behind this problem $\left(\mathcal{B}\right)$ is locating the tumour source by recovering the initial density of the tumour cells. In other words, the inverse problem we want to investigate in this paper is seeking the initial value $u\left(x,t=0\right)=g_{0}\left(x\right)$ in a regular tissue $\Omega$, provided that (1.1) and (1.2) are satisfied.

Naturally, we impose here the standard measurement on the final data (1.2), which reads as

where $\varepsilon>0$ represents the deterministic noise level. In practice, one attempts to get the initial function $g_{0}\left(x\right)$ from this measured data $g_{T}^{\varepsilon}$ using some potential regularization.

In the following, we denote by $H_{\#}^{p}$ for $p\in\mathbb{N}$ the Hilbert spaces equipped with the standard norms and with the periodic boundary conditions posed in the domain. For $\sigma\geq 0$, we define the Gevrey classes $G_{\sigma}^{p/2}=\mathcal{D}\left(\mathcal{A}^{p/2}e^{\sigma\mathcal{A}^{1/2}}\right)$ where the operator $\mathcal{A}$ is defined in (1.1). These are also Hilbert spaces with respect to the inner product

and then with the corresponding norm

With this setting, it is worth mentioning the weak solvability of the forward problem of $\left(\mathcal{B}\right)$ where $F$ is real analytic. Note that cf. [2], the real analyticity of $F$ means that if it can be represented by $F\left(u\right)=\sum_{j=0}^{\infty}a_{j}u^{j}$ for $a_{j}\in\mathbb{R}$, then the corresponding majorising series $\sum_{j=0}^{\infty}\left|a_{j}\right|u^{j}$ is convergent for any $u\in\mathbb{R}$.

As a by-product of Theorem 2, we can show that $u\left(\cdot,t\right)\in G_{T^{*}}^{p/2}\left(\Omega\right)$ for all $t\geq T^{*}$ and then $F\left(u\right)\in G_{T^{*}}^{p/2}\left(\Omega\right)$ by virtue of the Taylor series expansion of $F$. Therefore, in this work we assume our final time of observation $T$ is such that $T<T^{*}$. Since $G_{T^{*}}^{p/2}\left(\Omega\right)\subset H^{p}\left(\Omega\right)$, it is reasonable to assume in $\left(\mathcal{B}\right)$ that

This paper is devoted to the convergence analysis of a modified cut-off regularization of the inverse problem $\left(\mathcal{B}\right)$. In this regard, we comply with the weak solvability of the forward model to take into account the Gevrey-Sobolev source conditions for the inverse problem. In other words, together with Assumption (1.4) we make use of the following source condition:

It is worth mentioning that in the forward process, we obtain the solution in $C\left(\left[0,T^{*}\right];H_{\#}^{p}\left(\Omega\right)\right)$ with $p>d/2$, which indicates the fact that the solution belongs to $C\left(\left[0,T\right];L^{\infty}\left(\Omega\right)\right)$. Instead of working with the real analyticity of $F$, we can further apply the mean value theorem to get

where for $u,v\in C\left(\left[0,T\right];L^{\infty}\left(\Omega\right)\right)$ we have denoted by

To this end, we will thus exploit Assumption (1.6) through the analysis of Section 2. We also assume that there exists a continuous function $L\left(M\right)>0$ such that

Naturally, observing the applications mentioned in Remark 1 we have

• •

  for the von Bertalanffy law: $L\left(M\right)=\left|a\right|+\left|b\right|\left(N+1\right)M^{N}$;

• •

  for the Gompertz law: $L\left(M\right)=\left|a\right|+\left|b\right|M$

• •

  for the de Pillis-Radunskaya law: $L\left(M\right)=\left|\frac{a}{b}\right|NM^{N-1}$ for $b\neq 0$;

• •

  for the Arrhenius law: $L\left(M\right)=\left|a\right|$ if $a<0$ and $L\left(M\right)=ae^{aM}$ if $a\geq 0$.

In this work, all the constants $C$ used here are independent of the measurement error $\varepsilon$. Nonetheless, their precise values may change from line to line and even within a single chain of estimates. On the other side, we use either the superscript or the subscript $\varepsilon$ to accentuate the possible dependence of the present error on the constants.

In the first step, we derive the error bounds when $L\left(M\right)=C$ is independent of $M$. Our proof relies on the way we verify a variational source condition. This interestingly contributes to one of their first applications to the convergence analysis of regularization of inverse problems for nonlinear PDEs. In this approach, we prove the usual logarithmic-type rate of convergence of the nonlinear spectral regularization. We remark that during the time evolving backward in the open set $\left(0,T\right)$, the nonlinear scheme yields the asymptotic Hölder rate. When $L\left(M\right)$ essentially depends on $M$, the convergence is slower due to the phenomenal growth of the quantity $L\left(M\right)$ involved in the Lipschitz property (1.6). Technically, the impediment of this growth, albeit its high-impact on the structure of the variational source condition, can be solved using a careful choice of a cut-off function for the nonlinearity $F$. In this sense, the regularized solution is sought in a proper open set decided by the measurement parameter $\varepsilon$. Eventually, a slower logarithmic-type convergence is also expected, which extends the results in [8, 17, 15] and references cited therein. Typically, this extension illuminates that the Gevrey regularity conventionally restricted to the convergence analysis in [17, 15] is applicable when solving a class of time-reversed PDEs. We also add that this work partly completes the gap of verifying variational source conditions for a class of nonlinear PDEs, which is still questioned in [8].

In the second theme, our concentration moves to a derivation of an iteration-based version of the nonlinear spectral scheme with a stronger version of (1.6). We end up with the convergence of the approximate scheme by exploring the choice of the so-called stabilization constant, which depends not only on the number of iterations, but also on the measurement $\varepsilon$.

Prior to the setting of our approach and to closing this section, we introduce what the variational source condition concerns in principle as it is exploited in the skeleton of our proof.

Consider the ill-posed operator equation $\mathcal{T}\left(f\right)=g$ where $\mathcal{T}$ maps from $\mathcal{D}\left(\mathcal{T}\right)\subset\mathbb{X}$ to $\mathbb{Y}$ with $\mathbb{X}$ and $\mathbb{Y}$ being Hilbert spaces. In this Hilbert setting, we denote by $f^{\dagger}\in\mathcal{D}\left(\mathcal{T}\right)$ the exact solution and by $g^{\varepsilon}\in\mathbb{Y}$ the noisy data satisfying $\left\|\mathcal{T}\left(f^{\dagger}\right)-g^{\varepsilon}\right\|_{\mathbb{Y}}\leq\varepsilon$, where $\varepsilon>0$ represents the deterministic noise level (cf. e.g (1.3)). There are several stable approximations to regularize such equations. One of the most effective methods is Tikhonov regularization in which the exact solution is approximated by a solution of the minimization problem, viz.

for some regularization parameter $\alpha:=\alpha\left(\varepsilon\right)>0$.

In regularization theory, one can prove that $\mathcal{R}_{\alpha}\left(g^{\varepsilon}\right)\stackrel{{\scriptstyle\alpha\searrow 0^{+}}}{{\longrightarrow}}\mathcal{T}^{-1}$ for suitable choices of $\alpha$ in some adequate topology. Furthermore, people usually want to specify rates of convergence, which turns out to be the question of finding the worst case error, since such rates are arbitrarily slow in general. Denote by $\psi:\left[0,\infty\right)\to\left[0,\infty\right)$ an index function if it is continuous and monotonically increasing with $\psi\left(0\right)=0$. In this sense, one aims to:

Nevertheless, we often need additional a priori assumptions on the exact solution $f^{\dagger}$ to gain the speed of convergence, which are called source conditions. In the literature, the former type of such conditions singles out that $f^{\dagger}=\psi\left(\mathcal{T}^{\prime}\left[f^{\dagger}\right]*\mathcal{T}^{\prime}\left[f^{\dagger}\right]\right)w$ for some $w\in\mathbb{X}$, renowned as the spectral source condition in the community of inverse and ill-posed problems for several years; see e.g. [11] for a prevailing background of source conditions. Here $\psi$ could be of the Hölder and logarithmic types, depending on every situation. More recently, a novel formulation for source conditions has been derived in the sense of variational inequality, which reads as

Cf. [6], it is in particular called as the variational source condition. Essentially, we can benefit from this source condition not only to simplify proofs for convergence rates in Hilbert frameworks, but also to extensively work in some Banach setting and in more general models with distinctive measurements (cf. e.g. [9, 4, 12]). Compared to the spectral source condition, it does not require the Fréchet derivative $\mathcal{T}^{\prime}$ and would thus be helpful in certain applications where the forward operator is not sufficiently smooth. Most importantly, variational source conditions have been shown to be well-adapted to bounded linear operators in Hilbert spaces; see [3]. On top of that, the variational source condition combined with some nonlinearity condition has been formulated in e.g. [16] to treat nonlinear operators.

Even though there have been enormous advantages over the spectral source conditions, variational source conditions could barely be verified in particular models in terms of partial differential equations. So far, with the aid of complex geometrical optics solutions the variational source condition holds true for an inverse medium scattering problem (cf. [7]) with $\psi$ in the logarithmic form when the solution belongs to some Sobolev ball. In this paper, we aim to show that it holds true for more complex and nonlinear models. It is worth mentioning that characterizations of the variational source conditions have been obtained in [8] where the degree of ill-posedness and the smoothness of solution are established for arbitrary families of subspaces. Denoting by $\mathcal{L}\left(\mathbb{X}\right)$ the set of all bounded linear transformations mapping from $\mathbb{X}$ onto itself, the characterizations are provided cf. [8, Theorem 2.1] in the following.

In line with the characterizations of variational source conditions, in this section we verify and improve the cut-off projection that has been postulated in [15, 18], using Assumptions (1.3), (1.4), (1.5), (1.6), (1.7). The nature of this approach is that one solves the problem in a finite dimensional subspace, which turns out to be a well-posed problem effectively controlled by the measurement error.

We denote by $\left\{E_{\lambda}\right\}$ the spectral family of the positive operator $\mathcal{A}$ defined in the backward problem $\left(\mathcal{B}\right)$ and the function $\lambda\mapsto\left\|E_{\lambda}v\right\|$ is called the spectral distribution function of $v\in L^{2}\left(\Omega\right)$. Thereby, with $C_{\varepsilon}>0$ being a cut-off parameter that will be chosen appropriately, we introduce the spectral cut-off projection $E_{\lambda_{\varepsilon}}:=\mathbf{1}_{\left(0,\lambda_{\varepsilon}\right]}\left(\mathcal{A}\right)$ where $\mathbf{1}_{\left(0,\lambda_{\varepsilon}\right]}$ denotes the characteristic function of the interval $\left(0,\lambda_{\varepsilon}\right]$ with $\left\lceil\lambda_{\varepsilon}\right\rceil=C_{\varepsilon}$. To establish the stable approximate problem of $\left(\mathcal{B}\right)$, we adapt the cut-off projection to both the nonlinearity $F$ and the final data $g_{T}^{\varepsilon}$. When doing so, we characterize the abstract Gevrey class $G_{\sigma}^{p/2}$ for $\sigma\geq 0$ and $p\in\mathbb{N}$ as

equipped with the norm

Henceforward, for each $\varepsilon>0$ we take into account the following approximate problem, denoted by $\left(\mathcal{B}_{\varepsilon}\right)$,

We define a mildly weak solution of $\left(\mathcal{B}_{\varepsilon}\right)$ (or a mild solution for short) to be a continuous mapping $u_{\varepsilon}:\left[0,T\right]\to H^{p}\left(\Omega\right)$ obeying the integral equation

Observe that in (2.1) we actually obtain an integral equation of the form $u_{\varepsilon}\left(t\right)=\mathcal{G}\left(u_{\varepsilon}\right)\left(t\right)$ where $\mathcal{G}$ is completely formulated by the right-hand side, mapping from $C\left(\left[0,T\right];H^{p}\left(\Omega\right)\right)$ onto itself. Therefore, the existence and uniqueness results for (2.1) can be done by the standard fixed-point argument, requiring that the number of fixed-point iterations must be larger than the stability magnitude usually involved in the approximate problem. Since these results are standard, we, for simplicity, refer the reader to the concrete reference [17] for the detailed notion of proof.

Observe again from (2.1) that we can define the cut-off operator $\mathcal{P}_{\varepsilon}\in\mathcal{L}\left(H^{p}\left(\Omega\right)\right)$ for the solution $u\left(\cdot,t_{0}\right)$ by fixing $t=t_{0}$. In particular, it is an orthogonal projection given by

We also remark that in the same spirit of (2.1) the nonlinear ill-posed operator $\mathcal{T}$ (see Subsection 1.3) can be written in the following closed form:

in which we define that $\mathcal{T}:\mathcal{D}\left(\mathcal{T}\right)\to H_{\#}^{p}\left(\Omega\right)$ with $\mathcal{D}\left(\mathcal{T}\right)=H_{\#}^{p}\left(\Omega\right)$.

From now on, we aim to verify a modified version of the spectral cut-off regularization $\mathcal{P}_{\varepsilon}$ in a high-order smoothness setting. As delved into the nonlinear case in [15], if we impose the Gevrey smoothness on $u\left(\cdot,t\right)$ for all $t\in\left[0,T\right]$, the verification of this approach is straightforward. We thereupon obtain the convergence for $t\geq 0$; compared to the standard cut-off method which yields the convergence only for $t>0$ in the nonlinear case. This regularity assumption is very strong and it does not seem applicable as discussed in the analysis of the forward model (cf. Theorem 2), saying that the Gevrey smoothness can only be available for $t>0$. It turns out that we need some modification of this cut-off projection in this paper. On the whole, we use the following scheme:

• •

  Since $u\left(\cdot,t\right)$ satisfies the Gevrey smoothness $G_{t}^{p/2}\left(\Omega\right)$ for $t>0$, we keep solving the backward problem $\mathcal{T}u\left(t_{0}\right)=g_{T}^{\varepsilon}$ for $t_{0}>0$ by the projection $\mathcal{P}_{\varepsilon}$ constructed from (2.1).

• •

  Assume that $u_{t}\left(\cdot,t\right)\in H^{p}\left(\Omega\right)$. We find $t_{\varepsilon}\in\left(0,T\right)$ such that $u\left(t_{\varepsilon}\right)$ is an approximation of $g_{0}$ in $H^{p}\left(\Omega\right)$.

The central point of this modification is that this time we use the Sobolev smoothness on $u_{t}$ in the neighborhood of $t=0$ to avoid the Gevrey smoothness on $g_{0}$. This assumption is consistent with the fact that $u_{t}\in L^{2}\left(0,T;L^{2}\left(\Omega\right)\right)$ obtained in Theorem 2 and thus attainable by the inclusion $C\left(0,T;H^{p}\left(\Omega\right)\right)\subset L^{2}\left(0,T;L^{2}\left(\Omega\right)\right)$. Note that the Sobolev smoothness imposed on both $g_{0}$ and $u\left(\cdot,t\right)$ in $H^{p}\left(\Omega\right)$ with $p>d/2$ is essential for the presence of the nonlinearity $F$.