Sparsity-based nonlinear reconstruction of optical parameters in two-photon photoacoustic computed tomography

Abstract

We present a new nonlinear optimization approach for the sparse reconstruction of single-photon absorption and two-photon absorption coefficients in the photoacoustic computed tomography (PACT). This framework comprises of minimizing an objective functional involving a least squares fit of the interior pressure field data corresponding to two boundary source functions, where the absorption coefficients and the photon density are related through a semi-linear elliptic partial differential equation (PDE) arising in PAT. Further, the objective functional consists of an L¹ regularization term that promotes sparsity patterns in absorption coefficients. The motivation for this framework primarily comes from some recent works related to solving inverse problems in acousto-electric tomography and current density impedance tomography. We provide a new proof of existence and uniqueness of a solution to the semi-linear PDE. Further, a proximal method, involving a Picard solver for the semi-linear PDE and its adjoint, is used to solve the optimization problem. Several numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

1. Introduction

The hybrid medical imaging problems have attracted the research community a lot in the last few decades. The idea behind hybrid imaging methods is to combine a high contrast modality and a high resolution modality to get images with high contrast and resolution simultaneously. High contrast modalities like electrical impedance tomography (EIT) are used primarily for imaging electrical, optical or elastic properties of biological tissues because these properties vary greatly between healthy and unhealthy tissues. On the other hand, modalities like magnetic resonance imaging (MRI) and ultrasound are used to provide better resolution. Therefore, the inversion process for hybrid imaging problems involves two steps coming from each modality discussed above. For a more detailed discussion on hybrid imaging techniques, please see the review articles [3, 17].

One of the hybrid imaging modalities is photoacoustic tomography (PAT) that couples electromagnetic waves together with ultrasound. PAT takes advantage of the photoacoustic effect to convert absorbed optical energy into acoustic waves. In PAT, near infrared (NIR) light propagates into a medium of interest and a fraction of the incoming light energy is absorbed, which results in local heating and subsequent cooling of the medium. Due to this heating and cooling phenomenon, acoustic waves are generated that are recorded at the boundary of the medium. The inverse problem is to reconstruct the diffusion, absorption and Grüneisen coefficients from these acoustic measurements, for more details on the subject see [2, 4, 5, 18, 23, 37, 38, 42, 43, 44] and references therein.

The PAT technology can be divided into two main categories based on the way images are formed, namely, photoacoustic microscopy (PAM) and photoacoustic computed tomography (PACT) [39, 41]. In PACT, unfocused light waves and a series of transducers are used to obtain cross-sectional images of an object from the photoacoustic data by solving an inverse problem. Moreover, the PACT mechanism provides a larger tomographic imaging penetration depth beyond one centimeter, but at the expense of inferior spatial resolution [41]. On the other hand, in PAM, focused laser waves are used along with the method of transverse raster-scanning to obtain cross-sectional images of an object from photoacoustic data in a direct way. However, PAM is known to provide high resolution within a tomographic imaging depth of several millimeters [39]. A modified PAM method, known as the acoustic resolution-PAM (AR-PAM), uses unfocused light on a larger region with similar intensity as that of the focused light to obtain greater tomographic imaging depth similar to PACT [39].

In recent years, a strong non-linear mechanism, called the two-photon absorption, has been observed and measured in the process of PAT reconstructions (see [21, 22, 6, 26, 45]. The two-photon absorption phenomenon is observed when an electron transfers to an excited state after simultaneously absorbing two photons. An imaging modality where one tries to recover optical properties of heterogeneous media (such as biological tissues) using the photoacoustic effect resulting from two photon absorption is known as two-photon photoacoustic tomography (2P-PACT) [21, 22, 35]. Even though the occurrence of two-photon absorption (in healthy biological tissues) is less frequent than single-photon absorption, two-photon absorption is extremely useful in practice. Similar to the PAT mechanism, 2P-PACT can be classified based on the different image formation ways: two-photon photoacoustic microscopy (2P-PAM) and two-photon photoacoustic computed tomography (2P-PACT). While, 2P-PAM is quite often used for deep tissue imaging, thick brain tissue imaging blood vessel imaging, liver biopsies (see [9, 14, 34, 40, 49, 51, 33] and references therein), there has not been any feasible demonstration of the application of 2P-PACT in medical imaging. However, since the one-photon and two-photon absorption effects are not easily separable from each other [6], a coupling of the two-photon effect due to unfocused strong light intensity mechanism, as in AR-PAM, and using an array of transducers to capture the photoacoustic wave data and solving an inverse problem to obtain the optical coefficients, as in PACT, is a promising and efficient way to use the 2P-PACT method in medical imaging (see [25] for a AR-PAM setup). Thus, it is of paramount importance to study mathematical numerical algorithms for obtaining reconstructions in 2P-PACT, which is the main aim of this paper.

The mathematical formulation of 2P-PACT was first introduced in [6, 26], where the authors consider an optically absorbing and scattering medium $\mathrm{\Omega }\subset {ℝ}^n(n\ge 2)$. Denoting the density of photons at a point x ∈ Ω as u(x), it was shown that u(x) solves the following semi-linear diffusion equation

$$\begin{gathered} -\nabla \cdot (D(x)\nabla u(x))+\sigma (x)u(x)+\mu (x)|u(x)|u(x)=0,\text{in}\mathrm{\Omega }, \\ u(x)=g(x),\text{on}\partial \mathrm{\Omega }, \end{gathered}$$ (1)

where D(x) denotes the diffusion coefficient, σ(x) and μ(x) represent the single-photon and the two-photon absorption coefficients respectively, and the function g(x) is the illumination pattern on the boundary ∂Ω. The term μ(x)|u(x)| is the total two-photon absorption coefficient, where the absolute value of u is taken to ensure that the total two-photon absorption coefficient is non-negative [26].

The medium Ω heats up due to absorption of some portion of incoming photons that results in thermal expansion of the medium. The medium cools down after photons leave the medium and this results in contraction of the medium, which gives rise to acoustic waves. This effect is known as the photoacoustic effect. This photoacoustic effect generates an acoustic wave pressure field ${\mathcal{H}}^{\sigma ,\mu }$ is given by (see [5, 13])

$${\mathcal{H}}^{\sigma ,\mu }(x)=\mathrm{\Gamma }(x)[\sigma (x)u(x)+\mu (x)|u(x)|u(x)],\text{for}x\in \mathrm{\Omega },$$ (2)

where Γ is the Grüneisen coefficient that determines the efficiency of the photoacoustic effect. The aim is to recover the optical properties of the medium Ω from the measured acoustic wave signals on the surface of the medium. In this process, the first step involves the recovery of the initial acoustic wave pressure field ${\mathcal{H}}^{\sigma ,\mu }$ from measured data, as usually done in a standard PAT. In the second step of 2P-PACT, the goal is to reconstruct the optical coefficients D, σ, μ and Γ from the information of internal data ${\mathcal{H}}^{\sigma ,\mu }$. This step is usually known as the quantitative step. Recently, the experimental aspect of 2P-PACT have been studied by several authors and it has been shown that the effect of two-photon absorption can be measured accurately, we refer to [21, 22, 35, 46, 47, 48] for detailed discussions. Thus, we assume that the first step in the 2P-PACT process has been accomplished to obtain the initial acoustic wave pressure field ${\mathcal{H}}^{\sigma ,\mu }$. For the second step of recovery of the optical coefficients, detailed mathematical and numerical analysis has been done in very few works [6, 26, 36]. It has been shown in [26] that simultaneous reconstruction of all the four coefficients D, σ, μ, Γ is not possible. In [6, 26], the authors show that given D, Γ, one of σ and μ can be reconstructed with internal data corresponding to one boundary illumination pattern and reconstruction of both coefficients require two sets of internal datum. The authors also present two reconstruction algorithms for reconstructing σ, μ.

There are three major drawbacks of the existing reconstruction algorithms for 2P-PACT: First, four sets of internal datum are used for reconstructing two coefficients. While this gives better reconstructions, it is not conforming with the theoretical requirement of only two sets of internal datum. Secondly, in the presence of 5% noise in the data, the reconstructions of μ exhibit severe artifacts. Thirdly, there is no evidence of the algorithms performing well to reconstruct complex objects with high contrast such as holes and inclusions. In this article, we aim at using a robust computational framework that has the ability to provide high contrast and high resolution reconstructions of objects with holes and inclusions. The framework is based on a non-linear PDE-constrained optimization technique, developed recently [1, 16, 29] to study the aforementioned hybrid inverse problem for 2P-PACT. We start by formulating a minimization problem where we aim to determine σ and μ given the interior acoustic wave pressure field ${\mathcal{H}}^{\sigma ,\mu }$. Additionally, we also assume that the variations in the values of absorption coefficients from known background absorption coefficients demonstrate sparsity patterns. These patterns arise frequently in several tomographic imaging scenarios, for e.g. in blood vessel tomographic reconstructions [24]. One could use a Total Variation (TV) regularization for sparse reconstructions and denoising. However, though TV regularization is more appropriate for piecewise smooth patterns, for general objects like blood vessels, where optical coefficients are not piecewise constants, L¹ regularization has been observed to outperform TV regularization for obtaining sparsity (for e.g. see [12]). Thus, sparsity is incorporated in our model through an L¹ regularization term in our objective functional. An H¹ regularization term is also introduced in the functional that helps reducing artifacts. We provide a comprehensive theoretical analysis of our optimization framework. We provide a new proof for the existence of solutions of (1) with higher regularity, under the assumption that g ≥ 0, using a fixed point approach. We also prove the existence of minimizers of our minimization problem. We solve the optimization problem using a variable inertial proximal scheme that efficiently handles the non-differentiable L¹ regularization term in the objective functional. Finally, we demonstrate the applicability of our reconstruction approach by implementing scheme to several examples.

The article is organized as follows: In Section 2, we formulate the minimization problem for the 2P-PACT reconstruction problem. In Section 3, we present some theoretical results about our optimization problem and we also characterize the optimality system. The numerical schemes to solve the forward problem and the optimization problem are discussed in Section 4. In Section 5, we present simulation results of our 2P-PACT framework. A section on conclusions completes our work.

2. A minimization problem

In this section, we describe the minimization problem corresponding to the 2P-PACT reconstruction problem. We assume Ω to be bounded domain in ${ℝ}^2$. The authors in [26] show that, under the assumptions of the boundary function g ≥ 0, there exists a non-negative solution u of (1) in $H_g^1(\mathrm{\Omega })$. Since g represents the density of photons, g is non-negative. Therefore, instead of the photon propagation equation (1), we consider the following boundary value problem

$$\begin{gathered} -\nabla \cdot (D(x)\nabla u(x))+\sigma (x)u(x)+\mu (x)u^2(x)=0,\text{in}\mathrm{\Omega }, \\ u(x)=g(x)\text{on}\partial \mathrm{\Omega } \end{gathered}$$ (3)

as the model for photon propagation in Ω. We assume that the diffusion coefficient D ∈ W^1,∞(Ω) is known. Throughout the article, we assume that the absorption coefficients σ and μ belong to the function spaces $L_{ad}^{\sigma }$ and $L_{ad}^{\mu }$ respectively, where

$$L_{ad}^{\sigma }=\left\{q(x)\in H^1(\mathrm{\Omega }):a_{\sigma }\le q(x)\le b_{\sigma },\forall x\in \mathrm{\Omega },a_{\sigma },b_{\sigma }>0\right\},$$

$$L_{ad}^{\mu }=\left\{q(x)\in H^1(\mathrm{\Omega }):a_{\mu }\le q(x)\le b_{\mu },\forall x\in \mathrm{\Omega },a_{\mu },b_{\mu }>0\right\}.$$

Then the aim is to recover both absorption coefficients σ and μ from the knowledge of two sets boundary illumination functions g₁, g₂ and the corresponding initial acoustic wave pressure field ${\mathcal{H}}_1^{\sigma ,\mu }$, ${\mathcal{H}}_2^{\sigma ,\mu }$, where

$${\mathcal{H}}^{\sigma ,\mu }(x)=\mathrm{\Gamma }(x)\left[\sigma (x)u(x)+\mu (x)u^2(x)\right],\text{for}x\in \mathrm{\Omega }.$$ (4)

For a known diffusion coefficient D, the equation (1) can be represented as follows

$$\mathcal{L}(u,\sigma ,\mu ,g)=0.$$ (5)

We will use an optimization based approach to reconstruct the coefficients σ(x) and μ(x). We start by defining the following cost functional

$$\begin{gathered} J\left(\sigma ,\mu ,u_1,u_2\right)=\sum _{j=1}^2\frac{{\alpha }_j}{2}{\Vert {\mathcal{H}}_j^{\sigma ,\mu }-G_j^{\delta }\Vert }^2+\frac{{\xi }_1}{2}{\Vert \sigma -{\sigma }_b\Vert }_{H^1(\mathrm{\Omega })}^2+\frac{{\xi }_2}{2}{\Vert \mu -{\mu }_b\Vert }_{H^1(\mathrm{\Omega })}^2 \\ +{\gamma }_1{\Vert \sigma -{\sigma }_b\Vert }_{L^1}+{\gamma }_2{\Vert \mu -{\mu }_b\Vert }_{L^1}, \end{gathered}$$ (6)

where u₁, u₂ satisfy (1) with boundary source functions g₁, g₂ respectively, σ_b, μ_b are known background absorption coefficients and $G_j^{\delta }$, j = 1, 2 are the (possibly noisy) measured initial acoustic wave pressure fields.

We now consider the following constrained minimization problem associated to the above cost functional

$$\underset{\sigma ,\mu }{\min }J\left(\sigma ,\mu ,u_1,u_2\right),$$ (7)

$$\text{s.t.}\mathcal{L}\left(u_1,\sigma ,\mu ,g_1\right)=0\text{,}$$ (8)

$$\mathcal{L}\left(u_2,\sigma ,\mu ,g_2\right)=0.$$ (P)

The first term in the functional (6) represents a least-square data fitting term for obtaining σ, μ such that ${\mathcal{H}}_j^{\sigma ,\mu }\approx G_j^{\delta }$, j = 1, 2. The regularization terms ${\Vert \sigma -{\sigma }_b\Vert }_{L^1}$ and ${\Vert \mu -{\mu }_b\Vert }_{L^1}$ in the above functional (6) implement L¹ regularization of the minimization problem that helps promote sparsity patterns in the reconstruction of absorption coefficients. The use of such L¹ regularization terms has been shown to obtain high contrast in the reconstructions [16, 29]. The H¹ regularization terms ${\Vert \sigma -{\sigma }_b\Vert }_{H^1}^2$ and ${\Vert \mu -{\mu }_b\Vert }_{H^1}^2$ help in denoising and removal of artifacts, thus, promoting high resolution.

3. Theory of the minimization problem

In this section, we analyze the existence of a solution to the minimization problem (7) and, further, characterize this solution through a first-order optimality system. We refer to this minimization problem as the 2P-PACT sparse reconstruction problem (2PPACT-SR). We begin our discussion with the analysis of the solution of (3). The existence of solution $u\in H_g^1(\mathrm{\Omega })$ for the boundary value problem (3) has been established in [26] under the assumptions that the coefficients D, σ, μ are bounded above and below by some positive constants and the boundary function g is the restriction of a continuous function $\phi \in C^0(\overline{\mathrm{\Omega }})$. The authors also showed the existence of a regular solution $u\in H_g^3(\mathrm{\Omega })$ under extra assumptions D, σ, μ are in H¹(Ω) and g comes from $\phi \in C^3(\overline{\mathrm{\Omega }})$. Further, the authors show that u is non-negative corresponding to a non-negative boundary function g is non-negative.

To prove the existence of minimizer of (6), we need u ∈ H²(Ω). For this purpose, we impose weaker assumptions on the coefficients of (3) and boundary function g compared to the assumptions used in [26]. We present a new proof to the existence and uniqueness of solution u ∈ H²(Ω) for the boundary value problem (3). We first recall the following well known fixed point theorem, for reference see [11, Theorem 4, Section 9.2].

Theorem 3.1 (Schaefer’s Fixed Point Theorem).

Suppose A : X → X is a continuous and compact mapping. Assume further that the set

$$\{u\in X:u=\lambda A[u]forsome0\le \lambda \le 1\}$$

is bounded. Then A has a fixed point.

The following theorem gives the existence and uniqueness of solution u ∈ H²(Ω) of (3).

Theorem 3.2.

Let Ω be a bounded domain in ${ℝ}^2$. Assume D(x) ∈ W^1,∞(Ω), $(\sigma (x),\mu (x))\in L_{ad}^{\sigma }\times L_{ad}^{\mu }$ and g ∈ H^3/2(∂Ω) are given. Then the boundary value problem (3) has a unique solution u in $H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$. Further, any weak solution u of (3) is also a strong solution, that is, u ∈ H²(Ω).

Proof.

In order to solve above equation (3), we start by reducing it to a homogeneous boundary value problem by putting u = v + φ, where φ ∈ H²(Ω) is a possible extension of g from boundary ∂Ω to whole Ω. Then, we can verify that the function v satisfies the equation:

$$-\nabla \cdot (D(x)\nabla v(x))+\vartheta (x)v+\mu (x)v^2=f(x),\text{in}\mathrm{\Omega },$$ (9)

$$v(x)=0,\text{on}\partial \mathrm{\Omega }.$$ (10)

where ϑ = σ + 2μφ and f = ∇ · (D(x)∇φ) − σφ − μφ².

For a given $v\in H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$, define

$$F(x):=-\mu (x)v^2(x)+f(x).$$

Using conditions on φ, D, σ and μ together with v ∈ L⁴(Ω), we see F ∈ L²(Ω). Hence there exists a unique $w\in H_0^1(\mathrm{\Omega })$ (dependent on v) satisfying the following linear boundary value problem, see [7, Chapter 9] and [20, Chapter 3, Section 7]

$$-\nabla \cdot (D(x)\nabla w(x))+\vartheta (x)w(x)=F(x),\text{in}\mathrm{\Omega },$$

$$w(x)=0,\mathrm{on}\partial \mathrm{\Omega }$$

with the estimate

$${\Vert w\Vert }_{H^2(\mathrm{\Omega })}\le C{\Vert F\Vert }_{L^2(\mathrm{\Omega })}$$

for some constant C (dependent only on coefficient functions and the domain Ω).

This motivates us to define the the operator $A:H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })\to H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$ given by A[v] = w, where w and v are related in the same manner as above. Further, we have

$${\Vert A[v]\Vert }_{H^2(\mathrm{\Omega })}\le C{\Vert F\Vert }_{L^2(\mathrm{\Omega })}\le C\left({\Vert v\Vert }_{L^4(\mathrm{\Omega })}+{\Vert f\Vert }_{L^2(\mathrm{\Omega })}\right).$$ (11)

Note that any fixed point of A will solve (3) which means to obtain a solution of (3) it is enough to verify the conditions of Theorem 3.1 for A, i.e., we need to show that the operator A is continuous, compact and the set $\left\{v\in H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega }):v=\lambda A[v]\text{for}\text{some}0\le \lambda \le 1\right\}$ is bounded.

To show continuity of A, let us start with a sequence

$$v_k\to v,\text{in}{H}_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$$

then by the inequality (11), we have

$$\underset{k}{\sup }{\Vert w_k\Vert }_{H^2(\mathrm{\Omega })}<\infty ,\text{where}{w}_k=A\left[v_k\right],\text{for}k=1,\dots$$

Thus there is a subsequence ${\left\{w_{k_j}\right\}}_{j=1}^{\infty }$ and a function $w\in H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$ with

$$w_{k_j}\to w,\text{in}{H}_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega }).$$

Now,

$${\int }_{\mathrm{\Omega }}\left(D\left(\nabla w_{k_j}\cdot \nabla \chi \right)+\vartheta w_{k_j}\chi \right)dx=-{\int }_{\mathrm{\Omega }}\left(\mu v_{k_j}^2\chi -f\chi \right)dx,\forall \chi \in H_0^1(\mathrm{\Omega }).$$

Taking the limit k_j → ∞ we get

$${\int }_{\mathrm{\Omega }}(D(\nabla w\cdot \nabla \chi )+\vartheta w\chi )dx=-{\int }_{\mathrm{\Omega }}\left(\mu v^2\chi -f\chi \right)dx,\forall \chi \in H_0^1(\mathrm{\Omega }).$$

Hence w = A[v]. This shows the continuity of A. The compactness of A also follows by a similar argument, indeed if {v_k} is a bounded sequence in $H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$, the estimate (11) shows ${\left\{A\left[v_k\right]\right\}}_{k=1}^{\infty }$ is bounded in H²(Ω) and hence possess a strongly convergent subsequence. The only thing remains to prove is the boundedness of the set:

$$Y=\left\{v\in H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega }):v=\lambda A[v]\text{for}\text{some}0\le \lambda \le 1\right\}.$$

Let $v\in H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$ such that

$$v=\lambda A[v],\text{for}\text{some}0\le \lambda \le 1.$$

Then $v/\lambda =A[v]\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$ and

$$-\nabla \cdot (D(x)\nabla v(x))+\vartheta (x)v(x)=-\lambda \mu v^2+\lambda f,\text{a.e.}\text{in}\mathrm{\Omega }.$$

Multiplying the above relation with v and integrating over Ω to get

$$\begin{gathered} {\int }_{\mathrm{\Omega }}D{|\nabla v|}^2+v{|v|}^2=-{\int }_{\mathrm{\Omega }}\lambda \mu v^{3}dx+{\int }_{\mathrm{\Omega }}\lambda fvdx \\ \le {\int }_{\mathrm{\Omega }}fvdx={\int }_{\mathrm{\Omega }}\left(\frac{1}{ϵ}f\right)(ϵv)dx,\text{for}\text{any}ϵ>0 \\ \le \frac{{ϵ}^2}{2}{\int }_{\mathrm{\Omega }}{v}^{2}dx+\frac{1}{2{ϵ}^2}{\int }_{\mathrm{\Omega }}{f}^{2}dx. \end{gathered}$$

This gives

$${\int }_{\mathrm{\Omega }}D{|\nabla v|}^2+\left(\vartheta -\frac{{ϵ}^2}{2}\right){|v|}^2\le \frac{1}{2{ϵ}^2}{\int }_{\mathrm{\Omega }}{f}^{2}dx.$$

Choose an ϵ > 0 such that $\left(\vartheta -\frac{{ϵ}^2}{2}\right)$ is bounded below by positive constant. Using this information together with the fact D is bounded below by a positive constant, we verified that the set Y is bounded. Hence by Schaefer’s Theorem 3.1, we conclude that the operator A has a fixed point $v\in H^2(\mathrm{\Omega })\cap H_g^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$.

To show the uniqueness of the solution u, let u₁ and u₂ be two non-negative solutions of the boundary value problem (3). Then w = u₁ − u₂ satisfies the following boundary value problem

$$\begin{gathered} -\nabla \cdot (D(x)\nabla w(x))+\sigma (x)w(x)+\mu (x)w(x)\left(u_1(x)+u_2(x)\right)=0,\text{in}\mathrm{\Omega }, \\ w(x)=0,\text{on}\partial \mathrm{\Omega }. \end{gathered}$$

Multiplying above equation by w and integrating by part, we get

$${\int }_{\mathrm{\Omega }}D(x){(\nabla w(x))}^2+\sigma (x)w^2(x)+\mu (x)w^2(x)\left(u_1(x)+u_2(x)\right)dx=0.$$

Since all coefficients are positive and solutions u₁, u₂ are non-negative therefore the above relation entails w ≡ 0. This proves the uniqueness of solution for boundary value problem (3). □

Remark 3.1.

The result in Theorem 3.2 ensures that the initial acoustic wave pressure field ${\mathcal{H}}^{\sigma ,\mu }$ given by (4) belongs to L²(Ω) ∩ L⁴(Ω). Thus, the functional J given by (6) is well-defined.

The solvability of the 2PPACT-SR inversion problem depends on the type of Dirichlet boundary data g_j, j = 1, 2. In this context, we have the following lemma from [26]

Lemma 3.1 (Boundary data).

Let g_i, i = 1, 2 be two sets of boundary conditions with g_i > 0 and g₁ − g₂ > 0. Then u₁ ≠ u₂ almost everywhere in Ω and one can uniquely reconstruct (σ, μ) from the two sets of initial acoustic wave pressure fields ${\mathcal{H}}_i^{\sigma ,\mu }$, i = 1, 2.

Next, we state the following lemma about the Fréchet differentiability of the mapping u(σ, μ) which will be needed later. For proof of this lemma, we refer to [26, Proposition 2.5].

Lemma 3.2.

The map u(σ, μ) defined by (1) is Fréchet differentiable with respect to σ and μ as a mapping from $L_{ad}^{\sigma }\times L_{ad}^{\mu }$ to $H_g^1(\mathrm{\Omega })$.

Using Lemma 3.2, we introduce the reduced cost functional

$$\widehat{J}(\sigma ,\mu )=J\left(\sigma ,\mu ,u_1(\sigma ,\mu ),u_2(\sigma ,\mu )\right),$$ (12)

where u_i(σ, μ), i = 1, 2 denotes the unique solution of (5) given σ, μ and g_i, i = 1, 2. The constrained optimization problem (7) can be formulated as an unconstrained one as follows

$$\underset{(\sigma ,\mu )\in L_{ad}^{\sigma }\times L_{ad}^{\mu }}{\min }\widehat{J}(\sigma ,\mu ).$$ (13)

We next investigate the existence of a minimizer to the 2PPACT-SR problem (7).

Proposition 1.

Let g₁, g₂ ∈ H^1/2(Ω). Then there exists a quadruplet $\left({\sigma }^{*},{\mu }^{*},u_1^{*},u_2^{*}\right)\in L_{ad}^{\sigma }\times L_{ad}^{\mu }\times H_{g_1}^1(\mathrm{\Omega })\times H_{g_2}^1(\mathrm{\Omega })$ such that $u_i^{*},i=1,2$ are solutions to $\mathcal{L}\left(\sigma ,\mu ,u_i,g_i\right)=0,i=1,2$ and (σ^∗, μ^∗) minimizes $\widehat{J}$ in $L_{ad}^{\sigma }\times L_{ad}^{\mu }$.

Proof.

We observe that $\widehat{J}$ is bounded below. This implies there exists a minimizing sequence $\left({\sigma }_m,{\mu }_m\right)\in L_{ad}^{\sigma }\times L_{ad}^{\mu }$. Since $\widehat{J}$ is coercive in $L_{ad}^{\sigma }\times L_{ad}^{\mu }$, we have that the sequence (σ_m, μ_m) is bounded. Since $L_{ad}^{\sigma }\times L_{ad}^{\mu }$ is a closed subspace of a Hilbert space, it is reflexive. Thus, the sequence (σ_m, μ_m) has a weakly convergent subsequence $\left({\sigma }_{m_l},{\mu }_{m_l}\right)\rightharpoonup \left({\sigma }^{*},{\mu }^{*}\right)$. Consequently, the sequences $u_i\left({\sigma }_{m_l},{\mu }_{m_l}\right)\rightharpoonup u^{*}$ in $H^2(\mathrm{\Omega })\subset H_{g_i}^1(\mathrm{\Omega }),i=1,2$. Due to the fact that H²(Ω) is compactly embedded in $H_{g_i}^1(\mathrm{\Omega })$, we have $u_i\left({\sigma }_{m_l},{\mu }_{m_l}\right)\to u^{*}\in H_{g_l}^1(\mathrm{\Omega })$. Again, since H²(Ω) is compactly embedded in L⁴(Ω), we additionally have $u_i\left({\sigma }_{m_l},{\mu }_{m_l}\right)\to u^{*}\in L^4(\mathrm{\Omega })$. We next aim at showing that $u^{*}=u\left({\sigma }^{*},{\mu }^{*}\right)\in H_{g_i}^1(\mathrm{\Omega })$. For this purpose, we consider the weak formulation of the solution of (1). The first term in the weak formulation we need to consider is ${\langle {\sigma }_{m_l}{u}_i\left({\sigma }_{m_l},{\mu }_{m_l}\right),\psi \rangle }_{L^2(\mathrm{\Omega })}$. By the preceding discussion, we have ${\langle {\sigma }_{m_l}{u}_i\left({\sigma }_{m_l},{\mu }_{m_l}\right),\psi \rangle }_{L^2(\mathrm{\Omega })}\to {\langle {\sigma }^{*}{u}_i^{*},\psi \rangle }_{L^2(\mathrm{\Omega })}$. The second term we need to analyze is ${\langle {\mu }_{m_l}{u}_i^2\left({\sigma }_{m_l},{\mu }_{m_l}\right),\psi \rangle }_{L^2(\mathrm{\Omega })}$. Since, ${\mu }_{m_l}\rightharpoonup {\mu }^{*}$ in L²(Ω) and $u_i\left({\sigma }_{m_l},{\mu }_{m_l}\right)\to u^{*}\in L^4(\mathrm{\Omega })$, we have ${\langle {\mu }_{m_l}{u}_i^2\left({\sigma }_{m_l},{\mu }_{m_l}\right),\psi \rangle }_{L^2(\mathrm{\Omega })}\to {\langle {\mu }^{*}{\left(u_i^{*}\right)}^2,\psi \rangle }_{L^2(\mathrm{\Omega })}$.

Thus, $\left({\sigma }^{*},{\mu }^{*},u_i^{*}\right)$ solves (1) with boundary condition g_i and by continuity of the map u(σ, μ), we have u^∗ = u(σ^∗, μ^∗). Since $\widehat{J}$ is sequentially weakly lower semi-continuous, we have that $\left({\sigma }^{*},{\mu }^{*},u_1^{*},u_2^{*}\right)$ minimizes $\widehat{J}$ in $L_{ad}^{\sigma }\times L_{ad}^{\mu }\times H_{g_1}^1(\mathrm{\Omega })\times H_{g_2}^1(\mathrm{\Omega })$ □

3.1. Characterization of local minima

To characterize the solution of our optimization problem through first-order optimality conditions, we write the reduced functional $\widehat{J}$ as follows

$$\widehat{J}={\widehat{J}}_1+{\widehat{J}}_2,{\widehat{J}}_i:L_{ad}^{\sigma }\times L_{ad}^{\mu }\to {ℝ}^{+},i=1,2,$$

where

$$\begin{gathered} {\widehat{J}}_1(\sigma ,\mu )=\sum _{j=1}^2\frac{{\alpha }_j}{2}{\Vert {\mathcal{H}}_j^{\sigma ,\mu }-G_j^{\delta }\Vert }^2+\frac{{\xi }_1}{2}{\Vert \sigma -{\sigma }_b\Vert }_{H^1(\mathrm{\Omega })}^2+\frac{{\xi }_2}{2}{\Vert \mu -{\mu }_b\Vert }_{H^1(\mathrm{\Omega })}^2, \\ {\widehat{J}}_2(\sigma ,\mu )={\gamma }_1{\Vert \sigma -{\sigma }_b\Vert }_{L^1}+{\gamma }_2{\Vert \mu -{\mu }_b\Vert }_{L^1}. \end{gathered}$$ (14)

Remark 3.2.

The functional ${\widehat{J}}_1$ is smooth and possibly non-convex, while ${\widehat{J}}_2$ is non-smooth and convex.

The following property can be proved using arguments in [19].

Proposition 2.

The reduced functional ${\widehat{J}}_1(\sigma ,\mu )$ is weakly lower semi-continuous, bounded below and Fréchet differentiable with respect to σ, μ.

Next, we are going to define the subdifferential of a non-smooth functional.

Definition 3.1 (Subdifferential).

If $\widehat{J}$ is finite at a point (σ, μ), the Fréchet subdifferential of $\widehat{J}$ at (σ, μ) is defined as follows [10]

$$\partial \widehat{J}(\overline{\sigma },\overline{\mu }):=\left\{\varphi \in {\left(L_{ad}^{\sigma }\times L_{ad}^{\mu }\right)}^{*}:\underset{(\sigma ,\mu )\to (\overline{\sigma },\overline{\mu })}{\lim \inf }\frac{\widehat{J}(\sigma ,\mu )-\widehat{J}(\overline{\sigma },\overline{\mu })-\langle \varphi ,(\sigma ,\mu )-(\overline{\sigma },\overline{\mu })\rangle }{\Vert (\overline{\sigma },\overline{\mu })-(\sigma ,\mu ){\Vert }_2}\ge 0\right\},$$ (15)

where ${\left(L_{ad}^{\sigma }\times L_{ad}^{\mu }\right)}^{*}$ is the dual space of $L_{ad}^{\sigma }\times L_{ad}^{\mu }$. An element $\varphi \in \partial \widehat{J}(\sigma ,\mu )$ is called a subdifferential of $\widehat{J}$ at (σ, μ).

In our setting, we have the following

$$\partial \widehat{J}(\sigma ,\mu )={\nabla }_{(\sigma ,\mu )}{\widehat{J}}_1(\sigma ,\mu )+\partial {\widehat{J}}_2(\sigma ,\mu ),$$

since ${\widehat{J}}_1$ is Fréchet differentiable by Prop. 2. Moreover, for each α > 0, it holds that

$$\partial (\alpha \widehat{J})=\alpha \partial \widehat{J}.$$

The following proposition gives a necessary condition for a local minimum of $\widehat{J}$ (see [29]).

Proposition 3 (Necessary condition).

If $\widehat{J}={\widehat{J}}_1+{\widehat{J}}_2$, with ${\widehat{J}}_1$, ${\widehat{J}}_2$ given by (14), attains a local minimum at $\left({\sigma }^{*},{\mu }^{*}\right)\in L_{ad}^{\sigma }\times L_{ad}^{\mu }$, then

$$\mathit{0}\in \partial \widehat{J}\left({\sigma }^{*},{\mu }^{*}\right),$$

or equivalently

$$-{\nabla }_{(\sigma ,\mu )}{\widehat{J}}_1\left({\sigma }^{*},{\mu }^{*}\right)\in \partial {\widehat{J}}_2\left({\sigma }^{*},{\mu }^{*}\right).$$

The following variational inequality holds for each $\lambda \in \partial {\widehat{J}}_2\left({\sigma }^{*},{\mu }^{*}\right)$ (see [32]).

$$\langle \nabla {\widehat{J}}_1\left({\sigma }^{*},{\mu }^{*}\right)+\lambda ,(\sigma ,\mu )-\left({\sigma }^{*},{\mu }^{*}\right)\rangle \ge 0,\forall (\sigma ,\mu )\in L_{ad}^{\sigma }\times L_{ad}^{\mu }.$$ (16)

Using the definition of ${\widehat{J}}_2$ in (14) and the fact that $L_{ad}^{\sigma }\times L_{ad}^{\mu }$ is reflexive, the inclusion $\lambda \in \partial {\widehat{J}}_2\left({\sigma }^{*},{\mu }^{*}\right)$ gives the following characterization of space of λ

$$\lambda =\left({\lambda }_1,{\lambda }_2\right),{\lambda }_i\in {\mathrm{\Lambda }}_{ad}^i:=\left\{\kappa \in L^2(\mathrm{\Omega }):0\le \kappa \le {\gamma }_i,\text{a.e.}\text{in}\mathrm{\Omega }\right\},i=1,2.$$

A pointwise analysis of the variational inequality (16) leads to the existence of a non-negative functions ${\lambda }_{i,a}^{*},{\lambda }_{i,b}^{*}\in L^2(\mathrm{\Omega }),i=1,2$ that correspond to Lagrange multipliers for the inequality constraints in $L_{ad}^{\sigma }\times L_{ad}^{\mu }$. We, thus, have the following first-order optimality system.

Proposition 4 (First-order necessary conditions).

The optimal solution of the minimization problem (13) can be characterized by the existence of $({\lambda }_1^{*},{\lambda }_2^{*},{\lambda }_{1,a}^{*},{\lambda }_{2,a}^{*},{\lambda }_{1,b}^{*},{\lambda }_{2,b}^{*})\in {\left({\mathrm{\Lambda }}_{ad}\right)}^2\times {\left(L^2(\mathrm{\Omega })\right)}^4$ such that

$${\nabla }_{\sigma }{\widehat{J}}_1\left({\sigma }^{*},{\mu }^{*}\right)+{\lambda }_1^{*}+{\lambda }_{1,b}^{*}-{\lambda }_{1,a}^{*}=0,$$ (17)

$${\nabla }_{\mu }{\widehat{J}}_1\left({\sigma }^{*},{\mu }^{*}\right)+{\lambda }_2^{*}+{\lambda }_{2,b}^{*}-{\lambda }_{2,a}^{*}=0,$$ (18)

$${\lambda }_{1,b}^{*}\ge 0,b-{\sigma }^{*}\ge 0,\langle {\lambda }_{1,b}^{*},b-{\sigma }^{*}\rangle =0,$$ (19)

$${\lambda }_{1,a}^{*}\ge 0,{\sigma }^{*}-a\ge 0,\langle {\lambda }_{1,a},{\sigma }^{*}-a\rangle =0,$$ (20)

$${\lambda }_{2,b}^{*}\ge 0,b-{\mu }^{*}\ge 0,\langle {\lambda }_{2,b}^{*},b-{\mu }^{*}\rangle =0,$$ (21)

$${\lambda }_{2,a}^{*}\ge 0,{\mu }^{*}-a\ge 0,\langle {\lambda }_{2,a},{\mu }^{*}-a\rangle =0,$$ (22)

$${\lambda }_1^{*}={\gamma }_{1}a.e.on\left\{x\in \mathrm{\Omega }:{\sigma }^{*}(x)>0\right\},$$ (23)

$${\lambda }_2^{*}={\gamma }_{2}a.e.on\left\{x\in \mathrm{\Omega }:{\mu }^{*}(x)>0\right\},$$ (24)

$$0\le {\lambda }_1^{*}\le {\gamma }_{1}a.e.on\left\{x\in \mathrm{\Omega }:{\sigma }^{*}(x)=0\right\},$$ (25)

$$0\le {\lambda }_2^{*}\le {\gamma }_{2}a.e.on\left\{x\in \mathrm{\Omega }:{\mu }^{*}(x)=0\right\}.$$ (26)

The conditions (19)–(26) are known as the complementarity conditions for $\left({\sigma }^{*},{\mu }^{*},{\lambda }_1^{*},{\lambda }_2^{*}\right)$.

To determine the gradient ${\nabla }_{\sigma }{\widehat{J}}_1$, ${\nabla }_{\mu }{\widehat{J}}_1$, we use the adjoint approach (see for e.g., [27, 28]). This gives the following reduced gradients of ${\widehat{J}}_1$

$$\begin{gathered} {\nabla }_{\sigma }{\widehat{J}}_1\left({\sigma }^{*},{\mu }^{*}\right)={\alpha }_1\left({\mathcal{H}}_1^{{\sigma }^{*},{\mu }^{*}}-G_1^{\delta }\right)\mathrm{\Gamma }{u}_1+{\alpha }_2\left({\mathcal{H}}_2^{{\sigma }^{*},{\mu }^{*}}-G_2^{\delta }\right)\mathrm{\Gamma }{u}_2+u_1{v}_1+u_2{v}_2+{\xi }_1{\sigma }^{*} \\ {\nabla }_{\mu }{\widehat{J}}_1\left({\sigma }^{*},{\mu }^{*}\right)={\alpha }_1\left({\mathcal{H}}_1^{{\sigma }^{*},{\mu }^{*}}-G_1^{\delta }\right)\mathrm{\Gamma }{u}_1^2+{\alpha }_2\left({\mathcal{H}}_2^{{\sigma }^{*},{\mu }^{*}}-G_2^{\delta }\right)\mathrm{\Gamma }{u}_2^2+u_1^2{v}_1+u_2^2{v}_2+{\xi }_2{\mu }^{*} \end{gathered}$$ (27)

where u₁, u₂ satisfy the forward equations $\mathcal{L}\left(u_1,{\sigma }^{*},{\mu }^{*},g_1\right)=0$, $\mathcal{L}\left(u_2,{\sigma }^{*},{\mu }^{*},g_2\right)=0$, respectively, and v₁, v₂ satisfy the adjoint equations

$$\begin{gathered} -\nabla \cdot \left(D\nabla v_1\right)+{\sigma }^{*}{v}_1+2{\mu }^{*}{u}_1{v}_1=-{\alpha }_1\mathrm{\Gamma }\left({\sigma }^{*}{u}_1+{\mu }^{*}{u}_1^2-G_1^{\delta }\right)\cdot \left({\sigma }^{*}+2u_1\right)\text{in}\mathrm{\Omega }, \\ {v}_1=0,\text{on}\partial \mathrm{\Omega } \end{gathered}$$ (28)

$$\begin{gathered} -\nabla \cdot \left(D\nabla v_2\right)+{\sigma }^{*}{v}_2+2{\mu }^{*}{u}_2{v}_2=-{\alpha }_2\mathrm{\Gamma }\left({\sigma }^{*}{u}_2+{\mu }^{*}{u}_2^2-G_2^{\delta }\right)\cdot \left({\sigma }^{*}+2\left|u_2\right|\right)\text{in}\mathrm{\Omega }, \\ {v}_2=0,\text{on}\partial \mathrm{\Omega }. \end{gathered}$$ (29)

The complementarity conditions (19)–(26) can be rewritten in a compact form as follows. Define

$$\begin{gathered} c_1^{*}={\lambda }_1^{*}+{\lambda }_{1,b}^{*}-{\lambda }_{1,a}^{*}, \\ {c}_2^{*}={\lambda }_2^{*}+{\lambda }_{2,b}^{*}-{\lambda }_{2,a}^{*}. \end{gathered}$$ (30)

Then the triplets $\left({\lambda }_1^{*},{\lambda }_{1,a}^{*},{\lambda }_{1,b}^{*}\right)$, $\left({\lambda }_2^{*},{\lambda }_{2,a}^{*},{\lambda }_{2,b}^{*}\right)$ are obtained by solving the following equations

$$\begin{gathered} {\lambda }_i^{*}=\min \left({\gamma }_i,\max \left(0,c_i^{*}\right)\right), \\ {\lambda }_{i,a}^{*}=-\min \left(0,c_i^{*}+{\gamma }_i\right), \\ {\lambda }_{i,b}^{*}=\max \left(0,c_i^{*}-{\gamma }_i\right), \end{gathered}$$ (31)

for i = 1,2 (see [32]). For each $k\in {ℝ}^{+}$, define the following quantity

$$\begin{gathered} E_1\left({\sigma }^{*},c_1^{*}\right)={\sigma }^{*}-\max \left\{0,{\sigma }^{*}+k\left(c_1^{*}-{\gamma }_1\right)\right\}+\max \left\{0,{\sigma }^{*}-b+k\left(c_1^{*}-{\gamma }_1\right)\right\} \\ -\min \left\{0,{\sigma }^{*}+k\left(c_1^{*}+{\gamma }_1\right)\right\}+\min \left\{0,{\sigma }^{*}-a+k\left(c_1^{*}+{\gamma }_1\right)\right\}. \end{gathered}$$

$$\begin{gathered} E_2\left({\mu }^{*},c_2^{*}\right)={\mu }^{*}-\max \left\{0,{\mu }^{*}+k\left(c_2^{*}-{\gamma }_2\right)\right\}+\max \left\{0,{\mu }^{*}-b+k\left(c_2^{*}-{\gamma }_2\right)\right\} \\ -\min \left\{0,{\mu }^{*}+k\left(c_2^{*}+{\gamma }_2\right)\right\}+\min \left\{0,{\mu }^{*}-a+k\left(c_2^{*}+{\gamma }_2\right)\right\}. \end{gathered}$$

The following lemma determines the complementarity conditions (19)–(26) in terms of E₁, E₂ (see [32, Lemma 2.2]).

Lemma 3.3.

The complementarity conditions (19)–(26) are equivalent to the following

$$E_1\left({\sigma }^{*},c_1^{*}\right)=0=E_2\left({\mu }^{*},c_2^{*}\right),$$ (32)

where c_i, i = 1, 2 are defined in (30).

Using the gradients in (27) and Lemma 3.3, the optimality conditions (28)–(26) for the 2PPACT-SR problem can be rewritten as follows

Proposition 5.

A local minimizer (u₁, u₂, σ^∗, μ^∗) of the problem (7) can be characterized by the existence of $\left(v_1,v_2,c_1^{*},c_2^{*}\right)\in H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega })\times L_{ad}^{\sigma }\times L_{ad}^{\mu }$, such that the following system is satisfied

$$\begin{gathered} -\nabla \cdot \left(D\nabla u_1\right)+{\sigma }^{*}{u}_1+{\mu }^{*}{u}_1^2=0,in\mathrm{\Omega }, \\ {u}_1=g_1,on\partial \mathrm{\Omega }, \\ -\nabla \cdot \left(D\nabla v_1\right)+{\sigma }^{*}{v}_1+2{\mu }^{*}{u}_1{v}_1=-{\alpha }_1\mathrm{\Gamma }\left({\sigma }^{*}{u}_1+{\mu }^{*}{u}_1^2-G_1^{\delta }\right)\cdot \left({\sigma }^{*}+2u_1\right)in\mathrm{\Omega }, \\ {v}_1=0,on\partial \mathrm{\Omega }, \\ -\nabla \cdot \left(D\nabla u_2\right)+{\sigma }^{*}{u}_2+{\mu }^{*}{u}_2^2=0,in\mathrm{\Omega }, \\ {u}_2=g_2,on\partial \mathrm{\Omega }, \\ -\nabla \cdot \left(D\nabla v_2\right)+{\sigma }^{*}{v}_2+2{\mu }^{*}{u}_2{v}_2=-{\alpha }_2\mathrm{\Gamma }\left({\sigma }^{*}{u}_2+{\mu }^{*}{u}_2^2-G_2^{\delta }\right)\cdot \left({\sigma }^{*}+2u_2\right)in\mathrm{\Omega }, \\ {v}_2=0,on\partial \mathrm{\Omega }, \\ {\alpha }_1\left({\mathcal{H}}_1^{{\sigma }^{*},{\mu }^{*}}-G_1^{\delta }\right)\mathrm{\Gamma }{u}_1+{\alpha }_2\left({\mathcal{H}}_2^{{\sigma }^{*},{\mu }^{*}}-G_2^{\delta }\right)\mathrm{\Gamma }{u}_2+u_1{v}_1+u_2{v}_2+{\xi }_1{\sigma }^{*}=0, \\ {\alpha }_1\left({\mathcal{H}}_1^{{\sigma }^{*},{\mu }^{*}}-G_1^{\delta }\right)\mathrm{\Gamma }{u}_1^2+{\alpha }_2\left({\mathcal{H}}_2^{{\sigma }^{*},{\mu }^{*}}-G_2^{\delta }\right)\mathrm{\Gamma }{u}_2^2+u_1^2{v}_1+u_2^2{v}_2+{\xi }_2{\mu }^{*}=0, \\ {E}_1\left({\sigma }^{*},c_1^{*}\right)=0, \\ {E}_2\left({\mu }^{*},c_2^{*}\right)=0. \end{gathered}$$ (33)

4. Numerical schemes for solving the 2PPACT-SR inverse problem

4.1. Picard type method to solve the forward problem

In this section we propose a Picard type iterative scheme to solve the semi-linear boundary value problem (3). The algorithm is given as follows

Algorithm 4.1 (Picard-type algorithm).

1. Input: Initial guess u₀, D, σ, μ, g, N and TOL

   Initialize: err₀ = 1, k = 0

2. While err_k > TOL and k < N do

3. Solve the following linear elliptic boundary value problem

   $$\begin{gathered} -\nabla \cdot \left(D(x)\nabla u_{k+1}(x)\right)+\sigma (x)u_{k+1}(x)+\mu (x)u_k(x)u_{k+1}(x)=0,in\mathrm{\Omega }, \\ {u}_{k+1}(x)=g(x),on\partial \mathrm{\Omega } \end{gathered}$$

   to get u_k+1 for k ≥ 0

4. err_k+1 = ||u_k+1 – u_k||₂

5. k = k + 1

6. end

We now show the convergence of the Picard algorithm 4.1 to the solution of (1).

Theorem 4.1.

Let D, σ, μ be non-negative functions in L^∞(Ω) and g be non-negative function in C⁰(∂Ω). Then the iterative sequence {u_k}, we obtained from the above Picard’s method, converges in H¹(Ω) and the limit u is a solution of the following semi-linear elliptic boundary value problem

$$\begin{gathered} -\nabla \cdot (D(x)\nabla u(x))+\sigma (x)u(x)+\mu (x)u^2(x)=0,in\mathrm{\Omega }, \\ u(x)=g(x),on\partial \mathrm{\Omega }. \end{gathered}$$

Proof.

By completeness of H¹(Ω) to show the convergence of sequence {u_k} in H¹(Ω), we only need to show that the sequence {u_k} is a Cauchy sequence in H¹(Ω). To achieve this goal, we will show the following contraction type relation for any k ≥ 1

$${\Vert u_{k+1}-u_k\Vert }_{H^1(\mathrm{\Omega })}\le \gamma {\Vert u_k-u_{k-1}\Vert }_{H^1(\mathrm{\Omega })}\le \cdots \le {\gamma }^k{\Vert u_1-u_0\Vert }_{H^1(\mathrm{\Omega })},\text{for}\text{some}\gamma <1.$$

We start with u₂ and u₁, recall from above Picard’s type algorithm 4.1 that the iterates u₁ and u₂ satisfy the following two BVP’s respectively

$$\begin{gathered} -\nabla \cdot \left(D(x)\nabla u_1(x)\right)+\sigma (x)u_1(x)+\mu (x)u_0(x)u_1(x)=0,\text{in}\mathrm{\Omega }, \\ {u}_1(x)=g(x),\text{on}\partial \mathrm{\Omega }. \end{gathered}$$ (34)

$$\begin{gathered} -\nabla \cdot \left(D(x)\nabla u_1(x)\right)+\sigma (x)u_2(x)+\mu (x)u_1(x)u_2(x)=0,\text{in}\mathrm{\Omega }, \\ {u}_2(x)=g(x),\text{on}\partial \mathrm{\Omega }. \end{gathered}$$ (35)

Then by direct substitution, we see that the difference $\overline{u}=u_2-u_1$ solves

$$\begin{gathered} -\nabla \cdot (D(x)\nabla \overline{u}(x))+\sigma (x)\overline{u}(x)+\mu (x)\overline{u}=\mu u_2\left(u_0-u_1\right),\text{in}\mathrm{\Omega }, \\ \overline{u}(x)=0,\mathrm{on}\partial \mathrm{\Omega }. \end{gathered}$$

With the help of regularity estimates for elliptic boundary value problem, we get

$${\Vert \overline{u}\Vert }_{H^1(\mathrm{\Omega })}\le {\Vert \overline{u}\Vert }_{H^2(\mathrm{\Omega })}\le C{\Vert \mu u_2\left(u_0-u_1\right)\Vert }_{L^2(\mathrm{\Omega })}.$$

Consider the right hand side of the above inequality

$$\begin{gathered} {\Vert \mu u_2\left(u_0-u_1\right)\Vert }_{L^2(\mathrm{\Omega })}={\left({\int }_{\mathrm{\Omega }}{\left|\mu \right|}^2{\left|u_2\right|}^2{\left|u_0-u_1\right|}^{2}dx\right)}^{\frac{1}{2}} \\ \le \underset{\tilde{C}}{\underbrace{\Vert \mu {\Vert }_{L^{\infty }}\Vert g{\Vert }_{L^{\infty }}}}{\Vert \left(u_0-u_1\right)\Vert }_{L^2(\mathrm{\Omega })}. \end{gathered}$$

Using this inequality, we have

$${\Vert u_2-u_1\Vert }_{H^1(\mathrm{\Omega })}\le {\Vert u_2-u_1\Vert }_{H^2(\mathrm{\Omega })}\le \underset{\gamma }{\underbrace{C\tilde{C}}}{\Vert \left(u_0-u_1\right)\Vert }_{L^2(\mathrm{\Omega })}\le \gamma {\Vert \left(u_0-u_1\right)\Vert }_{H^1(\mathrm{\Omega })}.$$

By exactly same argument, we get

$${\Vert u_{k+1}-u_k\Vert }_{H^1(\mathrm{\Omega })}\le \gamma {\Vert u_k-u_{k-1}\Vert }_{H^1(\mathrm{\Omega })}.$$

Thus, we have the required relation

$${\Vert u_{k+1}-u_k\Vert }_{H^1(\mathrm{\Omega })}\le {\gamma }^k{\Vert \left(u_0-u_1\right)\Vert }_{H^1(\mathrm{\Omega })}.$$

We can make γ < 1 by choosing appropriate g and μ. Hence, the sequence {u_k} is a Cauchy sequence in H¹(Ω) and hence converges to a limit u in H¹(Ω). To complete the proof of our theorem, the only thing remain to show is that u solve

$$\begin{gathered} -\nabla \cdot (D(x)\nabla u(x))+\sigma (x)u(x)+\mu (x)u^2(x)=0,\text{in}\mathrm{\Omega }, \\ u(x)=g(x),\text{on}\partial \mathrm{\Omega }. \end{gathered}$$

We know each u_k satisfies

$${\int }_{\mathrm{\Omega }}D\nabla u_k\cdot \nabla \phi dx+{\int }_{\mathrm{\Omega }}\sigma u_k\phi dx+{\int }_{\mathrm{\Omega }}\mu u_{k-1}{u}_k\phi dx=0,\text{for}\text{all}\phi \in C_0^{\infty }(\mathrm{\Omega }).$$

The convergence of u_k → u in H¹(Ω) implies the convergence $\nabla u_k\to \nabla u$ in L²(Ω) and the convergence $\sigma u_k\to \sigma u$ in H¹(Ω) as k → ∞. Additionally, the strong convergence of {u_k} in H¹(Ω) will guarantee the weak convergence of u_k−1u_k ⇀ u² in L²(Ω). Thus we have

$${\int }_{\mathrm{\Omega }}D\nabla u_k\cdot \nabla \phi dx+{\int }_{\mathrm{\Omega }}\sigma u_k\phi dx+{\int }_{\mathrm{\Omega }}\mu u_{k-1}{u}_k\phi dx\to {\int }_{\mathrm{\Omega }}D\nabla u\cdot \nabla \phi dx+{\int }_{\mathrm{\Omega }}\sigma u\phi dx+{\int }_{\mathrm{\Omega }}\mu u^2\phi dx$$

for all $\phi \in C_0^{\infty }(\mathrm{\Omega })$. Therefore

$${\int }_{\mathrm{\Omega }}D\nabla u\cdot \nabla \phi dx+{\int }_{\mathrm{\Omega }}\sigma u\phi dx+{\int }_{\mathrm{\Omega }}\mu u^2\phi dx=0,\text{for}\text{all}\phi \in C_0^{\infty }(\mathrm{\Omega }).$$

This completes the proof of the theorem. □

4.2. Variable inertial proximal method for solving the optimality system

For solving the optimality system (33), we use a class of iterative schemes known as the proximal method. The fundamental idea behind a proximal scheme is to minimize an upper bound of the objective function ${\widehat{J}}_1$, and, thus, $\widehat{J}$, instead of directly minimizing the functional. Assuming that the gradients of $\widehat{J}$, given in (27) are Lipschitz continuous, the formulation of the proximal method depends on the proximal operators ${\mathbb{S}}_{{\gamma }_1}^{L_{ad}^{\sigma }}(\tilde{\sigma })$ and ${\text{S}}_{{\gamma }_2}^{L_{ad}^{\mu }}(\tilde{\mu })$, which are defined and characterized as follows

$$\underset{\sigma \in L_{ad}^{\sigma }}{\arg \min }\left\{{\gamma }_1{\Vert \sigma \Vert }_{L^1}+\frac{1}{2}{\Vert \sigma -\tilde{\sigma }\Vert }^2\right\}={\mathbb{S}}_{{\gamma }_1}^{L_{ad}^{\sigma }}(\tilde{\sigma })\text{for}\text{any}\tilde{\sigma }\in L^2(\mathrm{\Omega }),$$

$$\underset{\mu \in L_{ad}^{\mu }}{\arg \min }\left\{{\gamma }_2{\Vert \mu \Vert }_{L^1}+\frac{1}{2}{\Vert \mu -\tilde{\mu }\Vert }^2\right\}={\mathbb{S}}_{{\gamma }_2}^{L_{ad}^{\mu }}(\tilde{\mu })\text{for}\text{any}\tilde{\mu }\in L^2(\mathrm{\Omega }),$$

where the left-hand sides represents the proximal functions corresponding to σ and μ, respectively and the associated projected soft thresholding functions on the right-hand side are defined as follows

$${\mathbb{S}}_{{\gamma }_1}^{L_{ad}^{\sigma }}(\tilde{\sigma }):=\{\begin{array}{ll}\min \left\{\tilde{\sigma }-{\gamma }_1,b_{\sigma }\right\}\hfill & \text{on}\left\{(x,y)\in \mathrm{\Omega }:\tilde{\sigma }(x,y)>{\gamma }_1\right\}\hfill \\ 0\hfill & \text{on}\left\{(x,y)\in \mathrm{\Omega }:|\tilde{\sigma }(x,y)|\le {\gamma }_1\right\}\hfill \\ \max \left\{\tilde{\sigma }+{\gamma }_1,a_{\sigma }\right\}\hfill & \text{on}\left\{(x,y)\in \mathrm{\Omega }:\tilde{\sigma }(x,y)<-{\gamma }_1\right\}\hfill \end{array}.$$ (36)

$${\mathbb{S}}_{{\gamma }_2}^{L_{ad}^{\mu }}(\tilde{\mu }):=\{\begin{array}{ll}\min \left\{\tilde{\mu }-{\gamma }_2,b_{\mu }\right\}\hfill & \text{on}\left\{(x,y)\in \mathrm{\Omega }:\tilde{\mu }(x,y)>{\gamma }_2\right\}\hfill \\ 0\hfill & \text{on}\left\{(x,y)\in \mathrm{\Omega }:|\tilde{\mu }(x,y)|\le {\gamma }_2\right\}\hfill \\ \max \left\{\tilde{\mu }+{\gamma }_2,a_{\mu }\right\}\hfill & \text{on}\left\{(x,y)\in \mathrm{\Omega }:\tilde{\mu }(x,y)<-{\gamma }_2\right\}\hfill \end{array}.$$ (37)

Using (36)–(37), we get the following iterative schemes

$${\sigma }_{k+1}\leftarrow {\mathbb{S}}_{\frac{{\gamma }_1}{L_{\sigma }}}^{L_{ad}^{\sigma }}\left({\sigma }_k-\frac{1}{L}{\nabla }_{\sigma }{\widehat{J}}_1\left({\sigma }_k\right)\right),$$

$${\mu }_{k+1}\leftarrow {\mathbb{S}}_{\frac{\gamma 2}{L_{\mu }}}^{L_{ad}^{\mu }}\left({\mu }_k-\frac{1}{L}{\nabla }_{\mu }{\widehat{J}}_1\left({\mu }_k\right)\right),$$

starting from a given σ₀, where L is the Lischitz constants for the gradient function $\nabla {\widehat{J}}_1$. The above schemes form the iterative shrinkage-thresholding algorithm (ISTA) method. to accelerate the ISTA scheme described above, one can consider a sequence {t_k, v_k} such that

$$t_0=1,t_k:=1+\sqrt{1+4t_{k-1}^2}/2,$$ (38)

and

$$v_0^{\sigma }:={\sigma }_0,v_k^{\sigma }:={\sigma }_k+\frac{\left(t_{k-1}-1\right)}{t_k}\left({\sigma }_k-{\sigma }_{k-1}\right),$$ (39)

$$v_0^{\mu }:={\mu }_0,v_k^{\mu }:={\mu }_k+\frac{\left(t_{k-1}-1\right)}{t_k}\left({\mu }_k-{\mu }_{k-1}\right).$$ (40)

This gives us the following update for the optimization variable σ_k

$$\begin{gathered} {\sigma }_{k+1}\leftarrow {\mathbb{S}}_{\frac{{\gamma }_1}{L_{\sigma }}}^{L_{ad}^{\sigma }}\left({\sigma }_k-\frac{1}{L}{\nabla }_{\sigma }{\widehat{J}}_1\left(v_k^{\sigma }\right)\right), \\ {\mu }_{k+1}\leftarrow {\mathbb{S}}_{\frac{{\gamma }_2}{L_{\mu }}}^{L_{\mathit{ad}}^{\mu }}\left({\mu }_k-\frac{1}{L}{\nabla }_{\mu }{\widehat{J}}_1\left(v_k^{\mu }\right)\right). \end{gathered}$$ (41)

We note that the arguments of the proximal operators in (41) represents a gradient step with fixed step size s = 1/L. Replacing $v_k^{\sigma }$, $v_k^{\mu }$ in (41) with (39), and assuming that ${\nabla }_{\sigma }{\widehat{J}}_1\left({\sigma }_k\right)\approx {\nabla }_{\sigma }{\widehat{J}}_1\left(v_k^{\sigma }\right)$ and ${\nabla }_{\mu }{\widehat{J}}_1\left({\mu }_k\right)\approx {\nabla }_{\mu }{\widehat{J}}_1\left(v_k^{\mu }\right)$, we obtain the following iterative scheme

$$\begin{gathered} {\sigma }_{k+1}\leftarrow {\mathbb{S}}_{{\gamma }_{1}s}^{L_{ad}^{\sigma }}\left({\sigma }_k-s{\nabla }_{\sigma }{\widehat{J}}_1\left({\sigma }_k\right)+{\theta }_k\left({\sigma }_k-{\sigma }_{k-1}\right)\right), \\ {\mu }_{k+1}\leftarrow {\mathbb{S}}_{{\gamma }_{2}s}^{L_{ad}^{\mu }}\left({\mu }_k-s{\nabla }_{\mu }{\widehat{J}}_1\left({\mu }_k\right)+{\theta }_k\left({\mu }_k-{\mu }_{k-1}\right)\right), \end{gathered}$$ (42)

where μ₋₁ = μ₀. The schemes given in (42) are known as the fast iterative shrinkage-thresholding algorithm (FISTA) proximal gradient method.

The computations done above are valid for any $L\ge L\left({\widehat{J}}_1\right)$. However, because the quantity s = 1/L represents the step size in a gradient update, we use a backtracking line search algorithm to determine the optimal step size in each iteration. We then compute an upper bound L_k that satisfies $L_k\ge L\left({\widehat{J}}_1\right)$ at each iteration step. This leads to the definition of the variable step size as s_k = 1/L_k. The variable step size causes the factor $\frac{\left(t_{k-1}-1\right)}{t_k}$ in (39) to be non-optimal and we replace it by the fixed inertial parameter θ and, thus, the resulting scheme is known as Variable Inertial Proximal Gradient Method (VIP) (see[16] for more detail). We summarize the VIP scheme in the algorithm below

Algorithm 4.2 (Variable inertial proximal (VIP) method).

1. Input: β, ${\widehat{J}}_1$, σ₀ = σ₋₁, μ₀ = μ₋₁, TOL, n > 1, L₀ > 0

   Initialize: $E_1^0=E_2^0=1,k=0$, choose θ ∈ (0,1) and c₁ < 2 and c₂ > 0;

2. While $\Vert E_1^{k-1}\Vert +\Vert E_2^{k-1}\Vert >TOL$ do

3. Compute ${\nabla }_{\sigma }{\widehat{J}}_1\left({\sigma }_k,{\mu }_k\right)$, ${\nabla }_{\mu }{\widehat{J}}_1\left({\sigma }_k,{\mu }_k\right)$

4. Backtracking: Find the smallest non-negative integer i such that with

    $\tilde{L}=n^i{L}_{k-1}$

   $$\begin{gathered} {\widehat{J}}_1(\tilde{\sigma },\tilde{\mu })\le {\widehat{J}}_1\left({\sigma }_k,{\mu }_k\right)+\langle {\nabla }_{\sigma }{\widehat{J}}_1\left({\sigma }_k,{\mu }_k\right),\tilde{\sigma }-{\sigma }_k\rangle +\langle {\nabla }_{\mu }{\widehat{J}}_1\left({\sigma }_k,{\mu }_k\right),\tilde{\mu }-{\mu }_k\rangle \\ +\frac{\tilde{L}}{2}\left({\Vert \tilde{\sigma }-{\sigma }_k\Vert }^2+{\Vert \tilde{\mu }-{\mu }_k\Vert }^2\right) \end{gathered}$$

    where $\tilde{\sigma }={\mathbb{S}}_{{\gamma }_{1}s}^{L_{ad}^{\sigma }}\left({\sigma }_k-s\left({\nabla }_{\sigma }{\widehat{J}}_1\right)\left({\sigma }_k,{\mu }_k\right)+\theta \left({\sigma }_k-{\sigma }_{k-1}\right)\right)$

    $\tilde{\mu }={\mathbb{S}}_{{\gamma }_{2}s}^{L_{ad}^{\mu }}\left({\sigma }_k-s\left({\nabla }_{\mu }{\widehat{J}}_1\right)\left({\sigma }_k,{\mu }_k\right)+\theta \left({\mu }_k-{\mu }_{k-1}\right)\right),$

    $s=c_1(1-\theta )/(\tilde{L}+2c_2),$

5. Set $L_k=\tilde{L}\text{and}{s}_k=c_1(1-\theta )/\left(L_k+2c_2\right)$

6. ${\sigma }_{k+1}={\mathbb{S}}_{{\gamma }_1{s}_k}^{L_{ad}\sigma }\left({\sigma }_k-s_k\left({\nabla }_{\sigma }{\widehat{J}}_1\right)\left({\sigma }_k,{\mu }_k\right)+\theta \left({\sigma }_k-{\sigma }_{k-1}\right)\right)$

   ${\mu }_{k+1}={\mathbb{S}}_{{\gamma }_2{s}_k}^{L_{\mathit{ad}}^{\mu }}\left({\mu }_k-s_k\left({\nabla }_{\mu }{\widehat{J}}_1\right)\left({\sigma }_k,{\mu }_k\right)+\theta \left({\mu }_k-{\mu }_{k-1}\right)\right)$

7. $c_1^k=-\left({\nabla }_{\sigma }{\widehat{J}}_1\right)\left({\sigma }_k,{\mu }_k\right),c_2^k=-\left({\nabla }_{\mu }{\widehat{J}}_1\right)\left({\sigma }_k,{\mu }_k\right)$

8. $E_1^k=E\left({\sigma }_k,c_1^k\right),E_2^k=E\left({\mu }_k,c_2^k\right)$

9. k = k + 1

10. end

5. Numerical results

We first demonstrate the convergence of the Picard scheme given in Algorithm 4.1 for solving (1). We use the method of manufactured solutions to construct an exact solution for (1) with a non-zero source term f(x₁, x₂) on the right hand side. We set D(x₁, x₂) = 1.0, σ(x₁, x₂) = sin(x₁)sin(x₂), μ = 1. Further, we choose Ω = (0, 1) × (0, 1). The boundary condition is given as g(x₁, x₂) = sin(x₁)sin(x₂) and the right-hand side f(x₁, x₂) = 2sin(x₁)sin(x₂)+2(sin(x₁)sin(x₂))². With the preceding choices of the parameters, the exact solution is given as u_ex = sin(x₁)sin(x₂). The solution error is evaluated based on the following discrete L¹ norm

$${\Vert u\Vert }_1=h^2\sum _{i,j=0}^{N_x}\left|u_{i,j}\right|,$$

which we identify with $L_h^1$. The discrete L¹ error is defined as follows

$$Err={\Vert u-u_{ex}\Vert }_1.$$

Table 1 shows the results of experiments that demonstrate the convergence of the Picard algorithm. We see that the resulting order of convergence is $\mathcal{O}(h)$.

Table 1:

Convergence of the Picard algorithm given in Algorithm 4.1

Nx	Err	Order
25	1.70e-3	–
50	8.77e-4	0.96
100	4.39e-4	0.99
200	2.20e-4	1.00

We now present the results of numerical experiments obtained using the VIP scheme to solve the 2PPACT-SR reconstruction problem. We choose our domain in the experiments below as Ω = (−1, 1) × (−1, 1). We discretize Ω into 150 equally spaced points in both x and y directions. The boundary illuminations for solving (1) to generate two sets of initial acoustic wave pressure field data are chosen as g₁(x, y) = 1.0, g₂(x, y) = 2.0. Such a choice of boundary conditions are consistent with Lemma 3.1 that ensure unique solvability of the 2PPACT-SR reconstruction problem. The range of the true values of σ and μ are chosen to lie in (0.1, 1.1) and (0.01, 0.11), respectively, except for the vascular phantom in test case 4. While these ranges are chosen for experimentation purposes, the ratio between the optical coefficients are based on the true experiments done in [14, 50]. Thus, upto a scaling factor, the chosen true values of the optical coefficients are similar to the experimentally observed values. The background values σ_b and μ_b are chosen to be 0.1 and 0.01 respectively, unless otherwise mentioned and D is chosen to be 0.1σ while generating the data with a known σ. For test case 4 with a brain vascular phantom, we choose the range of the values of σ ∈ (0.1, 0.2), which is consistent with the values for σ in brain blood vessels [30], and use the aforementioned ratios to set the values of μ and D. The weights of the functional J given in (6) are chosen as α₁ = α₂ = 1, ξ₁ = 0.01, ξ₂ = 0.01, γ₁ = 0.1, γ₂ = 0.1. We remark that large values of ξ₁, ξ₂ lead to smoothening of edges due to the presence of the H¹ regularization term whereas very small values lead to preservation of artifacts. Furthermore, large values of γ₁, γ₂ also lead to greater sparsity patterns in the reconstructions whereas very small values doesn’t help in removing the artifacts. Thus, the choices of the weights are purely experimental and we have observed that with values of ξ₁, ξ₂ ∈ (0.005, 0.1) and γ₁, γ₂ ∈ (0.01, 0.5), we obtain similar reconstructions. The value of the Grüneisen coefficient is chosen to be 1.0. To generate the data $G_i^{\delta }$, i = 1, 2, we first solve for u_i in (1) with given test values of σ, μ and boundary illumination data g_i on a finer mesh with N = 400 using the Picard iterative scheme given in Algorithm 4.1. We then compute $G_i^{\delta }$ on the finer mesh using the values of σ, μ, u_i from (4). Finally, we restrict $G_i^{\delta }$ onto the coarser mesh with N = 150 and use this as our given data.

In test case 1, we consider a phantom represented by a disk centered at (0.25, 0.25) and having radius 0.25. The value of σ inside the disk is .11 and outside is 0.1. The corresponding value of μ inside the disk is 0.11 and outside is 0.01. With this test case, we demonstrate the need for each of the regularization term in the functional J. Further, we also validate the convergence of the VIP algorithm. The results are shown in Figure 1.

Figure 1:

Test Case 1-Reconstructions of the disk phantom with the 2PPACT-SR framework

Figures 1a and 1b represent the exact phantoms for σ and μ, respectively. Figures 1d and 1g show the reconstructions without any regularization term in the functional J in (6). We observe the presence of strong artifacts in the reconstructions. With the H¹ regularization, the artifacts are severely reduced as seen in Figures 1e and 1h. One could choose a higher weight of the H¹ regularization term that would result in further removal of the artifacts. However, we observed that in addition to artifact suppression, it also resulted in smoothing of the edges and loss of contrast. The addition of the L¹ regularization term circumvents this issue as seen in Figures 1f and 1i. We also see the $\mathcal{O}\left(1/k^2\right)$ convergence of the relative error using the VIP algorithm in Figure 1c.

In test case 2, we consider a heart lung phantom for both σ and μ. For σ, the background value of the phantom is 0.1 that is perturbed into two ellipses that represent the lungs with value 1.1 and into a disk representing heart with value 0.5. The corresponding value of μ inside the ellipses is 0.11, in the disks is 0.05 and 0 elsewhere. The plots of the exact and the reconstructed phantoms are shown in Figure 2.

Figure 2:

Test Case 2-Reconstructions of the heart and lung phantom with the 2PPACT-SR framework

We again see from Figures 2b and 2e that the reconstructions of σ, μ are of high contrast and high resolution. To test the robustness of our method, we added 20% multiplicative Gaussian noise to the interior data ${\mathcal{H}}^{\sigma ,\mu }$ and use it for our 2PPACT-SR inversion algorithm. We also modify the value of the regularization parameters ξ₁ = 0.1, ξ₂ = 0.1, γ₁ = 0.3, γ₂ = 0.3, in order to counter the noisy data. The results can be seen in Figure 2c and 2f. We see that the reconstruction of σ contains a few artifacts but still is of good quality. The reconstruction of μ demonstrates very little artifacts. This shows that our 2PPACT-SR reconstruction framework is robust and accurate even in the presence of noisy data.

In test case 3, we consider σ as the Shepp-Logan phantom given in [31]. The background σ_b is chosen to be 0.4 in this case. We compute μ = 0.1σ and the background value of μ_b is chosen as 0.04. The plots of the exact and reconstructed phantoms are shown in Figure 3.

Figure 3:

Test Case 3-Reconstructions of the Shepp-Logan phantom with the 2PPACT-SR framework

We again see from Figures 3b and 3e that the 2PPACT-SR reconstruction framework gives superior quality reconstructions even for objects with high contrast values and with holes and inclusions. The reconstructions with 20% noise in the interior data are shown in Figures 3c and 3f with the modified regularization parameter values as in the previous test case. We see that the reconstructions are still of high quality with very less artifacts.

In the final test case, we consider a brain vascular phantom taken from [15]. The value of σ lies between 0.1 and 0.2 and the value of μ lies between 0.01 and 0.02. The exact and the reconstructed phantoms are shown in Figure 4.

Figure 4:

Test Case 4-Reconstructions of brain vascular phantom with the 2PPACT-SR framework

Figures 4b and 4d demonstrates the capability of our 2PPACT-SR framework to provided superior reconstructions in vascular imaging. Our method is able to capture the fine detail of the structures without loss of resolution or contrast. We also tested the performance of our algorithm to provide reconstructions in presence of different levels of noise in the interior data. The results are shown in Figure 5.

Figure 5:

Test Case 4-Reconstructions of brain vascular phantom with the 2PPACT-SR framework

Figures 5a–5c and 5d–5f show the reconstructions of the vascular phantom in presence of 5%, 10% and 20% white multiplicative Gaussian noise in the interior data. We observe that as the noise level increases, the artifacts, especially in the reconstruction of σ, also increases. However, our algorithm is able to provide high contrast and high resolution images for upto 10% noise. With 20% noisy data, our algorithm still provided high contrast and high resolution images for μ but the reconstruction of σ shows the presence of moderate speckled artifacts at the corner that reduces the resolution, but still has good contrast. This shows the robustness of our algorithm, even in the presence of large noise in the data.

Remark 5.1.

For the proposed nonlinear reconstruction framework, the underlying assumption that has been used in the reconstruction procedure is the availability of interior acoustic wave pressure field measurements in the whole of the domain Ω. However, in practice, such measurements are limited and sparse due to the restriction of the number of detectors that can be used. Thus, it is of paramount importance to devise a computational algorithm for recovering the optical coefficients from sparse interior data. For this purpose, we remark that in the part of the domain where the sparse interior data can be obtained, multiple sets of such measurements are available. Using this fact, one possible approach to obtain the optical coefficients from sparse data would be to first solve a data completion problem using a Nash games framework (see for e.g., [8]), where missing data is filled in from the given multiple sets of sparse data and then, one can apply our proposed method to obtain superior reconstructions.

6. Conclusions

In this work, we have presented a new reconstruction framework in 2P-PACT for determining the optical coefficients from two-photon photoacoustic data. The framework comprises of a PDE-constrained optimization problem that promotes sparsity patterns in the reconstructions of the single and two photon absorption coefficients. We present a new theoretical analysis of the existence and uniqueness of a solution to a semi-linear elliptic PDE arising in 2P-PACT. Further, we present a proximal scheme using a Picard solver for the semi-linear PDE and its adjoint to solve the optimization problem. Several numerical results demonstrate that the proposed framework is able to achieve reconstructions with high contrast and high resolution for objects including holes and inclusions.