Project Description

The main mathematical challenge  for a Multidimensional  Coefficent Inverse Problem (MCIP)

Any MCIP is both non-linear and ill-posed. On the other hand, the conventional way to solve an MCIP numerically is via the minimization of a least squares objective functional. This functional characterizes

the misfit between the data and ones "guess'' about the unknown coefficient. However, ill-posedness and nonlinearity cause the phenomenon of multiple local minima and ravines of these functionals. Therefore, any gradient-like method of the minimization of such a functional will likely converge to a solution represented by a local minimum and this solution will likely be far from the correct one. Furthermore, because of the ill-posedness, that functional might have many global minima, and there is no guarantee that any of them

would be close to the correct solution.  Because of local minima, conventional numerical methods for MCIPs such as, e.g., Newton-like methods, are locally convergent ones. Convergence of such an algorithm to the correct solution can be rigorously guaranteed only if its starting point actually represents a good first guess for the

solution, i.e., if it is located in a small neighborhood of that solution. However, in many applications a good initial guess is unknown. In particular, in the case of medical optical imaging human organs of ones interest, such as, e.g., brain and female breast, are optically heterogeneous, see, e.g. experimental results of [15].

Therefore, in order to improve the reliability of imaging techniques, one needs to solve a quite challenging mathematical problem: to develop globally convergent numerical methods.  Recently a new globally convergent numerical method for solution of coefficient inverse problem was developed in [2].  Although

only the acoustic wave equation was considered in [2], the methodology of this work is very broad and can be extended on MCIPs of this project. We call a numerical method for a globally convergent  if: (1) a theorem is proven, which ensures that this method leads to a good approximation for the correct solution of that

CIP, regardless on the availability of a priori given good first guess for that solution, and (2) this theorem is confirmed by numerical experiments.

Inverse problem in optical tomography -  definition of absorption coefficient.

It was shown experimentally that diffusion coefficient of light changes slowly in human tissues [15]. Hence we can consider  parabolic equation governing light propagation in human tissues [14].

Near infrared light with, originated by a laser light source, propagates in a diffuse manner in a biological tissue. In medical applications the the wavelength of light is between 500 and 900 nanometers [20]. In the case of a pulsed laser light propagation in biological tissues is governed by the parabolic PDE [14]. It is established experimentally [15,20] that cancerous tumors absorb light more than the surrounding tissue. The tumor/background absorption contrast is between 2:1 and 3:1 [15]. The absorption coefficient characterizes blood oxygenation.  Since malignant tumors are less oxygenated than healthy tissues, a hope of researchers is to detect these tumors on early stages using optical methods.  The field of medical optical imaging is rapidly growing nowadays, see [12,14,15,16,18,20] and references therein. Currently there are some images of breast tumors in vivo produced by researchers [16,18]. The goal is to image tumors of at least 5 mm size. So that they would be discovered on early stages. Optical imaging apparatus is viewed as a complementary tool to the X-rays mammography, similarly with the ultrasound imaging.

Main  ideas of the project

The ground breaking point of this project is the global rather than conventional local convergence of the

proposed numerical methods applied to medical optical imaging and imaging using electromagnetic waves. The global convergence topic is in its infant age with pioneering publication [2]. The most difficult step of the method of [2] is the numerical solution of a certain nonlinear integral differential equation.  The idea of the derivation of that integral differential equation has strong roots in the idea of applications of Carleman estimates to proof the uniqueness and stability theorems for MCIPs [5]. A radically new idea of the method of [2] is that it is not using least squares objective functionals.  Instead a fine structure of the underlying differential operator is used.  Therefore, the phenomenon of local minima is avoided. This project will develop further ideas and methods, which were initiated in [2]; also see more recent publications [3,4].

Global convergence theorems of [2,3,4] assert that the error of the reconstruction of the unknown coefficient by the globally convergent method is proportional to some parameters characterizing approximation errors of this method. At the same time, the most important claim of those theorems is that the solution obtained by the

globally convergent technique is close to the correct one. Therefore it is reasonable to refine this solution via subsequent application of an advanced locally convergent numerical method, which would not contain those approximation errors. In doing so, the solution obtained by the globally convergent part would be taken as the starting point for the local part. The resulting two-stage numerical procedure would still hold the global convergence property. Thus, the second breakthrough idea is to synthesize the globally convergent method

of [2] with the Finite Element Adaptive Method, in the view of applications to medical optical imaging and imaging using electromagnetic waves. First results of this synthesis for the case of the acoustic equation show a good performance [3,4]. In [1] a new adaptive globally convergent method was developed for the case of acoustic equation, where adaptivity technique was applied directly inside globally convergent algorithm. Results of [1] show that this new algorithm significantly enhance shape and contrast of the reconstruction. In the current project we are going develop new adaptive globally convergent method for the case of optical imaging.

                  Figure 1 

            Figure 1 shows results of [4].

          Fig.1a) shows the correct image with inclusions to be reconstructed. 

          Fig.1b) shows the image obtained using globally convergent numerical method which was taken as the first                   

          guess for the adaptivity. Locations of inclusions need to be improved.

          Fig.1c) presents adaptively refined mesh using a posteriori error estimator of [4].

          Fig. 1d) displays obtained image. Both locations of inclusions and 4:1 inclusions/background contrasts are            

          imaged correctly