Two inverse problems in elasticity are addressed with motivation from cellular biomechanics. The first application is computation of holding forces on a cell during its manipulation and the second application is estimation of a cell’s interior elastic mapping (i.e., estimation of inhomogeneous distribution of stiffness) using only boundary forces and displacements. It is clear from recent works that mechanical forces can play an important role in developmental biology. In this regard, we have developed a vision-based force-sensing technique to estimate forces that are acting on a cell while it is manipulated. This problem is connected to one inverse problem in elasticity known as Cauchy’s problem in elasticity. Geometric nonlinearity under noisy displacement data is accounted while developing the solution procedures for Cauchy’s problem. We have presented solution procedures to the Cauchy’s problem under noisy displacement data. Geometric nonlinearity is also considered in order to account large deformations that the mechanisms (grippers) undergo during the manipulation. The second inverse problem is connected to elastic mapping of the cell. We note that recent works in biomechanics have shown that the disease state can alter the gross stiffness of a cell. Therefore, the pertinent question that one can ask is that which portion (for example Nucleus, cortex, ER) of the elastic property of the cell is majorly altered by the disease state. Mathematically, this question (estimation of inhomogeneous properties of cell) can be answered by solving an inverse elastic boundary value problem using sets of force-displacements boundary measurements. We address the theoretical question of number of boundary data sets required to solve the inverse boundary value problem.

This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.