856 research outputs found
Force traction microscopy: An inverse problem with pointwise observations
Force Traction Microscopy is an inversion method that allows to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself. This classical biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled using a variational framework and then a finite elements discretization. In such a case, a variation of the error functional under suitable regularization is operated in view of its minimization. This setting naturally suggests the introduction of a new equation, based on the adjoint operator of the elasticity problem. In this paper we illustrate the rigorous theory of the two-dimensional and three dimensional problem, involving in the former case a distributed control and in the latter case a surface control. The pointwise observations require to exploit the theory of elasticity extended to forcing terms that are Borel measure
Well-posedness via Monotonicity. An Overview
The idea of monotonicity (or positive-definiteness in the linear case) is
shown to be the central theme of the solution theories associated with problems
of mathematical physics. A "grand unified" setting is surveyed covering a
comprehensive class of such problems. We elaborate the applicability of our
scheme with a number examples. A brief discussion of stability and
homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte
Unique Determination of Sound Speeds for Coupled Systems of Semi-linear Wave Equations
We consider coupled systems of semi-linear wave equations with different
sound speeds on a finite time interval and a bounded Lipschitz domain
in , with boundary . We show the coupled
systems are well posed for variable coefficient sounds speeds and short times.
Under the assumption of small initial data, we prove the source to solutions
map on associated with the nonlinear problem is
sufficient to determine the source-to-solution map for the linear problem. We
can then reconstruct the sound speeds in for the coupled nonlinear
wave equations under certain geometric assumptions. In the case of the full
source to solution map in this reconstruction could also be
accomplished under fewer geometric assumptions.Comment: minor update
The higher order regularity Dirichlet problem for elliptic systems in the upper-half space
We identify a large class of constant (complex) coefficient, second order
elliptic systems for which the Dirichlet problem in the upper-half space with
data in -based Sobolev spaces, , of arbitrary smoothness
, is well-posed in the class of functions whose nontangential maximal
operator of their derivatives up to, and including, order is
-integrable. This class includes all scalar, complex coefficient elliptic
operators of second order, as well as the Lam\'e system of elasticity, among
others
Constraint interface preconditioning for topology optimization problems
The discretization of constrained nonlinear optimization problems arising in
the field of topology optimization yields algebraic systems which are
challenging to solve in practice, due to pathological ill-conditioning, strong
nonlinearity and size. In this work we propose a methodology which brings
together existing fast algorithms, namely, interior-point for the optimization
problem and a novel substructuring domain decomposition method for the ensuing
large-scale linear systems. The main contribution is the choice of interface
preconditioner which allows for the acceleration of the domain decomposition
method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com
Null boundary controllability of a 1-dimensional heat equation with an internal point mass
We consider a linear hybrid system composed by two rods of equal length
connected by a point mass. We show that the system is null controllable with
Dirichlet and Neumann controls. The results are based on a careful spectral
spectral analysis together with the moment method.Comment: 12 pages, typos corrected, added references, matches version to be
submitted to Systems and Control Letter
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