33 research outputs found
On the simultanenous identification of the nonlinearity coefficient and the sound speed in the Westervelt equation
This paper considers the Westervelt equation, one of the most widely used
models in nonlinear acoustics, and seeks to recover two spatially-dependent
parameters of physical importance from time-trace boundary measurements.
Specifically, these are the nonlinearity parameter often referred
to as in the acoustics literature and the wave speed . The
determination of the spatial change in these quantities can be used as a means
of imaging. We consider identifiability from one or two boundary measurements
as relevant in these applications. For a reformulation of the problem in terms
of the squared slowness and the combined coefficient
we devise a frozen Newton method and prove
its convergence. The effectiveness (and limitations) of this iterative scheme
are demonstrated by numerical examples
On the solution of two-sided fractional ordinary differential equations of Caputo type
This paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain Lévy processes
Cell Detection by Functional Inverse Diffusion and Non-negative Group SparsityPart I: Modeling and Inverse Problems
In this two-part paper, we present a novel framework and methodology to
analyze data from certain image-based biochemical assays, e.g., ELISPOT and
Fluorospot assays. In this first part, we start by presenting a physical
partial differential equations (PDE) model up to image acquisition for these
biochemical assays. Then, we use the PDEs' Green function to derive a novel
parametrization of the acquired images. This parametrization allows us to
propose a functional optimization problem to address inverse diffusion. In
particular, we propose a non-negative group-sparsity regularized optimization
problem with the goal of localizing and characterizing the biological cells
involved in the said assays. We continue by proposing a suitable discretization
scheme that enables both the generation of synthetic data and implementable
algorithms to address inverse diffusion. We end Part I by providing a
preliminary comparison between the results of our methodology and an expert
human labeler on real data. Part II is devoted to providing an accelerated
proximal gradient algorithm to solve the proposed problem and to the empirical
validation of our methodology.Comment: published, 15 page
Stabilized variational formulation for direct solution of inverse problems in heat conduction and elasticity with discontinuities
We consider the design of finite element methods for inverse problems with
full-field data governed by elliptic forward operators. Such problems arise in
applications in inverse heat conduction, in mechanical property
characterization, and in medical imaging. For this class of problems, novel
finite element methods have been proposed (Barbone et al., 2010) that
give good performance, provided the solutions are in the H^1(Ω) function
space. The material property distributions being estimated can be discontinuous,
however, and therefore it is desirable to have formulations that can
accommodate discontinuities in both data and solution. Toward this end, we
present a mixed variational formulation for this class of problems that handles
discontinuities well. We motivate the mixed formulation by examining the
possibility of discretizing using a discontinuous discretization in an irreducible finite
element method, and discuss the limitations of that approach. We then derive a
new mixed formulation based on a least-square error in the constitutive
equation. We prove that the continuous variational formulations are well-posed
for applications in both inverse heat conduction and plane stress elasticity. We
derive a priori error bounds for discretization error, valid in the limit
of mesh refinement. We demonstrate convergence of the method with mesh
refinement in cases with both continuous and discontinuous solutions. Finally we
apply the formulation to measured data to estimate the elastic shear modulus
distributions in both tissue mimicking phantoms and in breast masses from data
collected in vivo
Mathematical perspectives on waves and currents
The behaviour of waves on the surface of a fluid has fascinated scientists for centuries. Attempts to describe the problem mathematically have revealed a rich geometric structure, as well as a number of celebrated equations. When considering a stochastic theory of water waves, it is therefore sensible to begin with a structure preserving methodology of introducing a stochastic noise into a fluid model. Within this thesis, the mathematical framework of semi-martingale driven variational principles is introduced, which reveals a new methodology of formulating problems for which we have a stochastic action integral. The inclusion of stochastic advection by Lie transport into the underlying fluid momentum equation will allow us to achieve a novel stochastic perturbation of water wave theory which preserves its geometric properties. A number of phenomena observable on the free surface of a fluid are challenging to describe using existing modelling approaches. In particular, the classical modelling approach requires modification to permit the introduction of thermal gradients or rotational flows. Through a new variational perspective involving the composition of two maps, the interaction between waves and thermal fronts in the upper ocean is studied. This approach involves a natural separation of waves and currents on the free surface as vertical oscillations around a horizontal two dimensional flow, and allows the consideration of wave-current interactions. Separately, currents are responsible for the advection of material suspended within the fluid. We consider the procession of an inertial object through a fluid domain, which involves fluid equations with the structure of a fractional order differential equation. It is shown that the most commonly applied such equation, the Maxey-Riley equation, is globally well-posed, a fact which was absent from the literature prior to this thesis.Open Acces
Initial-Boundary Value Problem for Fractional Partial Differential Equations of Higher Order
The initial-boundary value problem for partial differential equations of higher-order involving the Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established
A mixed finite element method for Darcy’s equations with pressure dependent porosity
In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy’s equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose
dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuska-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the
velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two
different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Cl´ement interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de ConcepciónUniversity of LausanneMinistry of Education, Youth and Sports of the Czech Republi
Existence and Uniqueness of Solutions for the System of Nonlinear Fractional Differential Equations with Nonlocal and Integral Boundary Conditions
In the present study, the nonlocal and integral boundary value problems for the system of nonlinear fractional differential equations involving the Caputo fractional derivative are investigated. Theorems on existence and uniqueness of a solution are established under some sufficient conditions on nonlinear terms. A simple example of application of the main result of this paper is presented