116 research outputs found
On the Inverse Problem for a Size-Structured Population Model
We consider a size-structured model for cell division and address the
question of determining the division (birth) rate from the measured stable size
distribution of the population. We formulate such question as an inverse
problem for an integro-differential equation posed on the half line. We develop
firstly a regular dependency theory for the solution in terms of the
coefficients and, secondly, a novel regularization technique for tackling this
inverse problem which takes into account the specific nature of the equation.
Our results rely also on generalized relative entropy estimates and related
Poincar\'e inequalities
Tangent Graeffe Iteration
Graeffe iteration was the choice algorithm for solving univariate polynomials
in the XIX-th and early XX-th century. In this paper, a new variation of
Graeffe iteration is given, suitable to IEEE floating-point arithmetics of
modern digital computers. We prove that under a certain generic assumption the
proposed algorithm converges. We also estimate the error after N iterations and
the running cost. The main ideas from which this algorithm is built are:
classical Graeffe iteration and Newton Diagrams, changes of scale
(renormalization), and replacement of a difference technique by a
differentiation one. The algorithm was implemented successfully and a number of
numerical experiments are displayed
Geometrical Loci and CFTs via the Virasoro Symmetry of the mKdV-SG hierarchy: an excursus
We will describe the appearance of specific algebraic KdV potentials as a
consequence of a requirement on a integro-differential expression. This
expression belongs to a class generated by means of Virasoro vector fields
acting on the KdV field. The ``almost'' rational KdV fields are described in
terms of a geometrical locus of complex points. A class of solutions of this
locus has recently appeared as a description of any conformal Verma module
without degeneration.Comment: LaTex, 9 page
Online Local Volatility Calibration by Convex Regularization with Morozov's Principle and Convergence Rates
We address the inverse problem of local volatility surface calibration from
market given option prices. We integrate the ever-increasing flow of option
price information into the well-accepted local volatility model of Dupire. This
leads to considering both the local volatility surfaces and their corresponding
prices as indexed by the observed underlying stock price as time goes by in
appropriate function spaces. The resulting parameter to data map is defined in
appropriate Bochner-Sobolev spaces. Under this framework, we prove key
regularity properties. This enable us to build a calibration technique that
combines online methods with convex Tikhonov regularization tools. Such
procedure is used to solve the inverse problem of local volatility
identification. As a result, we prove convergence rates with respect to noise
and a corresponding discrepancy-based choice for the regularization parameter.
We conclude by illustrating the theoretical results by means of numerical
tests.Comment: 23 pages, 5 figure
Bi-Hamiltonian Aspects of a Matrix Harry Dym Hierarchy
We study the Harry Dym hierarchy of nonlinear evolution equations from the
bi-Hamiltonian view point. This is done by using the concept of an S-hierarchy.
It allows us to define a matrix Harry Dym hierarchy of commuting Hamiltonian
flows in two fields that projects onto the scalar Harry Dym hierarchy. We also
show that the conserved densities of the matrix Harry Dym equation can be found
by means of a Riccati-type equation.Comment: Revised version, 22 pages; a section on reciprocal transformations
added. To appear in J. Math. Phys
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