1,309 research outputs found
A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
This work deals with the numerical solution of the monodomain and bidomain
models of electrical activity of myocardial tissue. The bidomain model is a
system consisting of a possibly degenerate parabolic PDE coupled with an
elliptic PDE for the transmembrane and extracellular potentials, respectively.
This system of two scalar PDEs is supplemented by a time-dependent ODE modeling
the evolution of the so-called gating variable. In the simpler sub-case of the
monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple
models for the membrane and ionic currents are considered, the
Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical
solutions of the bidomain and monodomain models exhibit wavefronts with steep
gradients, we propose a finite volume scheme enriched by a fully adaptive
multiresolution method, whose basic purpose is to concentrate computational
effort on zones of strong variation of the solution. Time adaptivity is
achieved by two alternative devices, namely locally varying time stepping and a
Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical
examples demonstrates thatthese methods are efficient and sufficiently accurate
to simulate the electrical activity in myocardial tissue with affordable
effort. In addition, an optimalthreshold for discarding non-significant
information in the multiresolution representation of the solution is derived,
and the numerical efficiency and accuracy of the method is measured in terms of
CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure
A Virtual Element Method for a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology
We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion
system of the cardiac electric field. To this system, we analyze an
-conforming discretization by means of VEM which can make use of
general polygonal meshes. Under standard assumptions on the computational
domain, we establish the convergence of the discrete solution by considering a
series of a priori estimates and by using a general compactness
criterion. Moreover, we obtain optimal order space-time error estimates in the
norm. Finally, we report some numerical tests supporting the theoretical
results
Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems
We present a fully adaptive multiresolution scheme for spatially
two-dimensional, possibly degenerate reaction-diffusion systems, focusing on
combustion models and models of pattern formation and chemotaxis in
mathematical biology. Solutions of these equations in these applications
exhibit steep gradients, and in the degenerate case, sharp fronts and
discontinuities. The multiresolution scheme is based on finite volume
discretizations with explicit time stepping. The multiresolution representation
of the solution is stored in a graded tree. By a thresholding procedure, namely
the elimination of leaves that are smaller than a threshold value, substantial
data compression and CPU time reduction is attained. The threshold value is
chosen optimally, in the sense that the total error of the adaptive scheme is
of the same slope as that of the reference finite volume scheme. Since chemical
reactions involve a large range of temporal scales, but are spatially well
localized (especially in the combustion model), a locally varying adaptive time
stepping strategy is applied. It turns out that local time stepping accelerates
the adaptive multiresolution method by a factor of two, while the error remains
controlled.Comment: 27 pages, 14 figure
Well-posedness results for triply nonlinear degenerate parabolic equations
We study the well-posedness of triply nonlinear degenerate
elliptic-parabolic-hyperbolic problem in a bounded domain with
homogeneous Dirichlet boundary conditions. The nonlinearities and
are supposed to be continuous non-decreasing, and the nonlinearity
falls within the Leray-Lions framework. Some restrictions
are imposed on the dependence of on
and also on the set where degenerates. A model case is
with which is strictly increasing except on a locally finite number of
segments, and which is of the Leray-Lions kind. We are
interested in existence, uniqueness and stability of entropy solutions. If
, we obtain a general continuous dependence result on data
and nonlinearities . Similar result
is shown for the degenerate elliptic problem which corresponds to the case of
and general non-decreasing surjective . Existence, uniqueness
and continuous dependence on data are shown when and
is continuous
Convergence of discrete duality finite volume schemes for the cardiac bidomain model
We prove convergence of discrete duality finite volume (DDFV) schemes on
distorted meshes for a class of simplified macroscopic bidomain models of the
electrical activity in the heart. Both time-implicit and linearised
time-implicit schemes are treated. A short description is given of the 3D DDFV
meshes and of some of the associated discrete calculus tools. Several numerical
tests are presented
Using machine learning on the sources of retinal images for diagnosis by proxy of diabetes mellitus and diabetic retinopathy
In current research in ophthalmology, images of the vascular system in the human retina are used as exploratory proxies for pathologies affecting different organs. This thesis addresses the analysis, using machine learning and computer vision techniques, of retinal images acquired with different techniques (Fundus retinographies, optical coherence tomography and optical coherence tomography angiography), with the objective of using them to assist diagnostic decision making in diabetes mellitus and diabetic retinopathy. This thesis explores the use of matrix factorization-based source extraction techniques, as the basis to transform the retinal images for classification. The proposed approach consists on preprocessing the images to enable the learning of an unsupervised parts-based representation prior to the classification. As a result of the use of interpretable models, with this approach we unveiled an important bias in the data. After correcting for the bias, promising results were still obtained which merit for further exploration
Entropy Solution for Anisotropic Reaction-Diffusion-Advection Systems with L1 Data
In this paper, we study the question of existence and uniqueness of entropy solutions for a system of nonlinear partial differential equations with general anisotropic diffusivity and transport effects, supplemented with no-flux boundary conditions, modeling the spread of an epidemic disease through a heterogeneous habitat.In this paper, we study the question of existence and uniqueness of entropy solutions for a system of nonlinear partial differential equations with general anisotropic diffusivity and transport effects, supplemented with no-flux bound ary conditions, modeling the spread of an epidemic disease through a heteroge neous habitat
Optimal frequency conversion in the nonlinear stage of modulation instability
We investigate multi-wave mixing associated with the strongly pump depleted
regime of induced modulation instability (MI) in optical fibers. For a complete
transfer of pump power into the sideband modes, we theoretically and
experimentally demonstrate that it is necessary to use a much lower seeding
modulation frequency than the peak MI gain value. Our analysis shows that a
record 95 % of the input pump power is frequency converted into the comb of
sidebands, in good quantitative agreement with analytical predictions based on
the simplest exact breather solution of the nonlinear Schr\"odinger equation
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