240 research outputs found
PDEs in Moving Time Dependent Domains
In this work we study partial differential equations defined in a domain that
moves in time according to the flow of a given ordinary differential equation,
starting out of a given initial domain. We first derive a formulation for a
particular case of partial differential equations known as balance equations.
For this kind of equations we find the equivalent partial differential
equations in the initial domain and later we study some particular cases with
and without diffusion. We also analyze general second order differential
equations, not necessarily of balance type. The equations without diffusion are
solved using the characteristics method. We also prove that the diffusion
equations, endowed with Dirichlet boundary conditions and initial data, are
well posed in the moving domain. For this we show that the principal part of
the equivalent equation in the initial domain is uniformly elliptic. We then
prove a version of the weak maximum principle for an equation in a moving
domain. Finally we perform suitable energy estimates in the moving domain and
give sufficient conditions for the solution to converge to zero as time goes to
infinity.Comment: pp 559-577. Without Bounds: A Scientific Canvas of Nonlinearity and
Complex Dynamics (2013) p. 36
Bayesian Image Quality Transfer with CNNs: Exploring Uncertainty in dMRI Super-Resolution
In this work, we investigate the value of uncertainty modeling in 3D
super-resolution with convolutional neural networks (CNNs). Deep learning has
shown success in a plethora of medical image transformation problems, such as
super-resolution (SR) and image synthesis. However, the highly ill-posed nature
of such problems results in inevitable ambiguity in the learning of networks.
We propose to account for intrinsic uncertainty through a per-patch
heteroscedastic noise model and for parameter uncertainty through approximate
Bayesian inference in the form of variational dropout. We show that the
combined benefits of both lead to the state-of-the-art performance SR of
diffusion MR brain images in terms of errors compared to ground truth. We
further show that the reduced error scores produce tangible benefits in
downstream tractography. In addition, the probabilistic nature of the methods
naturally confers a mechanism to quantify uncertainty over the super-resolved
output. We demonstrate through experiments on both healthy and pathological
brains the potential utility of such an uncertainty measure in the risk
assessment of the super-resolved images for subsequent clinical use.Comment: Accepted paper at MICCAI 201
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
The numerical control of the motion of a passive particle in a point vortex flow
This work reports numerical explorations in the advection of one passive tracer by point vortices
living in the unbounded plane. The main objective is to find the energy-optimal displacement
of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The
direct formulation of the corresponding control problems is presented for the case of N = 1, N = 2,
N = 3 and N = 4 vortices. The restrictions are due to (i) the ordinary differential equations that
govern the displacement of the passive particle around the point vortices, (ii) the available time
T to go from the initial position z0 to the final destination zf; and (iii) the maximum absolute
value umax that is imposed on the control variables. The resulting optimization problems are
solved numerically. The numerical results show the existence of nearly/quasi-optimal control.info:eu-repo/semantics/publishedVersio
Harmonic balance Navier-Stokes analysis of tidal stream turbine wave loads
ARCTIC, a novel incompressible Reynoldsâaveraged NavierâStokes finite volume code for the hydrodynamic analysis of open rotor unsteady loads is presented. One of its unique features is a harmonic balance solver enabling highâfidelity analyses of turbine periodic hydrodynamic loads with runtimes reduced by more than one order of magnitude over conventional timeâdomain CFD, and with negligible accuracy penalty. The strength of the new technology is demonstrated by analyzing with both harmonic balance and timeâdomain solvers the load fluctuations of a realistic tidal stream turbine. Such fluctuations are caused by a harmonic perturbation of the freestream velocity similar to that due to surface gravity waves
Reconstructing charge-carrier dynamics in porous silicon membranes from time-resolved interferometric measurements
We performed interferometric time-resolved simultaneous reflectance and transmittance measurements to investigate the carrier dynamics in pump-probe experiments on thin porous silicon membranes. The experimental data was analysed by using a method built on the Wentzel-Kramers-Brillouin approximation and the Drude model, allowing us to reconstruct the excited carriersâ non-uniform distribution in space and its evolution in time. The analysis revealed that the carrier dynamics in porous silicon, with ~50% porosity and native oxide chemistry, is governed by the Shockley-Read-Hall recombination process with a characteristic time constant of 375 picoseconds, whereas diffusion makes an insignificant contribution as it is suppressed by the high rate of scattering
Simulation of the Response of the Inner Hair Cell Stereocilia Bundle to an Acoustical Stimulus
Mammalian hearing relies on a cochlear hydrodynamic sensor embodied in the inner
hair cell stereocilia bundle. It is presumed that acoustical stimuli induce a
fluid shear-driven motion between the tectorial membrane and the reticular
lamina to deflect the bundle. It is hypothesized that ion channels are opened by
molecular gates that sense tension in tip-links, which connect adjacent stepped
rows of stereocilia. Yet almost nothing is known about how the fluid and bundle
interact. Here we show using our microfluidics model how each row of stereocilia
and their associated tip links and gates move in response to an acoustical input
that induces an orbital motion of the reticular lamina. The model confirms the
crucial role of the positioning of the tectorial membrane in hearing, and
explains how this membrane amplifies and synchronizes the timing of peak tension
in the tip links. Both stereocilia rotation and length change are needed for
synchronization of peak tip link tension. Stereocilia length change occurs in
response to accelerations perpendicular to the oscillatory fluid shear flow.
Simulations indicate that nanovortices form between rows to facilitate diffusion
of ions into channels, showing how nature has devised a way to solve the
diffusive mixing problem that persists in engineered microfluidic devices
On the wellposedness of some McKean models with moderated or singular diffusion coefficient
We investigate the well-posedness problem related to two models of nonlinear
McKean Stochastic Differential Equations with some local interaction in the
diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with
moderate interaction, previously studied by Meleard and Jourdain in [16], under
slightly weaker assumptions, by showing the existence and uniqueness of a weak
solution using a Sobolev regularity framework instead of a Holder one. Second,
we study the construction of a Lagrangian Stochastic model endowed with a
conditional McKean diffusion term in the velocity dynamics and a nondegenerate
diffusion term in the position dynamics
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