131 research outputs found
Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints: an extended version
We consider smooth optimization problems with a Hermitian positive
semi-definite fixed-rank constraint, where a quotient geometry with three
Riemannian metrics is used to represent this
constraint. By taking the nonlinear conjugate gradient method (CG) as an
example, we show that CG on the quotient geometry with metric is
equivalent to CG on the factor-based optimization framework, which is often
called the Burer--Monteiro approach. We also show that CG on the quotient
geometry with metric is equivalent to CG on the commonly-used embedded
geometry. We call two CG methods equivalent if they produce an identical
sequence of iterates . In addition, we show that if the limit point of
the sequence generated by an algorithm has lower rank, that is
has rank and the limit
point has rank , then the condition number of the Riemannian
Hessian with metric can be unbounded, but those of the other two metrics
stay bounded. Numerical experiments show that the Burer--Monteiro CG method has
slower local convergence rate if the limit point has a reduced rank, compared
to CG on the quotient geometry under the other two metrics. This slower
convergence rate can thus be attributed to the large condition number of the
Hessian near a minimizer
A Bubble Model for the Gating of Kv Channels
Voltage-gated Kv channels play fundamental roles in many biological
processes, such as the generation of the action potential. The gating mechanism
of Kv channels is characterized experimentally by single-channel recordings and
ensemble properties of the channel currents. In this work, we propose a bubble
model coupled with a Poisson-Nernst-Planck (PNP) system to capture the key
characteristics, particularly the delay in the opening of channels. The coupled
PNP system is solved numerically by a finite-difference method and the solution
is compared with an analytical approximation. We hypothesize that the
stochastic behaviour of the gating phenomenon is due to randomness of the
bubble and channel sizes. The predicted ensemble average of the currents under
various applied voltages across the channels is consistent with experimental
observations, and the Cole-Moore delay is captured by varying the holding
potential
Micro-Macro Modeling of Polymeric Fluids and Shear-Induced Microscopic Behaviors
This article delves into the micro-macro modeling of polymeric fluids,
considering various microscopic potential energies, including the classical
Hookean potential, as well as newly proposed modified Morse and Elastic-plastic
potentials. These proposed potentials encompass microscopic-scale bond-breaking
processes. The development of a thermodynamically consistent micro-macro model
is revisited, employing the energy variational method. To validate the model's
predictions, we conduct numerical simulations utilizing a deterministic
particle-FEM method. Our numerical findings shed light on the distinct
behaviors exhibited by polymer chains at the micro-scale in comparison to the
macro-scale velocity and induced shear stresses of fluids under shear flow.
Notably, we observe that polymer elongation, rotation, and bond breaking
contribute to the zero polymer-induced stress in the micro-macro model when
employing Morse and Elastic-plastic potentials. Furthermore, at high shear
rates, polymer rotation is found to induce shear-thinning behavior in the model
employing the classical Hookean potential
A phase field model for mass transport with semi-permeable interfaces
In this paper, a thermal-dynamical consistent model for mass transfer across
permeable moving interfaces is proposed by using the energy variation method.
We consider a restricted diffusion problem where the flux across the interface
depends on its conductance and the difference of the concentration on each
side. The diffusive interface phase-field framework used here has several
advantages over the sharp interface method. First of all, explicit tracking of
the interface is no longer necessary. Secondly, the interfacial condition can
be incorporated with a variable diffusion coefficient. A detailed asymptotic
analysis confirms the diffusive interface model converges to the existing sharp
interface model as the interface thickness goes to zero. A decoupled energy
stable numerical scheme is developed to solve this system efficiently.
Numerical simulations first illustrate the consistency of theoretical results
on the sharp interface limit. Then a convergence study and energy decay test
are conducted to ensure the efficiency and stability of the numerical scheme.
To illustrate the effectiveness of our phase-field approach, several examples
are provided, including a study of a two-phase mass transfer problem where
drops with deformable interfaces are suspended in a moving fluid.Comment: 20 pages, 15 figure
An energy stable C<sup>0</sup> finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density
In this paper, we focus on modeling and simulation of two-phase flow with
moving contact lines and variable density. A thermodynamically consistent
phase-field model with General Navier Boundary Condition is developed based on
the concept of quasi-incompressibility and the energy variational method. Then
a mass conserving and energy stable C0 finite element scheme is developed to
solve the PDE system. Various numerical simulation results show that the
proposed schemes are mass conservative, energy stable and the 2nd order for P1
element and 3rd order for P2 element convergence rate in the sense of L2 norm
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