9,093 research outputs found
A new class of efficient and robust energy stable schemes for gradient flows
We propose a new numerical technique to deal with nonlinear terms in gradient
flows. By introducing a scalar auxiliary variable (SAV), we construct efficient
and robust energy stable schemes for a large class of gradient flows. The SAV
approach is not restricted to specific forms of the nonlinear part of the free
energy, and only requires to solve {\it decoupled} linear equations with {\it
constant coefficients}. We use this technique to deal with several challenging
applications which can not be easily handled by existing approaches, and
present convincing numerical results to show that our schemes are not only much
more efficient and easy to implement, but can also better capture the physical
properties in these models. Based on this SAV approach, we can construct
unconditionally second-order energy stable schemes; and we can easily construct
even third or fourth order BDF schemes, although not unconditionally stable,
which are very robust in practice. In particular, when coupled with an adaptive
time stepping strategy, the SAV approach can be extremely efficient and
accurate
A Noether theorem for random locations
We propose a unified framework for random locations exhibiting some
probabilistic symmetries such as stationarity, self-similarity, etc. A theorem
of Noether's type is proved, which gives rise to a conservation law describing
the change of the density function of a random location as the interval of
interest changes. We also discuss the boundary and near boundary behavior of
the distributions of the random locations.Comment: 26 page
A fourth-order time-splitting Laguerre-Hermite pseudo-spectral method for Bose-Einstein condensates
A fourth-order time-splitting Laguerre-Hermite pseudospectral method is
introduced for Bose-Einstein condensates (BEC) in 3-D with cylindrical
symmetry.
The method is explicit, unconditionally stable, time reversible and time
transverse invariant. It conserves the position density, and is spectral
accurate in space and fourth-order accurate in time. Moreover, the new method
has two other important advantages: (i) it reduces a 3-D problem with
cylindrical symmetry to an effective 2-D problem; (ii) it solves the problem in
the whole space instead of in a truncated artificial computational domain. The
method is applied to vector Gross-Pitaevskii equations (VGPEs) for
multi-component BECs. Extensive numerical tests are presented for 1-D GPE, 2-D
GPE with radial symmetry, 3-D GPE with cylindrical symmetry as well as 3-D
VGPEs for two-component BECs to show the efficiency and accuracy of the new
numerical method.Comment: 21 pages, 4 figure
A pressure correction scheme for generalized form of energy-stable open boundary conditions for incompressible flows
We present a generalized form of open boundary conditions, and an associated
numerical algorithm, for simulating incompressible flows involving open or
outflow boundaries. The generalized form represents a family of open boundary
conditions, which all ensure the energy stability of the system, even in
situations where strong vortices or backflows occur at the open/outflow
boundaries. Our numerical algorithm for treating these open boundary conditions
is based on a rotational pressure correction-type strategy, with a formulation
suitable for spectral-element spatial discretizations. We have introduced
a discrete equation and associated boundary conditions for an auxiliary
variable. The algorithm contains constructions that prevent a numerical locking
at the open/outflow boundary. In addition, we have also developed a scheme with
a provable unconditional stability for a sub-class of the open boundary
conditions. Extensive numerical experiments have been presented to demonstrate
the performance of our method for several flow problems involving open/outflow
boundaries. We compare simulation results with the experimental data to
demonstrate the accuracy of our algorithm. Long-time simulations have been
performed for a range of Reynolds numbers at which strong vortices or backflows
occur at the open/outflow boundaries. We show that the open boundary conditions
and the numerical algorithm developed herein produce stable simulations in such
situations.Comment: 24 pages, 7 figures, 4 table
LG/CY Correspondence for Elliptic Orbifold Curves via Modularity
We prove the Landau-Ginzburg/Calabi-Yau correspondence between the
Gromov-Witten theory of each elliptic orbifold curve and its
Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the
correlation functions in these two enumerative theories are different
representations of the same set of quasi-modular forms, expanded around
different points on the upper-half plane. We relate these two representations
by the Cayley transform.Comment: v3: minor correction
A Tight Bound of Hard Thresholding
This paper is concerned with the hard thresholding operator which sets all
but the largest absolute elements of a vector to zero. We establish a {\em
tight} bound to quantitatively characterize the deviation of the thresholded
solution from a given signal. Our theoretical result is universal in the sense
that it holds for all choices of parameters, and the underlying analysis
depends only on fundamental arguments in mathematical optimization. We discuss
the implications for two domains:
Compressed Sensing. On account of the crucial estimate, we bridge the
connection between the restricted isometry property (RIP) and the sparsity
parameter for a vast volume of hard thresholding based algorithms, which
renders an improvement on the RIP condition especially when the true sparsity
is unknown. This suggests that in essence, many more kinds of sensing matrices
or fewer measurements are admissible for the data acquisition procedure.
Machine Learning. In terms of large-scale machine learning, a significant yet
challenging problem is learning accurate sparse models in an efficient manner.
In stark contrast to prior work that attempted the -relaxation for
promoting sparsity, we present a novel stochastic algorithm which performs hard
thresholding in each iteration, hence ensuring such parsimonious solutions.
Equipped with the developed bound, we prove the {\em global linear convergence}
for a number of prevalent statistical models under mild assumptions, even
though the problem turns out to be non-convex.Comment: V1 was submitted to COLT 2016. V2 fixes minor flaws, adds extra
experiments and discusses time complexity, V3 has been accepted to JML
On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes Phase Field Model
We construct a numerical scheme based on the scalar auxiliary variable (SAV)
approach in time and the MAC discretization in space for the
Cahn-Hilliard-Navier-Stokes phase field model, and carry out stability and
error analysis. The scheme is linear, second-order, unconditionally energy
stable and can be implemented very efficiently. We establish second-order error
estimates both in time and space for phase field variable, chemical potential,
velocity and pressure in different discrete norms. We also provide numerical
experiments to verify our theoretical results and demonstrate the robustness
and accuracy of the our scheme
Distributional Compatibility for Change of Measures
In this paper, we characterize compatibility of distributions and probability
measures on a measurable space. For a set of indices , we say that
the tuples of probability measures and distributions
are {compatible} if there exists a random variable
having distribution under for each . We first
establish an equivalent condition using conditional expectations for general
(possibly uncountable) . For a finite , it turns out that
compatibility of and depends on the
heterogeneity among compared with that among .
We show that, under an assumption that the measurable space is rich enough,
and are compatible if and only if
dominates in a notion of heterogeneity
order, defined via multivariate convex order between the Radon-Nikodym
derivatives of and with respect to some
reference measures.
We then proceed to generalize our results to stochastic processes, and
conclude the paper with an application to portfolio selection problems under
multiple constraints.Comment: 34 page
Diameter critical graphs
A graph is called diameter--critical if its diameter is , and the
removal of any edge strictly increases the diameter. In this paper, we prove
several results related to a conjecture often attributed to Murty and Simon,
regarding the maximum number of edges that any diameter--critical graph can
have. In particular, we disprove a longstanding conjecture of Caccetta and
H\"aggkvist (that in every diameter-2-critical graph, the average edge-degree
is at most the number of vertices), which promised to completely solve the
extremal problem for diameter-2-critical graphs.
On the other hand, we prove that the same claim holds for all higher
diameters, and is asymptotically tight, resolving the average edge-degree
question in all cases except diameter-2. We also apply our techniques to prove
several bounds for the original extremal question, including the correct
asymptotic bound for diameter--critical graphs, and an upper bound of
for the number of edges in a diameter-3-critical
graph
Topological Crystalline Insulator Nanostructures
Topological crystalline insulators are topological insulators whose surface
states are protected by the crystalline symmetry, instead of the time reversal
symmetry. Similar to the first generation of three-dimensional topological
insulators such as Bi2Se3 and Bi2Te3, topological crystalline insulators also
possess surface states with exotic electronic properties such as spin-momentum
locking and Dirac dispersion. Experimentally verified topological crystalline
insulators to date are SnTe, Pb1-xSnxSe, and Pb1-xSnxTe. Because topological
protection comes from the crystal symmetry, magnetic impurities or in-plane
magnetic fields are not expected to open a gap in the surface states in
topological crystalline insulators. Additionally, because they are cubic
structure instead of layered structure, branched structures or strong coupling
with other materials for large proximity effects are possible, which are
difficult with layered Bi2Se3 and Bi2Te3. Thus, additional fundamental
phenomena inaccessible in three-dimensional topological insulators can be
pursued. In this review, topological crystalline insulator SnTe nanostructures
will be discussed. For comparison, experimental results based on SnTe thin
films will be covered. Surface state properties of topological crystalline
insulators will be discussed briefly.Comment: Accepted Manuscript Nanoscale, 201
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