1,304 research outputs found
A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
In this paper, we develop regularized discrete least squares collocation and
finite volume methods for solving two-dimensional nonlinear time-dependent
partial differential equations on irregular domains. The solution is
approximated using tensor product cubic spline basis functions defined on a
background rectangular (interpolation) mesh, which leads to high spatial
accuracy and straightforward implementation, and establishes a solid base for
extending the computational framework to three-dimensional problems. A
semi-implicit time-stepping method is employed to transform the nonlinear
partial differential equation into a linear boundary value problem. A key
finding of our study is that the newly proposed mesh-free finite volume method
based on circular control volumes reduces to the collocation method as the
radius limits to zero. Both methods produce a large constrained least-squares
problem that must be solved at each time step in the advancement of the
solution. We have found that regularization yields a relatively
well-conditioned system that can be solved accurately using QR factorization.
An extensive numerical investigation is performed to illustrate the
effectiveness of the present methods, including the application of the new
method to a coupled system of time-fractional partial differential equations
having different fractional indices in different (irregularly shaped) regions
of the solution domain
On the analysis of mixed-index time fractional differential equation systems
In this paper we study the class of mixed-index time fractional differential
equations in which different components of the problem have different time
fractional derivatives on the left hand side. We prove a theorem on the
solution of the linear system of equations, which collapses to the well-known
Mittag-Leffler solution in the case the indices are the same, and also
generalises the solution of the so-called linear sequential class of time
fractional problems. We also investigate the asymptotic stability properties of
this class of problems using Laplace transforms and show how Laplace transforms
can be used to write solutions as linear combinations of generalised
Mittag-Leffler functions in some cases. Finally we illustrate our results with
some numerical simulations.Comment: 21 pages, 6 figures (some are made up of sub-figures - there are 15
figures or sub-figures
Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
In this work, we extend the fractional linear multistep methods in [C.
Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional
integral and derivative operators in the sense that the tempered fractional
derivative operator is interpreted in terms of the Hadamard finite-part
integral. We develop two fast methods, Fast Method I and Fast Method II, with
linear complexity to calculate the discrete convolution for the approximation
of the (tempered) fractional operator. Fast Method I is based on a local
approximation for the contour integral that represents the convolution weight.
Fast Method II is based on a globally uniform approximation of the trapezoidal
rule for the integral on the real line. Both methods are efficient, but
numerical experimentation reveals that Fast Method II outperforms Fast Method I
in terms of accuracy, efficiency, and coding simplicity. The memory requirement
and computational cost of Fast Method II are and ,
respectively, where is the number of the final time steps and is the
number of quadrature points used in the trapezoidal rule. The effectiveness of
the fast methods is verified through a series of numerical examples for
long-time integration, including a numerical study of a fractional
reaction-diffusion model
Identifying (BN)2-pyrene as a new class of singlet fission chromophores: significance of azaborine substitution
Singlet fission converts one photoexcited singlet state to two triplet excited states and raises photoelectric conversion efficiency in photovoltaic devices. However, only a handful of chromophores have been known to undergo this process, which greatly limits the application of singlet fission in photovoltaics. We hereby identify a recently synthesized diazadiborine-pyrene ((BN)2-pyrene) as a singlet fission chromophore. Theoretical calculations indicate that it satisfies the thermodynamics criteria for singlet fission. More importantly, the calculations provide a physical chemistry insight into how the BN substitution makes this happen. Both calculation and transient absorption spectroscopy experiment indicate that the chromophore has a better absorption than pentacene. The convenient synthesis pathway of the (BN)2-pyrene suggests an in situ chromophore generation in photovoltaic devices. Two more (BN)2-pyrene isomers are proposed as singlet fission chromophores. This study sets a step forward in the cross-link of singlet fission and azaborine chemistry
Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems
While many phenomena in physics and engineering are formally
high-dimensional, their long-time dynamics often live on a lower-dimensional
manifold. The present work introduces an autoencoder framework that combines
implicit regularization with internal linear layers and regularization
(weight decay) to automatically estimate the underlying dimensionality of a
data set, produce an orthogonal manifold coordinate system, and provide the
mapping functions between the ambient space and manifold space, allowing for
out-of-sample projections. We validate our framework's ability to estimate the
manifold dimension for a series of datasets from dynamical systems of varying
complexities and compare to other state-of-the-art estimators. We analyze the
training dynamics of the network to glean insight into the mechanism of
low-rank learning and find that collectively each of the implicit regularizing
layers compound the low-rank representation and even self-correct during
training. Analysis of gradient descent dynamics for this architecture in the
linear case reveals the role of the internal linear layers in leading to faster
decay of a "collective weight variable" incorporating all layers, and the role
of weight decay in breaking degeneracies and thus driving convergence along
directions in which no decay would occur in its absence. We show that this
framework can be naturally extended for applications of state-space modeling
and forecasting by generating a data-driven dynamic model of a spatiotemporally
chaotic partial differential equation using only the manifold coordinates.
Finally, we demonstrate that our framework is robust to hyperparameter choices
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