15,509 research outputs found
Resolving phase transitions with Discontinuous Galerkin methods
We demonstrate the applicability and advantages of Discontinuous Galerkin
(DG) schemes in the context of the Functional Renormalization Group (FRG). We
investigate the -model in the large limit. It is shown that the flow
equation for the effective potential can be cast into a conservative form. We
discuss results for the Riemann problem, as well as initial conditions leading
to a first and second order phase transition. In particular, we unravel the
mechanism underlying first order phase transitions, based on the formation of a
shock in the derivative of the effective potential.Comment: 19 pages, 9 figures, corrected typos, updated references, extended
explanation
Comparative study of semiclassical approaches to quantum dynamics
Quantum states can be described equivalently by density matrices, Wigner
functions or quantum tomograms. We analyze the accuracy and performance of
three related semiclassical approaches to quantum dynamics, in particular with
respect to their numerical implementation. As test cases, we consider the time
evolution of Gaussian wave packets in different one-dimensional geometries,
whereby tunneling, resonance and anharmonicity effects are taken into account.
The results and methods are benchmarked against an exact quantum mechanical
treatment of the system, which is based on a highly efficient Chebyshev
expansion technique of the time evolution operator.Comment: 32 pages, 8 figures, corrected typos and added references; version as
publishe
Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth
The theory of Forward-Backward Stochastic Differential Equations (FBSDEs)
paves a way to probabilistic numerical methods for nonlinear parabolic PDEs.
The majority of the results on the numerical methods for FBSDEs relies on the
global Lipschitz assumption, which is not satisfied for a number of important
cases such as the Fisher--KPP or the FitzHugh--Nagumo equations. Furthermore,
it has been shown in \cite{LionnetReisSzpruch2015} that for BSDEs with monotone
drivers having polynomial growth in the primary variable , only the
(sufficiently) implicit schemes converge. But these require an additional
computational effort compared to explicit schemes.
This article develops a general framework that allows the analysis, in a
systematic fashion, of the integrability properties, convergence and
qualitative properties (e.g.~comparison theorem) for whole families of modified
explicit schemes. The framework yields the convergence of some modified
explicit scheme with the same rate as implicit schemes and with the
computational cost of the standard explicit scheme.
To illustrate our theory, we present several classes of easily implementable
modified explicit schemes that can computationally outperform the implicit one
and preserve the qualitative properties of the solution to the BSDE. These
classes fit into our developed framework and are tested in computational
experiments.Comment: 49 pages, 3 figure
Loop corrections in spin models through density consistency
Computing marginal distributions of discrete or semidiscrete Markov random
fields (MRFs) is a fundamental, generally intractable problem with a vast
number of applications in virtually all fields of science. We present a new
family of computational schemes to approximately calculate the marginals of
discrete MRFs. This method shares some desirable properties with belief
propagation, in particular, providing exact marginals on acyclic graphs, but it
differs with the latter in that it includes some loop corrections; i.e., it
takes into account correlations coming from all cycles in the factor graph. It
is also similar to the adaptive Thouless-Anderson-Palmer method, but it differs
with the latter in that the consistency is not on the first two moments of the
distribution but rather on the value of its density on a subset of values. The
results on finite-dimensional Isinglike models show a significant improvement
with respect to the Bethe-Peierls (tree) approximation in all cases and with
respect to the plaquette cluster variational method approximation in many
cases. In particular, for the critical inverse temperature of the
homogeneous hypercubic lattice, the expansion of
around of the proposed scheme is exact up to the order,
whereas the two latter are exact only up to the order.Comment: 12 pages, 3 figures, 1 tabl
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