19,700 research outputs found
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
String equations in Whitham hierarchies: tau-functions and Virasoro constraints
A scheme for solving Whitham hierarchies satisfying a special class of string
equations is presented. The tau-function of the corresponding solutions is
obtained and the differential expressions of the underlying Virasoro
constraints are characterized. Illustrative examples of exact solutions of
Whitham hierarchies are derived and applications to conformal maps dynamics are
indicated.Comment: 26 pages, 2 figure
Solving the Einstein constraints in periodic spaces with a multigrid approach
Novel applications of Numerical Relativity demand for more flexible
algorithms and tools. In this paper, I develop and test a multigrid solver,
based on the infrastructure provided by the Einstein Toolkit, for elliptic
partial differential equations on spaces with periodic boundary conditions.
This type of boundary often characterizes the numerical representation of
cosmological models, where space is assumed to be made up of identical copies
of a single fiducial domain, so that only a finite volume (with periodic
boundary conditions at its edges) needs to be simulated. After a few tests and
comparisons with existing codes, I use the solver to generate initial data for
an infinite, periodic, cubic black-hole lattice.Comment: 25 pages, 15 figures. Fixed typos, added references and software
release informatio
Kernel Formula Approach to the Universal Whitham Hierarchy
We derive the dispersionless Hirota equations of the universal Whitham
hierarchy from the kernel formula approach proposed by Carroll and Kodama.
Besides, we also verify the associativity equations in this hierarchy from the
dispersionless Hirota equations and give a realization of the associative
algebra with structure constants expressed in terms of the residue formulas.Comment: 18 page
Genus-zero Whitham hierarchies in conformal-map dynamics
A scheme for solving quasiclassical string equations is developped to prove
that genus-zero Whitham hierarchies describe the deformations of planar domains
determined by rational conformal maps. This property is applied in normal
matrix models to show that deformations of simply-connected supports of
eigenvalues under changes of coupling constants are governed by genus-zero
Whitham hierarchies.Comment: 12 pages, 3 figure
Nonminimal supersymmetric standard model with lepton number violation
We carry out a detailed analysis of the nonminimal supersymmetric standard
model with lepton number violation. The model contains a unique trilinear
lepton number violating term in the superpotential which can give rise to
neutrino masses at the tree level. We search for the gauged discrete symmetries
realized by cyclic groups which preserve the structure of the associated
trilinear superpotential of this model, and which satisfy the constraints of
the anomaly cancellation. The implications of this trilinear lepton number
violating term in the superpotential and the associated soft supersymmetry
breaking term on the phenomenology of the light neutrino masses and mixing is
studied in detail. We evaluate the tree and loop level contributions to the
neutrino mass matrix in this model. We search for possible suppression
mechanism which could explain large hierarchies and maximal mixing angles.Comment: Latex file, 43 pages, 2 figure
Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems
We present a modification to the Berger and Oliger adaptive mesh refinement
algorithm designed to solve systems of coupled, non-linear, hyperbolic and
elliptic partial differential equations. Such systems typically arise during
constrained evolution of the field equations of general relativity. The novel
aspect of this algorithm is a technique of "extrapolation and delayed solution"
used to deal with the non-local nature of the solution of the elliptic
equations, driven by dynamical sources, within the usual Berger and Oliger
time-stepping framework. We show empirical results demonstrating the
effectiveness of this technique in axisymmetric gravitational collapse
simulations. We also describe several other details of the code, including
truncation error estimation using a self-shadow hierarchy, and the
refinement-boundary interpolation operators that are used to help suppress
spurious high-frequency solution components ("noise").Comment: 31 pages, 15 figures; replaced with published versio
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