337 research outputs found
Computational complexity and memory usage for multi-frontal direct solvers in structured mesh finite elements
The multi-frontal direct solver is the state-of-the-art algorithm for the
direct solution of sparse linear systems. This paper provides computational
complexity and memory usage estimates for the application of the multi-frontal
direct solver algorithm on linear systems resulting from B-spline-based
isogeometric finite elements, where the mesh is a structured grid. Specifically
we provide the estimates for systems resulting from polynomial
B-spline spaces and compare them to those obtained using spaces.Comment: 8 pages, 2 figure
The cost of continuity: performance of iterative solvers on isogeometric finite elements
In this paper we study how the use of a more continuous set of basis
functions affects the cost of solving systems of linear equations resulting
from a discretized Galerkin weak form. Specifically, we compare performance of
linear solvers when discretizing using B-splines, which span traditional
finite element spaces, and B-splines, which represent maximum
continuity. We provide theoretical estimates for the increase in cost of the
matrix-vector product as well as for the construction and application of
black-box preconditioners. We accompany these estimates with numerical results
and study their sensitivity to various grid parameters such as element size
and polynomial order of approximation . Finally, we present timing results
for a range of preconditioning options for the Laplace problem. We conclude
that the matrix-vector product operation is at most \slfrac{33p^2}{8} times
more expensive for the more continuous space, although for moderately low ,
this number is significantly reduced. Moreover, if static condensation is not
employed, this number further reduces to at most a value of 8, even for high
. Preconditioning options can be up to times more expensive to setup,
although this difference significantly decreases for some popular
preconditioners such as Incomplete LU factorization
A Continuation Multilevel Monte Carlo algorithm
We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for
weak approximation of stochastic models. The CMLMC algorithm solves the given
approximation problem for a sequence of decreasing tolerances, ending when the
required error tolerance is satisfied. CMLMC assumes discretization hierarchies
that are defined a priori for each level and are geometrically refined across
levels. The actual choice of computational work across levels is based on
parametric models for the average cost per sample and the corresponding weak
and strong errors. These parameters are calibrated using Bayesian estimation,
taking particular notice of the deepest levels of the discretization hierarchy,
where only few realizations are available to produce the estimates. The
resulting CMLMC estimator exhibits a non-trivial splitting between bias and
statistical contributions. We also show the asymptotic normality of the
statistical error in the MLMC estimator and justify in this way our error
estimate that allows prescribing both required accuracy and confidence in the
final result. Numerical results substantiate the above results and illustrate
the corresponding computational savings in examples that are described in terms
of differential equations either driven by random measures or with random
coefficients
RETHINKING THE ROLE OF STORMWATER MANAGEMENT ON CAMPUS IN COLLEGE PARK, MARYLAND
As part of an intense effort to clean up the Anacostia River and the Chesapeake Bay region, the Maryland Department of Environment will soon enforce new policies to increase the treatment of impervious areas. The University of Maryland’s College Park campus needs to identify potential projects in order to meet the pending stormwater regulations as part of the new municipal separate storm sewer systems (MS4) permit for UM-CP. This thesis investigates retrofits a poorly maintained stormwater pond that has maintained itself as a wetland. The 4.89-acre site is located in the north part of campus is a part of the Anacostia watershed and includes the pond, two parking lots, and a wet swale. This thesis proposes a stormwater retrofit that includes various state acceptable BMPs including: a constructed wetland, mirco-bioretentions, pervious concrete, and a bio-swale. The BMPs forms a treatment train that reducing runoff by 7%, capturing and treating 113% of a one-year storm of 2.63 inches. This redesign that would provide a range of environmental, recreational, and educational services. While the proposal is site-specific, the model can be adaptable for retrofitting centralized stormwater facilities and by other college campuses within the Chesapeake Bay watershed
Fully inverted single-digit nanometer domains in ferroelectric films
Achieving stable single-digit nanometer inverted domains in ferroelectric thin films is a fundamental issue that has remained a bottleneck for the development of ultrahigh density (>1 Tbit/in.^2) probe-based memory devices using ferroelectric media. Here, we demonstrate that such domains remain stable only if they are fully inverted through the entire ferroelectric film thickness, which is dependent on a critical ratio of electrode size to the film thickness. This understanding enables the formation of stable domains as small as 4 nm in diameter, corresponding to 10 unit cells in size. Such domain size corresponds to 40 Tbit/in.^2 data storage densitie
Resurgence, Conformal Blocks, and the Sum over Geometries in Quantum Gravity
In two dimensional conformal field theories the limit of large central charge
plays the role of a semi-classical limit. Certain universal observables, such
as conformal blocks involving the exchange of the identity operator, can be
expanded around this classical limit in powers of the central charge . This
expansion is an asymptotic series, so - via the same resurgence analysis
familiar from quantum mechanics - necessitates the existence of
non-perturbative effects. In the case of identity conformal blocks, these new
effects have a simple interpretation: the CFT must possess new primary
operators with dimension of order the central charge. This constrains the data
of CFTs with large central charge in a way that is similar to (but distinct
from) the conformal bootstrap. We study this phenomenon in three ways:
numerically, analytically using Zamolodchikov's recursion relations, and by
considering non-unitary minimal models with large (negative) central charge. In
the holographic dual to a CFT, the expansion in powers of is the
perturbative loop expansion in powers of . So our results imply that the
graviton loop expansion is an asymptotic series, whose cure requires the
inclusion of new saddle points in the gravitational path integral. In certain
cases these saddle points have a simple interpretation: they are conical
excesses, particle-like states with negative mass which are not in the physical
spectrum but nevertheless appear as non-manifold saddle points that control the
asymptotic behaviour of the loop expansion. This phenomenon also has an
interpretation in Chern-Simons theory, where the
non-perturbative effects are associated with the non-Teichm\"uller component of
the moduli space of flat connections.Comment: 37 pages, 7 figures; References adde
Gradient-based estimation of Manning's friction coefficient from noisy data
We study the numerical recovery of Manning's roughness coefficient for the
diffusive wave approximation of the shallow water equation. We describe a
conjugate gradient method for the numerical inversion. Numerical results for
one-dimensional model are presented to illustrate the feasibility of the
approach. Also we provide a proof of the differentiability of the weak form
with respect to the coefficient as well as the continuity and boundedness of
the linearized operator under reasonable assumptions using the maximal
parabolic regularity theory.Comment: 19 pages, 3 figure
Harmonic analysis of 2d CFT partition functions
We apply the theory of harmonic analysis on the fundamental domain of
to partition functions of two-dimensional conformal field
theories. We decompose the partition function of free bosons on a Narain
lattice into eigenfunctions of the Laplacian of worldsheet moduli space
, and of target space moduli space . This decomposition manifests
certain properties of Narain theories and ensemble averages thereof. We extend
the application of spectral theory to partition functions of general
two-dimensional conformal field theories, and explore its meaning in connection
to AdS gravity. An implication of harmonic analysis is that the local
operator spectrum is fully determined by a certain subset of degeneracies.Comment: 35+24 pages, v2: corrected a mistake in Sec 3, v3: minor errors fixe
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