94,289 research outputs found
Challenges in computational lower bounds
We draw two incomplete, biased maps of challenges in computational complexity
lower bounds
Probabilistic embeddings of the Fr\'echet distance
The Fr\'echet distance is a popular distance measure for curves which
naturally lends itself to fundamental computational tasks, such as clustering,
nearest-neighbor searching, and spherical range searching in the corresponding
metric space. However, its inherent complexity poses considerable computational
challenges in practice. To address this problem we study distortion of the
probabilistic embedding that results from projecting the curves to a randomly
chosen line. Such an embedding could be used in combination with, e.g.
locality-sensitive hashing. We show that in the worst case and under reasonable
assumptions, the discrete Fr\'echet distance between two polygonal curves of
complexity in , where , degrades
by a factor linear in with constant probability. We show upper and lower
bounds on the distortion. We also evaluate our findings empirically on a
benchmark data set. The preliminary experimental results stand in stark
contrast with our lower bounds. They indicate that highly distorted projections
happen very rarely in practice, and only for strongly conditioned input curves.
Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure
New numerical approaches for modeling thermochemical convection in a compositionally stratified fluid
Seismic imaging of the mantle has revealed large and small scale
heterogeneities in the lower mantle; specifically structures known as large low
shear velocity provinces (LLSVP) below Africa and the South Pacific. Most
interpretations propose that the heterogeneities are compositional in nature,
differing in composition from the overlying mantle, an interpretation that
would be consistent with chemical geodynamic models. Numerical modeling of
persistent compositional interfaces presents challenges, even to
state-of-the-art numerical methodology. For example, some numerical algorithms
for advecting the compositional interface cannot maintain a sharp compositional
boundary as the fluid migrates and distorts with time dependent fingering due
to the numerical diffusion that has been added in order to maintain the upper
and lower bounds on the composition variable and the stability of the advection
method. In this work we present two new algorithms for maintaining a sharper
computational boundary than the advection methods that are currently openly
available to the computational mantle convection community; namely, a
Discontinuous Galerkin method with a Bound Preserving limiter and a
Volume-of-Fluid interface tracking algorithm. We compare these two new methods
with two approaches commonly used for modeling the advection of two distinct,
thermally driven, compositional fields in mantle convection problems; namely,
an approach based on a high-order accurate finite element method advection
algorithm that employs an artificial viscosity technique to maintain the upper
and lower bounds on the composition variable as well as the stability of the
advection algorithm and the advection of particles that carry a scalar quantity
representing the location of each compositional field. All four of these
algorithms are implemented in the open source FEM code ASPECT
Challenges of continuous global optimization in molecular structure prediction
The molecular geometry, the three dimensional arrangement of atoms in space, is a major factor determining the properties and reactivity of molecules, biomolecules and macromolecules. Computation of stable molecular conformations can be done by locating minima on the potential energy surface (PES). This is a very challenging global optimization problem because of extremely large numbers of shallow local minima and complicated landscape of PES. This paper illustrates the mathematical and computational challenges on one important instance of the problem, computation of molecular geometry of oligopeptides, and proposes the use of the Extended Cutting Angle Method (ECAM) to solve this problem.ECAM is a deterministic global optimization technique, which computes tight lower bounds on the values of the objective function and fathoms those part of the domain where the global minimum cannot reside. As with any domain partitioning scheme, its challenge is an extremely large partition of the domain required for accurate lower bounds. We address this challenge by providing an efficient combinatorial algorithm for calculating the lower bounds, and by combining ECAM with a local optimization method, while preserving the deterministic character of ECAM.<br /
On Characterizing the Data Access Complexity of Programs
Technology trends will cause data movement to account for the majority of
energy expenditure and execution time on emerging computers. Therefore,
computational complexity will no longer be a sufficient metric for comparing
algorithms, and a fundamental characterization of data access complexity will
be increasingly important. The problem of developing lower bounds for data
access complexity has been modeled using the formalism of Hong & Kung's
red/blue pebble game for computational directed acyclic graphs (CDAGs).
However, previously developed approaches to lower bounds analysis for the
red/blue pebble game are very limited in effectiveness when applied to CDAGs of
real programs, with computations comprised of multiple sub-computations with
differing DAG structure. We address this problem by developing an approach for
effectively composing lower bounds based on graph decomposition. We also
develop a static analysis algorithm to derive the asymptotic data-access lower
bounds of programs, as a function of the problem size and cache size
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