2,074 research outputs found
Revisiting Matrix Product on Master-Worker Platforms
This paper is aimed at designing efficient parallel matrix-product algorithms
for heterogeneous master-worker platforms. While matrix-product is
well-understood for homogeneous 2D-arrays of processors (e.g., Cannon algorithm
and ScaLAPACK outer product algorithm), there are three key hypotheses that
render our work original and innovative:
- Centralized data. We assume that all matrix files originate from, and must
be returned to, the master.
- Heterogeneous star-shaped platforms. We target fully heterogeneous
platforms, where computational resources have different computing powers.
- Limited memory. Because we investigate the parallelization of large
problems, we cannot assume that full matrix panels can be stored in the worker
memories and re-used for subsequent updates (as in ScaLAPACK).
We have devised efficient algorithms for resource selection (deciding which
workers to enroll) and communication ordering (both for input and result
messages), and we report a set of numerical experiments on various platforms at
Ecole Normale Superieure de Lyon and the University of Tennessee. However, we
point out that in this first version of the report, experiments are limited to
homogeneous platforms
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
Towards Interpretable Deep Learning Models for Knowledge Tracing
As an important technique for modeling the knowledge states of learners, the
traditional knowledge tracing (KT) models have been widely used to support
intelligent tutoring systems and MOOC platforms. Driven by the fast
advancements of deep learning techniques, deep neural network has been recently
adopted to design new KT models for achieving better prediction performance.
However, the lack of interpretability of these models has painfully impeded
their practical applications, as their outputs and working mechanisms suffer
from the intransparent decision process and complex inner structures. We thus
propose to adopt the post-hoc method to tackle the interpretability issue for
deep learning based knowledge tracing (DLKT) models. Specifically, we focus on
applying the layer-wise relevance propagation (LRP) method to interpret
RNN-based DLKT model by backpropagating the relevance from the model's output
layer to its input layer. The experiment results show the feasibility using the
LRP method for interpreting the DLKT model's predictions, and partially
validate the computed relevance scores from both question level and concept
level. We believe it can be a solid step towards fully interpreting the DLKT
models and promote their practical applications in the education domain
Curriculum Guidelines for Undergraduate Programs in Data Science
The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program
met for the purpose of composing guidelines for undergraduate programs in Data
Science. The group consisted of 25 undergraduate faculty from a variety of
institutions in the U.S., primarily from the disciplines of mathematics,
statistics and computer science. These guidelines are meant to provide some
structure for institutions planning for or revising a major in Data Science
A constrained pressure-temperature residual (CPTR) method for non-isothermal multiphase flow in porous media
For both isothermal and thermal petroleum reservoir simulation, the
Constrained Pressure Residual (CPR) method is the industry-standard
preconditioner. This method is a two-stage process involving the solution of a
restricted pressure system. While initially designed for the isothermal case,
CPR is also the standard for thermal cases. However, its treatment of the
energy conservation equation does not incorporate heat diffusion, which is
often dominant in thermal cases. In this paper, we present an extension of CPR:
the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted
pressure-temperature system is solved in the first stage. In previous work, we
introduced a block preconditioner with an efficient Schur complement
approximation for a pressure-temperature system. Here, we extend this method
for multiphase flow as the first stage of CPTR. The algorithmic performance of
different two-stage preconditioners is evaluated for reservoir simulation test
cases.Comment: 28 pages, 2 figures. Sources/sinks description in arXiv:1902.0009
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
Estimating the outcome of spreading processes on networks with incomplete information: a mesoscale approach
Recent advances in data collection have facilitated the access to
time-resolved human proximity data that can conveniently be represented as
temporal networks of contacts between individuals. While this type of data is
fundamental to investigate how information or diseases propagate in a
population, it often suffers from incompleteness, which possibly leads to
biased conclusions. A major challenge is thus to estimate the outcome of
spreading processes occurring on temporal networks built from partial
information. To cope with this problem, we devise an approach based on
Non-negative Tensor Factorization (NTF) -- a dimensionality reduction technique
from multi-linear algebra. The key idea is to learn a low-dimensional
representation of the temporal network built from partial information, to adapt
it to take into account temporal and structural heterogeneity properties known
to be crucial for spreading processes occurring on networks, and to construct
in this way a surrogate network similar to the complete original network. To
test our method, we consider several human-proximity networks, on which we
simulate a loss of data. Using our approach on the resulting partial networks,
we build a surrogate version of the complete network for each. We then compare
the outcome of a spreading process on the complete networks (non altered by a
loss of data) and on the surrogate networks. We observe that the epidemic sizes
obtained using the surrogate networks are in good agreement with those measured
on the complete networks. Finally, we propose an extension of our framework
when additional data sources are available to cope with the missing data
problem
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