15,891 research outputs found
Numerical algebraic geometry for model selection and its application to the life sciences
Researchers working with mathematical models are often confronted by the
related problems of parameter estimation, model validation, and model
selection. These are all optimization problems, well-known to be challenging
due to non-linearity, non-convexity and multiple local optima. Furthermore, the
challenges are compounded when only partial data is available. Here, we
consider polynomial models (e.g., mass-action chemical reaction networks at
steady state) and describe a framework for their analysis based on optimization
using numerical algebraic geometry. Specifically, we use probability-one
polynomial homotopy continuation methods to compute all critical points of the
objective function, then filter to recover the global optima. Our approach
exploits the geometric structures relating models and data, and we demonstrate
its utility on examples from cell signaling, synthetic biology, and
epidemiology.Comment: References added, additional clarification
A scalable H-matrix approach for the solution of boundary integral equations on multi-GPU clusters
In this work, we consider the solution of boundary integral equations by
means of a scalable hierarchical matrix approach on clusters equipped with
graphics hardware, i.e. graphics processing units (GPUs). To this end, we
extend our existing single-GPU hierarchical matrix library hmglib such that it
is able to scale on many GPUs and such that it can be coupled to arbitrary
application codes. Using a model GPU implementation of a boundary element
method (BEM) solver, we are able to achieve more than 67 percent relative
parallel speed-up going from 128 to 1024 GPUs for a model geometry test case
with 1.5 million unknowns and a real-world geometry test case with almost 1.2
million unknowns. On 1024 GPUs of the cluster Titan, it takes less than 6
minutes to solve the 1.5 million unknowns problem, with 5.7 minutes for the
setup phase and 20 seconds for the iterative solver. To the best of the
authors' knowledge, we here discuss the first fully GPU-based
distributed-memory parallel hierarchical matrix Open Source library using the
traditional H-matrix format and adaptive cross approximation with an
application to BEM problems
Learning Algebraic Varieties from Samples
We seek to determine a real algebraic variety from a fixed finite subset of
points. Existing methods are studied and new methods are developed. Our focus
lies on aspects of topology and algebraic geometry, such as dimension and
defining polynomials. All algorithms are tested on a range of datasets and made
available in a Julia package
Clustering Complex Zeros of Triangular Systems of Polynomials
This paper gives the first algorithm for finding a set of natural
-clusters of complex zeros of a triangular system of polynomials
within a given polybox in , for any given . Our
algorithm is based on a recent near-optimal algorithm of Becker et al (2016)
for clustering the complex roots of a univariate polynomial where the
coefficients are represented by number oracles.
Our algorithm is numeric, certified and based on subdivision. We implemented
it and compared it with two well-known homotopy solvers on various triangular
systems. Our solver always gives correct answers, is often faster than the
homotopy solver that often gives correct answers, and sometimes faster than the
one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update
A robust adaptive algebraic multigrid linear solver for structural mechanics
The numerical simulation of structural mechanics applications via finite
elements usually requires the solution of large-size and ill-conditioned linear
systems, especially when accurate results are sought for derived variables
interpolated with lower order functions, like stress or deformation fields.
Such task represents the most time-consuming kernel in commercial simulators;
thus, it is of significant interest the development of robust and efficient
linear solvers for such applications. In this context, direct solvers, which
are based on LU factorization techniques, are often used due to their
robustness and easy setup; however, they can reach only superlinear complexity,
in the best case, thus, have limited applicability depending on the problem
size. On the other hand, iterative solvers based on algebraic multigrid (AMG)
preconditioners can reach up to linear complexity for sufficiently regular
problems but do not always converge and require more knowledge from the user
for an efficient setup. In this work, we present an adaptive AMG method
specifically designed to improve its usability and efficiency in the solution
of structural problems. We show numerical results for several practical
applications with millions of unknowns and compare our method with two
state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM
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