1,137 research outputs found
Numerical Verification of Affine Systems with up to a Billion Dimensions
Affine systems reachability is the basis of many verification methods. With
further computation, methods exist to reason about richer models with inputs,
nonlinear differential equations, and hybrid dynamics. As such, the scalability
of affine systems verification is a prerequisite to scalable analysis for more
complex systems. In this paper, we improve the scalability of affine systems
verification, in terms of the number of dimensions (variables) in the system.
The reachable states of affine systems can be written in terms of the matrix
exponential, and safety checking can be performed at specific time steps with
linear programming. Unfortunately, for large systems with many state variables,
this direct approach requires an intractable amount of memory while using an
intractable amount of computation time. We overcome these challenges by
combining several methods that leverage common problem structure. Memory is
reduced by exploiting initial states that are not full-dimensional and safety
properties (outputs) over a few linear projections of the state variables.
Computation time is saved by using numerical simulations to compute only
projections of the matrix exponential relevant for the verification problem.
Since large systems often have sparse dynamics, we use Krylov-subspace
simulation approaches based on the Arnoldi or Lanczos iterations. Our method
produces accurate counter-examples when properties are violated and, in the
extreme case with sufficient problem structure, can analyze a system with one
billion real-valued state variables
Bayesian Inference of Log Determinants
The log-determinant of a kernel matrix appears in a variety of machine
learning problems, ranging from determinantal point processes and generalized
Markov random fields, through to the training of Gaussian processes. Exact
calculation of this term is often intractable when the size of the kernel
matrix exceeds a few thousand. In the spirit of probabilistic numerics, we
reinterpret the problem of computing the log-determinant as a Bayesian
inference problem. In particular, we combine prior knowledge in the form of
bounds from matrix theory and evidence derived from stochastic trace estimation
to obtain probabilistic estimates for the log-determinant and its associated
uncertainty within a given computational budget. Beyond its novelty and
theoretic appeal, the performance of our proposal is competitive with
state-of-the-art approaches to approximating the log-determinant, while also
quantifying the uncertainty due to budget-constrained evidence.Comment: 12 pages, 3 figure
Fast matrix computations for pair-wise and column-wise commute times and Katz scores
We first explore methods for approximating the commute time and Katz score
between a pair of nodes. These methods are based on the approach of matrices,
moments, and quadrature developed in the numerical linear algebra community.
They rely on the Lanczos process and provide upper and lower bounds on an
estimate of the pair-wise scores. We also explore methods to approximate the
commute times and Katz scores from a node to all other nodes in the graph.
Here, our approach for the commute times is based on a variation of the
conjugate gradient algorithm, and it provides an estimate of all the diagonals
of the inverse of a matrix. Our technique for the Katz scores is based on
exploiting an empirical localization property of the Katz matrix. We adopt
algorithms used for personalized PageRank computing to these Katz scores and
theoretically show that this approach is convergent. We evaluate these methods
on 17 real world graphs ranging in size from 1000 to 1,000,000 nodes. Our
results show that our pair-wise commute time method and column-wise Katz
algorithm both have attractive theoretical properties and empirical
performance.Comment: 35 pages, journal version of
http://dx.doi.org/10.1007/978-3-642-18009-5_13 which has been submitted for
publication. Please see
http://www.cs.purdue.edu/homes/dgleich/publications/2011/codes/fast-katz/ for
supplemental code
Approximating spectral densities of large matrices
In physics, it is sometimes desirable to compute the so-called \emph{Density
Of States} (DOS), also known as the \emph{spectral density}, of a real
symmetric matrix . The spectral density can be viewed as a probability
density distribution that measures the likelihood of finding eigenvalues near
some point on the real line. The most straightforward way to obtain this
density is to compute all eigenvalues of . But this approach is generally
costly and wasteful, especially for matrices of large dimension. There exists
alternative methods that allow us to estimate the spectral density function at
much lower cost. The major computational cost of these methods is in
multiplying with a number of vectors, which makes them appealing for
large-scale problems where products of the matrix with arbitrary vectors
are relatively inexpensive. This paper defines the problem of estimating the
spectral density carefully, and discusses how to measure the accuracy of an
approximate spectral density. It then surveys a few known methods for
estimating the spectral density, and proposes some new variations of existing
methods. All methods are discussed from a numerical linear algebra point of
view
Efficient solution of parabolic equations by Krylov approximation methods
Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms
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