2,067 research outputs found
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Communication-optimal Parallel and Sequential Cholesky Decomposition
Numerical algorithms have two kinds of costs: arithmetic and communication,
by which we mean either moving data between levels of a memory hierarchy (in
the sequential case) or over a network connecting processors (in the parallel
case). Communication costs often dominate arithmetic costs, so it is of
interest to design algorithms minimizing communication. In this paper we first
extend known lower bounds on the communication cost (both for bandwidth and for
latency) of conventional (O(n^3)) matrix multiplication to Cholesky
factorization, which is used for solving dense symmetric positive definite
linear systems. Second, we compare the costs of various Cholesky decomposition
implementations to these lower bounds and identify the algorithms and data
structures that attain them. In the sequential case, we consider both the
two-level and hierarchical memory models. Combined with prior results in [13,
14, 15], this gives a set of communication-optimal algorithms for O(n^3)
implementations of the three basic factorizations of dense linear algebra: LU
with pivoting, QR and Cholesky. But it goes beyond this prior work on
sequential LU by optimizing communication for any number of levels of memory
hierarchy.Comment: 29 pages, 2 tables, 6 figure
The Double Sphere Camera Model
Vision-based motion estimation and 3D reconstruction, which have numerous
applications (e.g., autonomous driving, navigation systems for airborne devices
and augmented reality) are receiving significant research attention. To
increase the accuracy and robustness, several researchers have recently
demonstrated the benefit of using large field-of-view cameras for such
applications. In this paper, we provide an extensive review of existing models
for large field-of-view cameras. For each model we provide projection and
unprojection functions and the subspace of points that result in valid
projection. Then, we propose the Double Sphere camera model that well fits with
large field-of-view lenses, is computationally inexpensive and has a
closed-form inverse. We evaluate the model using a calibration dataset with
several different lenses and compare the models using the metrics that are
relevant for Visual Odometry, i.e., reprojection error, as well as computation
time for projection and unprojection functions and their Jacobians. We also
provide qualitative results and discuss the performance of all models
Fixed-Functionals of three-dimensional Quantum Einstein Gravity
We study the non-perturbative renormalization group flow of f(R)-gravity in
three-dimensional Asymptotically Safe Quantum Einstein Gravity. Within the
conformally reduced approximation, we derive an exact partial differential
equation governing the RG-scale dependence of the function f(R). This equation
is shown to possess two isolated and one continuous one-parameter family of
scale-independent, regular solutions which constitute the natural
generalization of RG fixed points to the realm of infinite-dimensional theory
spaces. All solutions are bounded from below and give rise to positive definite
kinetic terms. Moreover, they admit either one or two UV-relevant deformations,
indicating that the corresponding UV-critical hypersurfaces remain finite
dimensional despite the inclusion of an infinite number of coupling constants.
The impact of our findings on the gravitational Asymptotic Safety program and
its connection to new massive gravity is briefly discussed.Comment: 34 pages, 14 figure
- …