85,474 research outputs found
A quasi-optimal coarse problem and an augmented Krylov solver for the Variational Theory of Complex Rays
The Variational Theory of Complex Rays (VTCR) is an indirect Trefftz method
designed to study systems governed by Helmholtz-like equations. It uses wave
functions to represent the solution inside elements, which reduces the
dispersion error compared to classical polynomial approaches but the resulting
system is prone to be ill conditioned. This paper gives a simple and original
presentation of the VTCR using the discontinuous Galerkin framework and it
traces back the ill-conditioning to the accumulation of eigenvalues near zero
for the formulation written in terms of wave amplitude. The core of this paper
presents an efficient solving strategy that overcomes this issue. The key
element is the construction of a search subspace where the condition number is
controlled at the cost of a limited decrease of attainable precision. An
augmented LSQR solver is then proposed to solve efficiently and accurately the
complete system. The approach is successfully applied to different examples.Comment: International Journal for Numerical Methods in Engineering, Wiley,
201
Quantum annealing for systems of polynomial equations
Numerous scientific and engineering applications require numerically solving
systems of equations. Classically solving a general set of polynomial equations
requires iterative solvers, while linear equations may be solved either by
direct matrix inversion or iteratively with judicious preconditioning. However,
the convergence of iterative algorithms is highly variable and depends, in
part, on the condition number. We present a direct method for solving general
systems of polynomial equations based on quantum annealing, and we validate
this method using a system of second-order polynomial equations solved on a
commercially available quantum annealer. We then demonstrate applications for
linear regression, and discuss in more detail the scaling behavior for general
systems of linear equations with respect to problem size, condition number, and
search precision. Finally, we define an iterative annealing process and
demonstrate its efficacy in solving a linear system to a tolerance of
.Comment: 11 pages, 4 figures. Added example for a system of quadratic
equations. Supporting code is available at
https://github.com/cchang5/quantum_poly_solver . This is a post-peer-review,
pre-copyedit version of an article published in Scientific Reports. The final
authenticated version is available online at:
https://www.nature.com/articles/s41598-019-46729-
A Collection of Challenging Optimization Problems in Science, Engineering and Economics
Function optimization and finding simultaneous solutions of a system of
nonlinear equations (SNE) are two closely related and important optimization
problems. However, unlike in the case of function optimization in which one is
required to find the global minimum and sometimes local minima, a database of
challenging SNEs where one is required to find stationary points (extrama and
saddle points) is not readily available. In this article, we initiate building
such a database of important SNE (which also includes related function
optimization problems), arising from Science, Engineering and Economics. After
providing a short review of the most commonly used mathematical and
computational approaches to find solutions of such systems, we provide a
preliminary list of challenging problems by writing the Mathematical
formulation down, briefly explaning the origin and importance of the problem
and giving a short account on the currently known results, for each of the
problems. We anticipate that this database will not only help benchmarking
novel numerical methods for solving SNEs and function optimization problems but
also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on
Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE
Congress on Evolutionary Computation (at Sendai International Center, Sendai,
Japan, from 25th to 28th May, 2015.
AI Feynman: a Physics-Inspired Method for Symbolic Regression
A core challenge for both physics and artificial intellicence (AI) is
symbolic regression: finding a symbolic expression that matches data from an
unknown function. Although this problem is likely to be NP-hard in principle,
functions of practical interest often exhibit symmetries, separability,
compositionality and other simplifying properties. In this spirit, we develop a
recursive multidimensional symbolic regression algorithm that combines neural
network fitting with a suite of physics-inspired techniques. We apply it to 100
equations from the Feynman Lectures on Physics, and it discovers all of them,
while previous publicly available software cracks only 71; for a more difficult
test set, we improve the state of the art success rate from 15% to 90%.Comment: 15 pages, 2 figs. Our code is available at
https://github.com/SJ001/AI-Feynman and our Feynman Symbolic Regression
Database for benchmarking can be downloaded at
https://space.mit.edu/home/tegmark/aifeynman.htm
Cryptography from tensor problems
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler
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