13,830 research outputs found
Efficient Approaches for Enclosing the United Solution Set of the Interval Generalized Sylvester Matrix Equation
In this work, we investigate the interval generalized Sylvester matrix
equation and develop some
techniques for obtaining outer estimations for the so-called united solution
set of this interval system. First, we propose a modified variant of the
Krawczyk operator which causes reducing computational complexity to cubic,
compared to Kronecker product form. We then propose an iterative technique for
enclosing the solution set. These approaches are based on spectral
decompositions of the midpoints of , , and
and in both of them we suppose that the midpoints of and
are simultaneously diagonalizable as well as for the midpoints of
the matrices and . Some numerical experiments are given to
illustrate the performance of the proposed methods
Self-similar Singularity of a 1D Model for the 3D Axisymmetric Euler Equations
We investigate the self-similar singularity of a 1D model for the 3D
axisymmetric Euler equations, which is motivated by a particular singularity
formation scenario observed in numerical computation. We prove the existence of
a discrete family of self-similar profiles for this model and analyze their
far-field properties. The self-similar profiles we find agree with direct
simulation of the model and seem to have some stability
Subsquares Approach - Simple Scheme for Solving Overdetermined Interval Linear Systems
In this work we present a new simple but efficient scheme - Subsquares
approach - for development of algorithms for enclosing the solution set of
overdetermined interval linear systems. We are going to show two algorithms
based on this scheme and discuss their features. We start with a simple
algorithm as a motivation, then we continue with a sequential algorithm. Both
algorithms can be easily parallelized. The features of both algorithms will be
discussed and numerically tested.Comment: submitted to PPAM 201
Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities
Stochastic variational inequalities (SVI) model a large class of equilibrium
problems subject to data uncertainty, and are closely related to stochastic
optimization problems. The SVI solution is usually estimated by a solution to a
sample average approximation (SAA) problem. This paper considers the normal map
formulation of an SVI, and proposes a method to build asymptotically exact
confidence regions and confidence intervals for the solution of the normal map
formulation, based on the asymptotic distribution of SAA solutions. The
confidence regions are single ellipsoids with high probability. We also discuss
the computation of simultaneous and individual confidence intervals
Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof
We prove the existence of globally attracting solutions of the viscous
Burgers equation with periodic boundary conditions on the line for some
particular choices of viscosity and non-autonomous forcing. The attract- ing
solution is periodic if the forcing is periodic. The method is general and can
be applied to other similar partial differential equations. The proof is
computer assisted.Comment: 38 pages, 1 figur
Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof
We present a computer assisted method for proving the existence of globally
attracting fixed points of dissipative PDEs. An application to the viscous
Burgers equation with periodic boundary conditions and a forcing function
constant in time is presented as a case study. We establish the existence of a
locally attracting fixed point by using rigorous numerics techniques. To prove
that the fixed point is, in fact, globally attracting we introduce a technique
relying on a construction of an absorbing set, capturing any sufficiently
regular initial condition after a finite time. Then the absorbing set is
rigorously integrated forward in time to verify that any sufficiently regular
initial condition is in the basin of attraction of the fixed point.Comment: To appear in Topological Methods in Nonlinear Analysis, 201
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