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Error Analysis For Standard and GMRES-Based Iterative Refinement in Two and Three-Precisions
Full text linkWe give a concise summary of conditions for the convergence of iterative refinement and GMRES-based iterative refinement in three precisions, as well as the limiting forward errors and backward errors. All combinations of precisions of practical interest are included. As well as known results, we include new results for GMRES-based iterative refinement with the preconditioner applied at the working precision and the residual computed at the working precision
Backward error and condition number of a generalized Sylvester equation, with application to the stochastic Galerkin method
Full text linkThe governing equations of the stochastic Galerkin method can be formulated as a general- ized Sylvester equation. Therefore developing solvers for it is attracting a lot of attention from the uncertainty quantification community. In this regard Krylov subspace based iter- ative solvers, which are used for standard linear systems are being used for the generalized Sylvester equations as well. This is achieved by converting the generalized Sylvester equa- tion to a standard linear system using the Kronecker product. Accordingly the residual is used as a stopping criterion for the iterations, and the condition number of linear systems is used for the generalized Sylvester equations as well. For a linear system a small residual implies a small backward error, and hence using residual as a stopping criterion is justified. In this work we prove that this need not be the case for the generalized Sylvester equation. We introduce two definitions for the backward error, and then derive an upperbound on each of them. We also verify the predictions of the analysis using numerical experiments. For the special case of the stochastic Galerkin method we show that the upper bound on the backward error can be computed with minimal computational overhead, and hence it can be used as a stopping criterion in the iterative solvers. For the matrices from the stochastic Galerkin method we numerically demonstrate that the actual backward error can be upto 2 orders of magnitude higher than the relative residual. Finally by taking into account the structure of the equation we derive an expression for the condition number, and discuss an algorithm for their computation in the special case of the stochastic Galerkin method
Semidirect Products and Applications to Geometric Mechanics
Full text linkIn this thesis we provide an overview of themes in geometric mechanics and apply them to the study of adjoint and coadjoint orbits of a semidirect product, and to the two-body problem on a sphere.
Firstly, we show the existence of a geometrically defined bijection between the sets of adjoint and coadjoint orbits for a particular class of semidirect product. We demonstrate the bijection for the examples of the affine linear group and the Poincaré group. Additionally, we prove that any two orbits paired between this bijection are homotopy equivalent.
Secondly, a correspondence is found between the two-body problem on a three- dimensional sphere and the four-dimensional Lagrange top. This correspondence establishes an equivalence between the two problems after reduction, and allows us to treat both reduced problems simultaneously. We implement a semidirect product reduction by stages to exhibit the reduced spaces as coadjoint orbits of a special Euclidean group, and then reduce by a further symmetry to obtain a full reduced system. This allows us to fully classify the relative equilibria for both problems and describe their stability
Stability analysis of a chain of non-identical vehicles under bilateral cruise control
Full text linkBilateral cruise control (BCC) suppresses traffic
flow instabilities. Previously, for simplicity of analysis, vehicles in BCC traffic flow were assumed to be identical, i.e., using the same gains for control. In this study, we analyze the stability of an inhomogeneous vehicular chain in which the gains used by different vehicles are not the same. Not unexpectedly, mathematical analysis becomes more difficult, and leads to a quadratic eigenvalue problem. We study several different cases, and shows that a chain of vehicles under bilateral cruise control is stable even when the vehicles do not all have the same control system properties. Numerical simulations validate the analysis
Mixed Precision Block Fused Multiply-Add: Error Analysis and Application to GPU Tensor Cores
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Diameter of the Monster Graph
No full textIn this paper the diameter of the commuting involution graph for
the Monster simple group on the 2B involution conjugacy class is
shown to be 3. As a consequence we deduce that the Monster graph
has diameter 5 or 6.
AMS Classi�cation: 20D08, 20B
Exploiting Lower Precision Arithmetic in Solving Symmetric Positive Definite Linear Systems and Least Squares Problems
Full text linkWhat is the fastest way to solve a linear system in arithmetic of a given precision when is symmetric positive definite and otherwise unstructured? The usual answer is by Cholesky factorization, assuming that can be factorized. We develop an algorithm that can be faster, given an arithmetic of precision lower than the working precision as well as (optionally) one of higher precision. The arithmetics might, for example, be of precisions half, single, and double; half and double, possibly with quadruple; or single and double, possibly with quadruple. We compute a Cholesky factorization at the lower precision and use the factors as preconditioners in GMRES-based iterative refinement. To avoid breakdown of the factorization we shift the matrix by a small multiple of its diagonal. We explain why this is preferable to the common approach of shifting by a multiple of the identity matrix, We also incorporate scaling in order to avoid overflow and reduce the chance of underflow when working in IEEE half precision arithmetic. We extend the algorithm to solve a linear least squares problem with a well conditioned coefficient matrix by forming and solving the normal equations. In both algorithms most of the work is done at low precision provided that iterative refinement and the inner iterative solver converge quickly. We explain why replacing GMRES by the conjugate gradient method causes convergence guarantees to be lost, but show that this change has little effect on convergence in practice. Our numerical experiments confirm the potential of the new algorithms to provide faster solutions in environments that support multiple precisions of arithmetic
Attracting and repelling 2-body problems on a family of surfaces of constant curvature
Full text linkWe first provide a classification of the pure rotational motion of 2 particles on a sphere interacting via a repelling potential. This is achieved by providing a simple geometric equivalence between repelling particles and attracting particles, and relying on previous work on the similar classification for attracting particles. The second theme of the paper is to study the 2-body problem on a surface of constant curvature treating the curvature as a parameter, and with particular interest in how families of relative equilibria and their stability behave as the curvature passes through zero and changes sign. We consider two cases: firstly one where the particles are always attracting throughout the family, and secondly where they are attracting for negative curvature and repelling for positive curvature, interpolated by no interaction when the curvature vanishes.
Our analysis clarifies the role of curvature in the existence and stability of relative equilibria