10,379 research outputs found
A Gauss--Newton iteration for Total Least Squares problems
The Total Least Squares solution of an overdetermined, approximate linear
equation minimizes a nonlinear function which characterizes the
backward error. We show that a globally convergent variant of the Gauss--Newton
iteration can be tailored to compute that solution. At each iteration, the
proposed method requires the solution of an ordinary least squares problem
where the matrix is perturbed by a rank-one term.Comment: 14 pages, no figure
The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows
The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear
model reduction method that operates on fully discretized computational models.
It achieves dimension reduction by a Petrov--Galerkin projection associated
with residual minimization; it delivers computational efficency by a
hyper-reduction procedure based on the `gappy POD' technique. Originally
presented in Ref. [1], where it was applied to implicit nonlinear
structural-dynamics models, this method is further developed here and applied
to the solution of a benchmark turbulent viscous flow problem. To begin, this
paper develops global state-space error bounds that justify the method's design
and highlight its advantages in terms of minimizing components of these error
bounds. Next, the paper introduces a `sample mesh' concept that enables a
distributed, computationally efficient implementation of the GNAT method in
finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability
of GNAT for parameterized problems is highlighted with the solution of an
academic problem featuring moving discontinuities. Finally, the capability of
this method to reduce by orders of magnitude the core-hours required for
large-scale CFD computations, while preserving accuracy, is demonstrated with
the simulation of turbulent flow over the Ahmed body. For an instance of this
benchmark problem with over 17 million degrees of freedom, GNAT outperforms
several other nonlinear model-reduction methods, reduces the required
computational resources by more than two orders of magnitude, and delivers a
solution that differs by less than 1% from its high-dimensional counterpart
A Note on Separable Nonlinear Least Squares Problem
Separable nonlinear least squares (SNLS)problem is a special class of
nonlinear least squares (NLS)problems, whose objective function is a mixture of
linear and nonlinear functions. It has many applications in many different
areas, especially in Operations Research and Computer Sciences. They are
difficult to solve with the infinite-norm metric. In this paper, we give a
short note on the separable nonlinear least squares problem, unseparated scheme
for NLS, and propose an algorithm for solving mixed linear-nonlinear
minimization problem, method of which results in solving a series of least
squares separable problems.Comment: 3 pages; IEEE, 2011 International Conference on Future Computer
Sciences and Application (ICFCSA 2011), Jun. 18- 19, 2011, Hong Kon
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