2,124 research outputs found
An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations
An a posteriori verification method is proposed for the generalized
real-symmetric eigenvalue problem and is applied to densely clustered
eigenvalue problems in large-scale electronic state calculations. The proposed
method is realized by a two-stage process in which the approximate solution is
computed by existing numerical libraries and is then verified in a moderate
computational time. The procedure returns intervals containing one exact
eigenvalue in each interval. Test calculations were carried out for organic
device materials, and the verification method confirms that all exact
eigenvalues are well separated in the obtained intervals. This verification
method will be integrated into EigenKernel (https://github.com/eigenkernel/),
which is middleware for various parallel solvers for the generalized eigenvalue
problem. Such an a posteriori verification method will be important in future
computational science.Comment: 15 pages, 7 figure
Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method
The reduced basis method is a model reduction technique yielding substantial
savings of computational time when a solution to a parametrized equation has to
be computed for many values of the parameter. Certification of the
approximation is possible by means of an a posteriori error bound. Under
appropriate assumptions, this error bound is computed with an algorithm of
complexity independent of the size of the full problem. In practice, the
evaluation of the error bound can become very sensitive to round-off errors. We
propose herein an explanation of this fact. A first remedy has been proposed in
[F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced
basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012)
539--542.]. Herein, we improve this remedy by proposing a new approximation of
the error bound using the Empirical Interpolation Method (EIM). This method
achieves higher levels of accuracy and requires potentially less
precomputations than the usual formula. A version of the EIM stabilized with
respect to round-off errors is also derived. The method is illustrated on a
simple one-dimensional diffusion problem and a three-dimensional acoustic
scattering problem solved by a boundary element method.Comment: 26 pages, 10 figures. ESAIM: Mathematical Modelling and Numerical
Analysis, 201
Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program
Computer programs may go wrong due to exceptional behaviors, out-of-bound
array accesses, or simply coding errors. Thus, they cannot be blindly trusted.
Scientific computing programs make no exception in that respect, and even bring
specific accuracy issues due to their massive use of floating-point
computations. Yet, it is uncommon to guarantee their correctness. Indeed, we
had to extend existing methods and tools for proving the correct behavior of
programs to verify an existing numerical analysis program. This C program
implements the second-order centered finite difference explicit scheme for
solving the 1D wave equation. In fact, we have gone much further as we have
mechanically verified the convergence of the numerical scheme in order to get a
complete formal proof covering all aspects from partial differential equations
to actual numerical results. To the best of our knowledge, this is the first
time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with
arXiv:1112.179
Computing the Lambert W function in arbitrary-precision complex interval arithmetic
We describe an algorithm to evaluate all the complex branches of the Lambert
W function with rigorous error bounds in interval arithmetic, which has been
implemented in the Arb library. The classic 1996 paper on the Lambert W
function by Corless et al. provides a thorough but partly heuristic numerical
analysis which needs to be complemented with some explicit inequalities and
practical observations about managing precision and branch cuts.Comment: 16 pages, 4 figure
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
This paper introduces tensorial calculus techniques in the framework of
Proper Orthogonal Decomposition (POD) to reduce the computational complexity of
the reduced nonlinear terms. The resulting method, named tensorial POD, can be
applied to polynomial nonlinearities of any degree . Such nonlinear terms
have an on-line complexity of , where is the
dimension of POD basis, and therefore is independent of full space dimension.
However it is efficient only for quadratic nonlinear terms since for higher
nonlinearities standard POD proves to be less time consuming once the POD basis
dimension is increased. Numerical experiments are carried out with a two
dimensional shallow water equation (SWE) test problem to compare the
performance of tensorial POD, standard POD, and POD/Discrete Empirical
Interpolation Method (DEIM). Numerical results show that tensorial POD
decreases by times the computational cost of the on-line stage of
standard POD for configurations using more than model variables. The
tensorial POD SWE model was only slower than the POD/DEIM SWE model
but the implementation effort is considerably increased. Tensorial calculus was
again employed to construct a new algorithm allowing POD/DEIM shallow water
equation model to compute its off-line stage faster than the standard and
tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table
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