780 research outputs found
Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method
This paper addresses optimization problems constrained by partial
differential equations with uncertain coefficients. In particular, the robust
control problem and the average control problem are considered for a tracking
type cost functional with an additional penalty on the variance of the state.
The expressions for the gradient and Hessian corresponding to either problem
contain expected value operators. Due to the large number of uncertainties
considered in our model, we suggest to evaluate these expectations using a
multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that
this results in the gradient and Hessian corresponding to the MLMC estimator of
the original cost functional. Furthermore, we show that the use of certain
correlated samples yields a reduction in the total number of samples required.
Two optimization methods are investigated: the nonlinear conjugate gradient
method and the Newton method. For both, a specific algorithm is provided that
dynamically decides which and how many samples should be taken in each
iteration. The cost of the optimization up to some specified tolerance
is shown to be proportional to the cost of a gradient evaluation with requested
root mean square error . The algorithms are tested on a model elliptic
diffusion problem with lognormal diffusion coefficient. An additional nonlinear
term is also considered.Comment: This work was presented at the IMG 2016 conference (Dec 5 - Dec 9,
2016), at the Copper Mountain conference (Mar 26 - Mar 30, 2017), and at the
FrontUQ conference (Sept 5 - Sept 8, 2017
Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras
We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
Polynomial eigenvalue solver based on tropically scaled Lagrange linearization
We propose an algorithm to solve polynomial eigenvalue problems via linearization combining several ingredients:
a specific choice of linearization, which is constructed using input from tropical algebra and the notion of
well-separated tropical roots, an appropriate scaling applied to the linearization and a modified stopping criterion for the iterations that takes advantage of the properties of our scaled linearization.
Numerical experiments suggest that our polynomial eigensolver computes all the finite and well-conditioned eigenvalues to high relative accuracy even when they are very different in magnitude.status: publishe
Inversion of a block Löwner matrix
AbstractIn this paper, we give a fast algorithm to compute the parameters of an inversion formula for any nonsingular block Löwner matrix. The connection with matrix rational interpolation is given
Rank-deficient submatrices of Fourier matrices
AbstractWe consider the maximal rank-deficient submatrices of Fourier matrices with order a power of a prime number. We do this by considering a hierarchical subdivision of these matrices into low rank blocks. We also explore some connections with the fast Fourier transform (FFT), and with an uncertainty principle for Fourier transforms over finite Abelian groups
A generalization of Floater--Hormann interpolants
In this paper the interpolating rational functions introduced by Floater and
Hormann are generalized leading to a whole new family of rational functions
depending on , an additional positive integer parameter. For , the original Floater--Hormann interpolants are obtained. When we
prove that the new rational functions share a lot of the nice properties of the
original Floater--Hormann functions. Indeed, for any configuration of nodes,
they have no real poles, interpolate the given data, preserve the polynomials
up to a certain fixed degree, and have a barycentric-type representation.
Moreover, we estimate the associated Lebesgue constants in terms of the minimum
() and maximum () distance between two consecutive nodes. It turns out
that, in contrast to the original Floater-Hormann interpolants, for all we get uniformly bounded Lebesgue constants in the case of equidistant and
quasi-equidistant nodes configurations (i.e., when ). In such cases,
we also estimate the uniform and the pointwise approximation errors for
functions having different degree of smoothness.
Numerical experiments illustrate the theoretical results and show a better
error profile for less smooth functions compared to the original
Floater-Hormann interpolants.Comment: 29 page
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