7,195 research outputs found
Verification Under Increasing Dimensionality
Verification decisions are often based on second order statistics estimated from a set of samples. Ongoing growth of computational resources allows for considering more and more features, increasing the dimensionality of the samples. If the dimensionality is of the same order as the number of samples used in the estimation or even higher, then the accuracy of the estimate decreases significantly. In particular, the eigenvalues of the covariance matrix are estimated with a bias and the estimate of the eigenvectors differ considerably from the real eigenvectors. We show how a classical approach of verification in high dimensions is severely affected by these problems, and we show how bias correction methods can reduce these problems
Bootstrap Multigrid for the Laplace-Beltrami Eigenvalue Problem
This paper introduces bootstrap two-grid and multigrid finite element
approximations to the Laplace-Beltrami (surface Laplacian) eigen-problem on a
closed surface. The proposed multigrid method is suitable for recovering
eigenvalues having large multiplicity, computing interior eigenvalues, and
approximating the shifted indefinite eigen-problem. Convergence analysis is
carried out for a simplified two-grid algorithm and numerical experiments are
presented to illustrate the basic components and ideas behind the overall
bootstrap multigrid approach
Unbiased estimate of dynamic term structure models
Affine dynamic term structure models (DTSMs) are the standard finance representation of the yield curve. However, the literature on DTSMs has ignored the coefficient bias that plagues estimated autoregressive models of persistent time series. We introduce new simulation-based methods for reducing or even eliminating small-sample bias in empirical affine Gaussian DTSMs. With these methods, we show that conventional estimates of DTSM coefficients are severely biased, which results in misleading estimates of expected future short-term interest rates and long-maturity term premia. Our unbiased DTSM estimates imply risk-neutral rates and term premia that are more plausible from a macro-finance perspective.Interest rates
A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs
A self-learning algebraic multigrid method for dominant and minimal singular
triplets and eigenpairs is described. The method consists of two multilevel
phases. In the first, multiplicative phase (setup phase), tentative singular
triplets are calculated along with a multigrid hierarchy of interpolation
operators that approximately fit the tentative singular vectors in a collective
and self-learning manner, using multiplicative update formulas. In the second,
additive phase (solve phase), the tentative singular triplets are improved up
to the desired accuracy by using an additive correction scheme with fixed
interpolation operators, combined with a Ritz update. A suitable generalization
of the singular value decomposition is formulated that applies to the coarse
levels of the multilevel cycles. The proposed algorithm combines and extends
two existing multigrid approaches for symmetric positive definite eigenvalue
problems to the case of dominant and minimal singular triplets. Numerical tests
on model problems from different areas show that the algorithm converges to
high accuracy in a modest number of iterations, and is flexible enough to deal
with a variety of problems due to its self-learning properties.Comment: 29 page
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