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Correcting for unknown errors in sparse high-dimensional function approximation
Springer Science and Business Media LLC - Tập 142 - Trang 667-711 - 2019
Ben Adcock, Anyi Bao, Simone Brugiapaglia
We consider sparsity-based techniques for the approximation of high-dimensional functions from random pointwise evaluations. To date, almost all the works published in this field contain some a priori assumptions about the error corrupting the samples that are hard to verify in practice. In this paper, we instead focus on the scenario where the error is unknown. We study the performance of four sparsity-promoting optimization problems: weighted quadratically-constrained basis pursuit, weighted LASSO, weighted square-root LASSO, and weighted LAD-LASSO. From the theoretical perspective, we prove uniform recovery guarantees for these decoders, deriving recipes for the optimal choice of the respective tuning parameters. On the numerical side, we compare them in the pure function approximation case and in applications to uncertainty quantification of ODEs and PDEs with random inputs. Our main conclusion is that the lesser-known square-root LASSO is better suited for high-dimensional approximation than the other procedures in the case of bounded noise, since it avoids (both theoretically and numerically) the need for parameter tuning.
Perturbation analysis of generalized singular subspaces
Springer Science and Business Media LLC - Tập 79 Số 4 - Trang 615-641 - 1998
Jiguang Sun
Sur les bornes d'erreur a posteriori pour les éléments propres d'opérateurs linéaires
Springer Science and Business Media LLC - Tập 32 - Trang 233-246 - 1979
Françoise Chatelin
The theoretical framework of this study is presented in Sect. 1, with a review of practical numerical methods. The linear operatorT and its approximationT n are defined in the same Banach space, which is a very common situation. The notion of strong stability forT n is essential and cannot be weakened without introducing a numerical instability [2]. IfT (or its inverse) is compact, most numerical methods are strongly stable. Without compactness forT(T −1) they may not be strongly stable [20]. In Sect. 2 we establish error bounds valid in the general setting of a strongly stable approximation of a closedT. This is a generalization of Vainikko [24, 25] (compact approximation). Osborn [19] (uniform and collectivity compact approximation) and Chatelin and Lemordant [6] (strong approximation), based on the equivalence between the eigenvalues convergence with preservation of multiplicities and the collectively compact convergence of spectral projections. It can be summarized in the following way: λ, eigenvalue ofT of multiplicitym is approximated bym numbers,λ n is their arithmetic mean.λ-λ n and the gap between invariant subspaces are of orderε n =‖(T-T n)‖P. IfT n * converges toT *, pointwise inX *, the principal term in the error on ∣λ-λ n ∣ is $$\frac{1}{m}|tr (T - T_n )P|$$ . And for projection methods, withT n=π n T, we get the bound $$|tr (T - T_n )P| \leqq C ||(1 - \pi _n )P|| ||(1 - \pi _n^* )P*||$$ . It applies to the finite element method for a differential operator with a noncompact resolvent. Aposteriori error bounds are given, and thegeneralized Rayleigh quotient $$\frac{1}{m}tr TP_n $$ TP n appears to be an approximation of λ of the second order, as in the selfadjoint case [12]. In Sect. 3, these results are applied to the Galerkin method and its Sloan variant [22], and to approximate quadrature methods. The error bounds and the generalized Rayleigh quotient are numerically tested in Sect. 4.
A constructive method for starlike harmonic mappings
Springer Science and Business Media LLC - Tập 54 Số 2 - Trang 167-178 - 1989
Nickolas S. Sapidis, Panagiotis Kaklis, T Loukakis
A high order direct method for solving Poisson's equation in a disc
Springer Science and Business Media LLC - Tập 70 - Trang 501-506 - 1995
W. Sun
A direct method is developed for solving Poisson's equation on an annulus region with Hermite bicubic collocation approximation. In terms of FFT, the operation count is $O(N^2\log_2 N)$ on an $N \times N$ mesh.
Transformed rational Chebyshev approximation. II
Springer Science and Business Media LLC - Tập 12 - Trang 8-10 - 1968
Charles B. Dunham
DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods
Springer Science and Business Media LLC - Tập 125 - Trang 511-543 - 2013
Tomoaki Okayama, Ken’ichiro Tanaka, Takayasu Matsuo, Masaaki Sugihara
In this paper, the theoretical convergence rate of the trapezoidal rule combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. It is well known that the DE transformation enables the rule to achieve a much higher rate of convergence than the SE transformation, and the convergence rate has been analyzed and justified theoretically under a proper assumption. Here, it should be emphasized that the assumption is more severe than the one for the SE transformation, and there actually exist some examples such that the trapezoidal rule with the SE transformation achieves its usual rate, whereas the rule with DE does not. Such cases have been observed numerically, but no theoretical analysis has been given thus far. This paper reveals the theoretical rate of convergence in such cases, and it turns out that the DE’s rate is almost the same as, but slightly lower than that of the SE. By using the analysis technique developed here, the theoretical convergence rate of the Sinc approximation with the DE transformation is also given for a class of functions for which the SE transformation is suitable. The result is quite similar to above; the convergence rate in the DE case is slightly lower than in the SE case. Numerical examples which support those two theoretical results are also given.
Tschebyscheff-Approximationen in kleinen Intervallen II
Springer Science and Business Media LLC - Tập 2 - Trang 293-307 - 1960
H. Maehly, Ch. Witzgall
Convergence of theQR algorithm
Springer Science and Business Media LLC - - 1965
Beresford Ν. Parlett
A breakdown-free Lanczos type algorithm for solving linear systems
Springer Science and Business Media LLC - Tập 63 - Trang 29-38 - 1992
C. Brezinski, M. Redivo Zaglia, H. Sadok
Lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates. The various Lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae. If the denominator in the formula for one of these coefficients is zero, then a breakdown occurs in the algorithm, and it must be stopped. Such a breakdown is in fact due to the non-existence of some orthogonal polynomial. In this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method. The resulting algorithm, called MRZ, is equivalent to the nongeneric BIODIR algorithm (which is a look-ahead Lanczos type algorithm), but our derivation is much simpler.
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