Journal of Complexity最新文献

A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1,2,\dots$ grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus on the situation where the space of available information consists of all linear functionals, and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.

Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results.

We improve the best known upper bound for the bracketing number of d-dimensional axis-parallel boxes anchored in 0 (or, put differently, of lower left orthants intersected with the d-dimensional unit cube ${[0,1]}^d$). More precisely, we provide a better estimate for the cardinality of an algorithmic bracketing cover construction due to Eric Thiémard, which forms the core of his algorithm to approximate the star discrepancy of arbitrary point sets from Thiémard (2001) [22]. Moreover, the new upper bound for the bracketing number of anchored axis-parallel boxes yields an improved upper estimate for the bracketing number of arbitrary axis-parallel boxes in ${[0,1]}^d$. In our upper bounds all constants are fully explicit.

This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.

We study the numerical integration of functions from isotropic Sobolev spaces $W_p^s\left({[0,1]}^d\right)$ using finitely many function evaluations within randomized algorithms, aiming for the smallest possible probabilistic error guarantee $\epsilon >0$ at confidence level $1-\delta \in (0,1)$. For spaces consisting of continuous functions, non-linear Monte Carlo methods with optimal confidence properties have already been known, in few cases even linear methods that succeed in that respect. In this paper we promote a method called stratified control variates (SCV) and by it show that already linear methods achieve optimal probabilistic error rates in the high smoothness regime without the need to adjust algorithmic parameters to the uncertainty δ. We also analyse a version of SCV in the low smoothness regime where $W_p^s\left({[0,1]}^d\right)$ may contain functions with singularities. Here, we observe a polynomial dependence of the error on ${\delta }^{-1}$ in contrast to the logarithmic dependence in the high smoothness regime.

Building upon the exact methods presented in our earlier work (2022) [5], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While it does not necessarily return an optimal solution, we obtain promising results for all tested dimensions. For example, for moderate sizes $30\le n\le 240$, we obtain point sets in dimension 6 with $L_{\infty }$ star discrepancy up to 35% better than that of the first n points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We provide a comparison with an energy functional introduced by Steinerberger (2019) [31], showing that our heuristic performs better on all tested instances. Finally, our results give further empirical information on inverse star discrepancy conjectures.

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

We consider linear problems in the worst-case setting. That is, given a linear operator and a pool of admissible linear measurements, we want to approximate the operator uniformly on a convex and balanced set by means of algorithms using at most n such measurements. It is known that, in general, linear algorithms do not yield an optimal approximation. However, as we show here, an optimal approximation can always be obtained with a homogeneous algorithm. This is of interest for two reasons. First, the homogeneity allows us to extend any error bound on the unit ball to the full input space. Second, homogeneous algorithms are better suited to tackle problems on cones, a scenario far less understood than the classical situation of balls. We use the optimality of homogeneous algorithms to prove solvability for a family of problems defined on cones. We illustrate our results by several examples.

For objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters can be precomputed by semidefinite programming. This general framework can be applied to several scenarios, including the two-space problem and problems involving ${\ell }_2$-inaccurate data. It can also be applied to the problem of recovery from ${\ell }_1$-inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition.