arXiv subjects

Yuya Suzuki

Publications and source records attributed to Yuya Suzuki.

13 recordsLinked to original sources

arXiv2026

M\"obius-transformed trapezoidal rule for polynomial weights

This work studies numerical integration by the M\"obius-transformed trapezoidal rule, which combines the classical trapezoidal rule with a change of variables induced by a M\"obius transformation that maps the unit circle onto the real line. It is shown that this method achieves the optimal convergence rate for a polynomially weighted integral over the real line if the integrand lives in a related polynomially weighted Sobolev space with positive integer smoothness index. This result can also be generalized in a slightly weaker form for fractional smoothness indices via complex interpolation of function spaces. The algorithm only requires pointwise evaluations of the weight and the target integrand at prescribed nodes that do not depend on the integrand and weight in question. The established theoretical convergence rates are verified by numerical experiments.

math.NA↗

arXiv2025

Approximation of differential entropy in Bayesian optimal experimental design

Bayesian optimal experimental design provides a principled framework for selecting experimental settings that maximize obtained information. In this work, we focus on estimating the expected information gain in the setting where the differential entropy of the likelihood is either independent of the design or can be evaluated explicitly. This reduces the problem to maximum entropy estimation, alleviating several challenges inherent in expected information gain computation. Our study is motivated by large-scale inference problems, such as inverse problems, where the computational cost is dominated by expensive likelihood evaluations. We propose a computational approach in which the evidence density is approximated by a Monte Carlo or quasi-Monte Carlo surrogate, while the differential entropy is evaluated using standard methods without additional likelihood evaluations. We prove that this strategy achieves convergence rates that are comparable to, or better than, state-of-the-art methods for full expected information gain estimation, particularly when the cost of entropy evaluation is negligible. Moreover, our approach relies only on mild smoothness of the forward map and avoids stronger technical assumptions required in earlier work. We also present numerical experiments, which confirm our theoretical findings.

stat.ML↗

arXiv2025

Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces

Optimality of several quasi-Monte Carlo methods and suboptimality of the sparse-grid quadrature based on the univariate Gauss--Hermite rule is proved in the Sobolev spaces of mixed dominating smoothness of order $\alpha$, where the optimality is in the sense of worst-case convergence rate. For sparse-grid Gauss--Hermite quadrature, lower and upper bounds are established, with rates coinciding up to a logarithmic factor. The dominant rate is found to be only $N^{-\alpha/2}$ with $N$ function evaluations, although the optimal rate is known to be $N^{-\alpha}(\ln N)^{(d-1)/2}$. The lower bound is obtained by exploiting the structure of the Gauss--Hermite nodes and is independent of the quadrature weights; consequently, no modification of the weights can improve the rate $N^{-\alpha/2}$. In contrast, several quasi-Monte Carlo methods with a change of variables are shown to achieve the optimal rate, some up to, and one including, the logarithmic factor.

math.NA↗

arXiv2025

Lattice Rules Meet Kernel Cubature

Rank-1 lattice rules are a class of equally weighted quasi-Monte Carlo methods that achieve essentially linear convergence rates for functions in a reproducing kernel Hilbert space (RKHS) characterized by square-integrable first-order mixed partial derivatives. In this work, we explore the impact of replacing the equal weights in lattice rules with optimized cubature weights derived using the reproducing kernel. We establish a theoretical result demonstrating a doubled convergence rate in the one-dimensional case and provide numerical investigations of convergence rates in higher dimensions. We also present numerical results for an uncertainty quantification problem involving an elliptic partial differential equation with a random coefficient.

math.NA↗

arXiv2024

M\"obius-Transformed Trapezoidal Rule

We study numerical integration by combining the trapezoidal rule with a M\"obius transformation that maps the unit circle onto the real line. We prove that the resulting transformed trapezoidal rule attains the optimal rate of convergence if the integrand function lives in a weighted Sobolev space with a weight that is only assumed to be a positive Schwartz function decaying monotonically to zero close to infinity. Our algorithm only requires the ability to evaluate the weight at the selected nodes, and it does not require sampling from a probability measure defined by the weight nor information on its derivatives. In particular, we show that the M\"obius transformation, as a change of variables between the real line and the unit circle, sends a function in the weighted Sobolev space to a periodic Sobolev space with the same smoothness. Since there are various results available for integrating and approximating periodic functions, we also describe several extensions of the M\"obius-transformed trapezoidal rule, including function approximation via trigonometric interpolation, integration with randomized algorithms, and multivariate integration.

math.NA↗

arXiv2024

Construction of Optimal Algorithms for Function Approximation in Gaussian Sobolev Spaces

This paper studies function approximation in Gaussian Sobolev spaces over the real line and measures the error in a Gaussian-weighted $L^p$-norm. We construct two linear approximation algorithms using $n$ function evaluations that achieve the optimal or almost optimal rate of worst-case convergence in a Gaussian Sobolev space of order $\alpha$. The first algorithm is based on scaled trigonometric interpolation and achieves the optimal rate $n^{-\alpha}$ up to a logarithmic factor. This algorithm can be constructed in almost-linear time with the fast Fourier transform. The second algorithm is more complicated, being based on spline smoothing, but attains the optimal rate $n^{-\alpha}$.

math.NA↗

arXiv2023

Randomizing the trapezoidal rule gives the optimal RMSE rate in Gaussian Sobolev spaces

Randomized quadratures for integrating functions in Sobolev spaces of order $α\ge 1$, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of $O(n^{-α-1/2})$ is proven, where $n$ denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory.

math.NA↗

arXiv2023

Sub-optimality of Gauss--Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness

The sub-optimality of Gauss--Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order $α$, where the optimality is in the sense of worst-case error. For Gauss--Hermite quadrature, we obtain matching lower and upper bounds, which turn out to be merely of the order $n^{-α/2}$ with $n$ function evaluations, although the optimal rate for the best possible linear quadrature is known to be $n^{-α}$. Our proof of the lower bound exploits the structure of the Gauss--Hermite nodes; the bound is independent of the quadrature weights, and changing the Gauss--Hermite weights cannot improve the rate $n^{-α/2}$. In contrast, we show that a suitably truncated trapezoidal rule achieves the optimal rate up to a logarithmic factor.

math.NA↗

arXiv2022

Approximation in Hilbert spaces of the Gaussian and related analytic kernels

We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian kernel $K(x, y) = \exp(-\tfrac{1}{2}\varepsilon^2(x-y)^2)$. For weighted power series kernels we derive almost matching upper and lower bounds on the worst-case error. When applied to the Gaussian kernel, our results state that, up to a sub-exponential factor, the $n$th minimal error decays as $(\varepsilon/2)^n (n!)^{-1/2}$. The proofs are based on weighted polynomial interpolation and classical polynomial coefficient estimates that we use to bound the Hilbert space norm of a weighted polynomial fooling function.

math.NA↗

arXiv2022

Scaled lattice rules for integration on $\mathbb{R}^d$ achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces

We introduce a new method to approximate integrals $\int_{\mathbb{R}^d} f(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}$ which simply scales lattice rules from the unit cube $[0,1]^d$ to properly sized boxes on $\mathbb{R}^d$, hereby achieving higher-order convergence that matches the smoothness of the integrand function $f$ in a certain Sobolev space of dominating mixed smoothness. Our method only assumes that we can evaluate the integrand function $f$ and does not assume a particular density nor the ability to sample from it. In particular, for the theoretical analysis we show a new result that the method of adding Bernoulli polynomials to a function to make it "periodic" on a box without changing its integral value over the box, is equivalent to an orthogonal projection from a well chosen Sobolev space of dominating mixed smoothness to an associated periodic Sobolev space of the same dominating mixed smoothness, which we call a Korobov space. We note that the Bernoulli polynomial method is often not used because of its excessive computational complexity and also here we only make use of it in our theoretical analysis. We show that our new method of applying scaled lattice rules to increasing boxes can be interpreted as orthogonal projections with decreasing projection error. Such a method would not work on the unit cube since then the committed error caused by non-periodicity of the integrand would be constant, but for integration on the Euclidean space we can use the certain decay towards zero when the boxes grow. Hence we can bound the truncation error as well as the projection error and show higher-order convergence in applying scaled lattice rules for integration on Euclidean space. We illustrate our theoretical analysis by numerical experiments which confirm our findings.

math.NA↗

arXiv2020

Strang splitting in combination with rank-$1$ and rank-$r$ lattices for the time-dependent Schrödinger equation

We approximate the solution for the time dependent Schrödinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of functions on rank-1 or rank-r lattice points with unitary Fourier transforms. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method. We prove that the numerical scheme proposed converges quadratically with respect to the time step size, given that the potential is in a Korobov space with the smoothness parameter greater than $9/2$. Particularly, we prove that the required degree of smoothness is independent of the dimension of the problem. We demonstrate our new method by comparing with results using sparse grids from [12], with several numerical examples showing large advantage for our new method and pushing the examples to higher dimensionality. The proposed method has two distinctive features from a numerical perspective: (i) numerical results show the error convergence of time discretization is consistent even for higher-dimensional problems; (ii) by using the rank-$1$ lattice points, the solution can be efficiently computed (and further time stepped) using only $1$-dimensional Fast Fourier Transforms.

math.NA↗

arXiv2020

From Natural Language Instructions to Complex Processes: Issues in Chaining Trigger Action Rules

Automation services for complex business processes usually require a high level of information technology literacy. There is a strong demand for a smartly assisted process automation (IPA: intelligent process automation) service that enables even general users to easily use advanced automation. A natural language interface for such automation is expected as an elemental technology for the IPA realization. The workflow targeted by IPA is generally composed of a combination of multiple tasks. However, semantic parsing, one of the natural language processing methods, for such complex workflows has not yet been fully studied. The reasons are that (1) the formal expression and grammar of the workflow required for semantic analysis have not been sufficiently examined and (2) the dataset of the workflow formal expression with its corresponding natural language description required for learning workflow semantics did not exist. This paper defines a new grammar for complex workflows with chaining machine-executable meaning representations for semantic parsing. The representations are at a high abstraction level. Additionally, an approach to creating datasets is proposed based on this grammar.

cs.AI↗

arXiv2019

Rank-1 lattices and higher-order exponential splitting for the time-dependent Schr\"odinger equation

In this paper, we propose a numerical method to approximate the solution of the time-dependent Schr\"odinger equation with periodic boundary condition in a high-dimensional setting. We discretize space by using the Fourier pseudo-spectral method on rank-$1$ lattice points, and then discretize time by using a higher-order exponential operator splitting method. In this scheme the convergence rate of the time discretization depends on properties of the spatial discretization. We prove that the proposed method, using rank-$1$ lattice points in space, allows to obtain higher-order time convergence, and, additionally, that the necessary condition on the space discretization can be independent of the problem dimension $d$. We illustrate our method by numerical results from 2 to 8 dimensions which show that such higher-order convergence can really be obtained in practice.

math.NA↗