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Kosuke Suzuki

Publications and source records attributed to Kosuke Suzuki.

At least 37 records · Page 2Linked to original sources
arXiv2022

Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters, $γ_j$, satisfy the summability condition $\sum_{j=1}^{\infty}γ_j^{1/α}<\infty$, where $α$ is a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces. In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition $\sum_{j=1}^{\infty}γ_j^{1/α}<\infty$ holds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.

math.NA↗
arXiv2021

Redox oscillations in 18650-type lithium-ion cell revealed by in operando Compton scattering imaging

Compton scattering imaging using high-energy synchrotron x-rays allows the visualization of the spatio-temporal lithiation state in lithium-ion batteries probed in-operando. Here, we apply this imaging technique to the commercial 18650-type cylindrical lithium-ion battery. Our analysis of the lineshapes of the Compton scattering spectra taken from different electrode layers reveals the emergence of inhomogeneous lithiation patterns during the charge-discharge cycles. Moreover, these patterns exhibit oscillations in time where the dominant period corresponds to the time scale of the charging curve.

physics.app-ph↗
arXiv2019

Identification of ferrimagnetic orbitals preventing spinel degradation by charge ordering in Li$_x$Mn$_2$O$_4$

Spinel Li$_x$Mn$_2$O$_4$ is a key cathode material that is used extensively in commercial Li-ion batteries. A challenge with this material has been that the capacity of the battery fades with cycling, an effect that can be traced to the presence of an anti-ferromagnetic insulator phase in the fully lithiated LiMn$_2$O$_4$ (LMO) and the associated charge disproportionation that drives distortions of the MnO$_6$ octahedra. Here, by combining x-ray magnetic Compton scattering experiments with parallel first-principles computations, we show that the anti-ferromagnetic phase of LMO is surrounded by a robust ferrimagnetic metallic phase, which becomes stable when even a small amount of Li is removed from or added to the charge-ordered LMO. In this surprising ferrimagnetic state, charge-ordering and octahedral distortions are found to be strongly suppressed. We identify the nature of the ferrimagnetic orbitals involved through theoretical and experimental analyses of the magnetic Compton scattering spectra.

cond-mat.mtrl-sci↗
arXiv2019

Recent advances in higher order quasi-Monte Carlo methods

In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007, 2008) who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration.

math.NA↗
arXiv2018

Lattice rules in non-periodic subspaces of Sobolev spaces

We investigate quasi-Monte Carlo (QMC) integration over the $s$-dimensional unit cube based on rank-1 lattice point sets in weighted non-periodic Sobolev spaces $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$ and their subspaces of high order smoothness $α>1$, where $\boldsymbolγ$ denotes a set of the weights. A recent paper by Dick, Nuyens and Pillichshammer has studied QMC integration in half-period cosine spaces with smoothness parameter $α>1/2$ consisting of non-periodic smooth functions, denoted by $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{cos}})$, and also in the sum of half-period cosine spaces and Korobov spaces with common parameter $α$, denoted by $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$. Motivated by the results shown there, we first study embeddings and norm equivalences on those function spaces. In particular, for an integer $α$, we provide their corresponding norm-equivalent subspaces of $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$. This implies that $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$ is strictly smaller than $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$ as sets for $α\geq 2$, which solves an open problem by Dick, Nuyens and Pillichshammer. Then we study the worst-case error of tent-transformed lattice rules in $\mathcal{H}(K_{2,\boldsymbolγ,s}^{\mathrm{sob}})$ and also the worst-case error of symmetrized lattice rules in an intermediate space between $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$ and $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$. We show that the almost optimal rate of convergence can be achieved for both cases, while a weak dependence of the worst-case error bound on the dimension can be obtained for the former case.

math.NA↗
arXiv2018

Digital net properties of a polynomial analogue of Frolov's construction

Frolov's cubature formula on the unit hypercube has been considered important since it attains an optimal rate of convergence for various function spaces. Its integration nodes are given by shrinking a suitable full rank $\mathbb{Z}$-lattice in $\mathbb{R}^d$ and taking all points inside the unit cube. The main drawback of these nodes is that they are hard to find computationally, especially in high dimensions.In such situations, quasi-Monte Carlo (QMC) rules based on digital nets have proven to be successful. However, there is still no construction known that leads to QMC rules which are optimal in the same generality as Frolov's. In this paper we investigate a polynomial analog of Frolov's cubature formula, which we expect to be important in this respect. This analog is defined in a field of Laurent series with coefficients in a finite field. A similar approach was previously studied in [M.~B.~Levin. Adelic constructions of low discrepancy sequences. Online Journal of Analytic Combinatorics. Issue 5, 2010.]. We show that our construction is a $(t,m,d)$-net, which also implies bounds on its star-discrepancy and the error of the corresponding cubature rule. Moreover, we show that our cubature rule is a QMC rule, whereas Frolov's is not, and provide an algorithm to determine its integration nodes explicitly. To this end we need to extend the notion of $(t,m,d)$-nets to fit the situation that the points can have infinite digit expansion and develop a duality theory. Additionally, we adapt the notion of admissible lattices to our setting and prove its significance.

math.NA↗
arXiv2017

In-operando quantitation of Li concentration for commercial Li-ion rechargeable battery using high-energy X-ray Compton scattering

Compton scattering is one of the promising probe to quantitate of the Li under in-operando condition, since high-energy X-rays which have high penetration power into the materials are used as incident beam and Compton scattered energy spectrum have specific line-shape by the elements. We develop in-operando quantitation method of Li composition in the electrodes by using line-shape (Sparameter) analysis of Compton scattered energy spectrum. In this study, we apply S-parameter analysis to commercial coin cell Li-ion rechargeable battery and obtain the variation of S-parameters during charge/discharge cycle at positive and negative electrodes. By using calibration curves for Li composition in the electrodes, we determine the change of Li composition of positive and negative electrodes through S-parameters, simultaneously.

physics.app-ph↗
arXiv2017

Quasi-Monte Carlo integration for twice differentiable functions over a triangle

We study quasi-Monte Carlo integration for twice differentiable functions defined over a triangle. We provide an explicit construction of infinite sequences of points including one by Basu and Owen (2015) as a special case, which achieves the integration error of order $N^{-1}(\log N)^3$ for any $N\geq 2$. Since a lower bound of order $N^{-1}$ on the integration error holds for any linear quadrature rule, the upper bound we obtain is best possible apart from the $\log N$ factor. The major ingredient in our proof of the upper bound is the dyadic Walsh analysis of twice differentiable functions over a triangle under a suitable recursive partitioning.

math.NA↗
arXiv2016

Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is the $\mathbb{Z}$-lattice in $\mathbb{R}^d$ generated by the Vandermonde matrix associated to the roots of the $d$-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev-Frolov lattices in axis-parallel boxes when $d = 2^n$ for a non-negative integer $n$, since the points are used for the nodes of Frolov's cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. The existing enumeration algorithm for such points by Kacwin, Oettershagen and Ullrich is efficient up to dimension $d=16$. In this paper we suggest a new enumeration algorithm of such points for $d=2^n$, efficient up to $d=32$.

math.NA↗
arXiv2016

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points $N$ and the weights, can be constructed using a fast component-by-component algorithm in at most $O(sN(\log N)^2)$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.

math.NA↗
arXiv2016

Optimal order quadrature error bounds for infinite-dimensional higher order digital sequences

Quasi-Monte Carlo (QMC) quadrature rules using higher order digital nets and sequences have been shown to achieve the almost optimal rate of convergence of the worst-case error in Sobolev spaces of arbitrary fixed smoothness $α\in \mathbb{N}$, $α\geq 2$. In a recent paper by the authors, it was proved that randomly-digitally-shifted order $2α$ digital nets in prime base $b$ achieve the best possible rate of convergence of the root mean square worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$ for $N=b^m$, where $N$ and $s$ denote the number of points and the dimension, respectively, which implies the existence of an optimal order QMC rule. More recently, the authors provided an explicit construction of such an optimal order QMC rule by using Chen-Skriganov's digital nets in conjunction with Dick's digit interlacing composition. These results were for fixed number of points. In this paper we give a more general result on an explicit construction of optimal order QMC rules for arbitrary fixed smoothness $α\in \mathbb{N}$ including the endpoint case $α=1$. That is, we prove that the projection of any infinite-dimensional order $2α+1$ digital sequence in prime base $b$ onto the first $s$ coordinates achieves the best possible rate of convergence of the worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$ for $N=b^m$. The explicit construction presented in this paper is not only easy to implement but also extensible in both $N$ and $s$.

math.NA↗
arXiv2016

Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions

We investigate multivariate integration for a space of infinitely times differentiable functions $\mathcal{F}_{s, \boldsymbol{u}} := \{f \in C^\infty [0,1]^s \mid \| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} < \infty \}$, where $\| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} := \sup_{\boldsymbolα = (α_1, \dots, α_s) \in \mathbb{N}_0^s} \|f^{(\boldsymbolα)}\|_{L^1}/\prod_{j=1}^s u_j^{α_j}$, $f^{(\boldsymbolα)} := \frac{\partial^{|\boldsymbolα|}}{\partial x_1^{α_1} \cdots \partial x_s^{α_s}}f$ and $\boldsymbol{u} = \{u_j\}_{j \geq 1}$ is a sequence of positive decreasing weights. Let $e(n,s)$ be the minimal worst-case error of all algorithms that use $n$ function values in the $s$-variate case. We prove that for any $\boldsymbol{u}$ and $s$ considered $e(n,s) \leq C(s) \exp(-c(s)(\log{n})^2)$ holds for all $n$, where $C(s)$ and $c(s)$ are constants which may depend on $s$. Further we show that if the weights $\boldsymbol{u}$ decay sufficiently fast then there exist some $1 < p < 2$ and absolute constants $C$ and $c$ such that $e(n,s) \leq C \exp(-c(\log{n})^p)$ holds for all $s$ and $n$. These bounds are attained by quasi-Monte Carlo integration using digital nets. These convergence and tractability results come from those for the Walsh space into which $\mathcal{F}_{s, \boldsymbol{u}}$ is embedded.

math.NA↗
arXiv2016

Lattice based integration algorithms: Kronecker sequences and rank-1 lattices

We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness $\mathring{\mathbf{B}}^s_{p,θ}$, which also comprise the concept of Sobolev spaces of dominating mixed smoothness $\mathring{\mathbf{H}}^s_{p}$ as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from $T_N(\mathbb{Z}^d) \cap [0,1)^d$, where $T_N$ is a real invertible generator matrix of size $d$. For such rules the worst-case error can be bounded in terms of the Zaremba index of the lattice $\mathbb{X}_N=T_N(\mathbb{Z}^d)$. We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to $\log N$-factors for arbitrary smoothness $s$. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.

math.NA↗
arXiv2016

WAFOM on abelian groups for quasi-Monte Carlo point sets

In this paper, we study quasi-Monte Carlo (QMC) rules for numerical integration. J. Dick proved a Koksma-Hlawka type inequality for $α$-smooth integrands and gave an explicit construction of QMC rules achieving the optimal rate of convergence in that function class. From this inequality, Matsumoto et al. introduced Walsh figure of merit (WAFOM) $\mathrm{WAFOM}(P)$ for an $\mathbb{F}_2$-digital net $P$ as a quickly computable quality criterion for $P$ as a QMC point set. The key ingredient for obtaining WAFOM is the Dick weight, a generalization of the Hamming weight and the Niederreiter-Rosenbloom-Tsfasman (NRT) weight. We extend the notions of the Dick weight and WAFOM for digital nets over a general finite abelian group $G$, and show that this version of WAFOM satisfies Koksma-Hlawka type inequality when $G$ is cyclic. We give a MacWilliams-type identity on the weight enumerator polynomials for the Dick weight, by which we can compute the minimum Dick weight as well as WAFOM. We give a lower bound of WAFOM of order $N^{-C'_G(\log N)/s}$ and an upper bound of lowest WAFOM of order $N^{-C_G(\log N)/s}$ for given $(G,N,s)$ if $(\log N)/s$ is sufficiently large, where $N$ is the cardinality of the point set $P$, $P$ is a quadrature rule in $[0,1)^s$, and $C'_G$ and $C_G$ are constants depending only on the cardinality of $G$. These bounds generalize the bounds given by Yoshiki and others given for $G=\mathbb{F}_2$.

math.NA↗
arXiv2016

An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands

In a recent paper by the authors, it is shown that there exists a quasi-Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions. In this paper we provide an explicit construction of such an optimal order QMC rule. Our approach is to exploit both the decay and the sparsity of the Walsh coefficients of the reproducing kernel simultaneously. This can be done by applying digit interlacing composition due to Dick to digital nets with large minimum Hamming and Niederreiter-Rosenbloom-Tsfasman metrics due to Chen and Skriganov. To our best knowledge, our construction gives the first QMC rule which achieves the best possible convergence in this function space.

math.NA↗
arXiv2016

Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness

We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $α\in \mathbb{N}$, $α\ge 2$, defined over the $s$-dimensional unit cube. We prove that randomly digitally shifted order $β$ digital nets can achieve the convergence of the root mean square worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$ when $β\ge 2α$. The exponent of the logarithmic term, i.e., $(s-1)/2$, is improved compared to the known result by Baldeaux and Dick, in which the exponent is $sα/2$. Our result implies the existence of a digitally shifted order $β$ digital net achieving the convergence of the worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$, which matches a lower bound on the convergence rate of the worst-case error for any cubature rule using $N$ function evaluations and thus is best possible.

math.NA↗