arXiv subjects

Yoshihito Kazashi

Publications and source records attributed to Yoshihito Kazashi.

At least 19 recordsLinked to original sources

arXiv2026

Existence of Dynamical Low-Rank Approximation for SDEs with Locally Lipschitz Coefficients

Numerical simulations of high-dimensional stochastic differential equations (SDEs), which are increasingly employed in real-world applications, can be unaffordable in terms of computational time and memory. A possible solution is the deployment of reduced order methods (ROMs) that provide fast simulations with good accuracy when dealing with low-rank problems. In the context of SDEs, the Dynamical Low-Rank Approximation (DLRA) already showed remarkable results, in terms of approximation and computational efficiency of computational time because of being completely computed "on-the-fly". In this article, we extend the framework of DLRA for SDEs proposed in our primary work arXiv:2308.11581 by considering locally Lipschitz drift and diffusion with linear-growth bound, by showing the existence of DLRA for this setting.

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arXiv2026

Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part II: Stochastic discretization

In this second article (Part II), we analyze the numerical algorithms for the Dynamical Low-Rank Approximation (DLRA) of Stochastic Differential Equations (SDEs) introduced in Part I arXiv:2601.21428 under the perspective of the stochastic discretization. Specifically, we employ a Monte Carlo method with $M$ samples to approximate the stochastic space and all the related quantities of interest. Consequently, these algorithms produce noisy interacting particle systems whose error analysis is not standard. Assuming high moments and subgaussian tails of the initial condition, in the case of elliptic diffusion, we provide convergence results for the DLR Projector Splitting for SDEs presented in Part I. When the fully discretized Gramian is of full rank for all the time evolution, then one observes a convergence rate close to the usual Monte Carlo one, i.e. $O(\frac{1}{\sqrt{M}})$, for large $M$. If a regularization of this matrix is employed at each time-step, the rate is close to $O(\frac{1}{\sqrt[3]{M}})$ for large $M$. On the other hand, in case the regularization occurs only when the smallest singular value of the Gramian is smaller than a prescribed positive threshold, the stochastic convergence rate is located between the aforementioned results. Furthermore, in the case of general diffusion, we provide an easy-to-implement convergent algorithm. Concerning the DLR Euler-Maruyama and the DLR Projector Splitting for Euler-Maruyama (EM), these results are not straightforward to derive and a brief discussion on why this is a challenging task is provided, too. Numerical simulations will complete this analysis.

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arXiv2026

Dynamical Low-Rank Filters for Data Assimilation

We propose dynamical low-rank (DLR) type filters for data-assimilation problems based on stochastic differential equations (SDEs). In detail, first we derive a DLRA filter for minimizing jointly the mean and covariance error, as well as a strategy to efficiently include the relevant orthogonal directions. This last approach allows the main subspace to evolve also according to the observation operator. Those procedures naturally extend to a Kalman-Bucy type filter when dealing with linear drift, and to ensemble methods, too, resulting also suitable for problems described by nonlinear drift and possible non-Gaussian distribution. Moreover, we further propose a preliminary particle-type DLRA filter that shows potentiality in nonlinear settings. Numerical simulations show the efficacy of these procedures in relevant applications, opening up to further studies in these filtering directions.

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arXiv2026

Further Approaches of Dynamical Low-Rank Approximation for SDEs

In this article, we propose two other DLRA-type dynamics for stochastic differential equations (SDEs) than the one studied in arXiv:2308.11581, derived from a minimization of functionals and (informally) from a Stratonovich formulation, respectively. The former approach resembles the DLRA for SDE system proposed in arXiv:1803.00499. Providing the differentiability of the diffusion, the latter procedure registers an additional term in the drift. Indeed, its derivation exploits the Stratonovich formulation to write stochastic processes on manifold, and, hence, possesses a term that depends on the geometry of the manifold itself. These developments open the debate on which formalism is more suitable and what DLRA for SDEs really is.

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arXiv2026

Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part I: Time discretization

In this work (Part I), we study three time-discretization schemes for the Dynamical Low-Rank Approximation (DLRA) of high-dimensional stochastic differential equations (SDEs). Specifically, we consider the Dynamically Orthogonal (DO) method for DLRA proposed and analyzed in arXiv:2308.11581v4, which approximates the true solution by a linear combination of few products between deterministic orthonormal modes and stochastic modes, both time-dependent. The first scheme considered consists in a forward discretization in time of both deterministic and stochastic components, in a Euler-Maruyama style. Its convergence is proven subject to a time-step restriction dependent on the smallest singular value of the Gram matrix associated to the stochastic modes, which, on its turn, is shown to be always positive, provided that the SDE under study is driven by a non-degenerate noise. The second and the third schemes, on the other hand, are staggered ones, alternating updates of the deterministic and the stochastic modes in half steps, and have a projector splitting nature. We show stability of the second scheme and prove convergence with constants independent of the smallest singular value. The third scheme works better in practice, although our theoretical convergence bounds are worse than those for the second one. Computational experiments support our theoretical results. In this work we do not consider the discretization in probability, which will be the topic of Part II.

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arXiv2026

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 $α$, 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 $N^{-α/2}$ with $N$ function evaluations, although the optimal rate is known to be $N^{-α}(\ln N)^{(d-1)/2}$. 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 $N^{-α/2}$. 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.

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arXiv2026

Multigrid Monte Carlo Revisited: Theory and Bayesian Inference

Gaussian random fields play an important role in many areas of science and engineering. In practice, they are often simulated by sampling from a high-dimensional multivariate normal distribution, which arises from the discretisation of a suitable precision operator. Existing methods such as Cholesky factorization and Gibbs sampling become prohibitively expensive on fine meshes due to their high computational cost. In this work, we revisit the Multigrid Monte Carlo (MGMC) algorithm developed by Goodman & Sokal (Physical Review D 40.6, 1989) in the quantum physics context. While the authors of this paper conclude that MGMC does not overcome critical slowing down in simulations of field theories near phase transitions, we demonstrate here that it has the potential to significantly accelerate sampling in spatial statistics. The class of Gaussian Random Fields we consider includes those with Matérn covariance, but is more general in that it also allows for non-stationary covariance functions. To show that MGMC can overcome the limitation of existing methods, we establish a grid-size-independent convergence theory based on the link between linear solvers and samplers for multivariate normal distributions, drawing on standard multigrid convergence arguments. We then apply this theory to linear Bayesian inverse problems. This application is achieved by extending the standard multigrid theory to operators with a low-rank perturbation. Moreover, we develop a novel bespoke random smoother which takes care of the low-rank updates that arise in constructing posterior moments. In particular, we prove that Multigrid Monte Carlo is algorithmically optimal in the limit of the grid-size going to zero. Numerical results support our theory, demonstrating that Multigrid Monte Carlo can be significantly more efficient than alternative methods when applied in a Bayesian setting.

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arXiv2025

$L_2$-approximation using randomized lattice algorithms

We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $α> 1/2$ and product weights $γ_1,γ_2,\ldots\in [0,1]$. Building upon the deterministic lattice algorithm by Kuo, Sloan, and Woźniakowski (2006), we incorporate a randomized quadrature rule by Dick, Goda, and Suzuki (2022) to accelerate the convergence rate. This randomization involves drawing the number of points for function evaluations randomly, and selecting a good generating vector for rank-1 lattice points using the randomized component-by-component algorithm. We prove that our randomized algorithm achieves a worst-case root mean squared $L_2$-approximation error of order $M^{-α(2α+1)/(4α+1)+\varepsilon}$ for an arbitrarily small $\varepsilon > 0$, where $M$ denotes the maximum number of function evaluations, and that the error bound is independent of the dimension $d$ if the weights satisfy $\sum_{j=1}^\infty γ_j^{1/α} < \infty$. Our upper bound converges faster than a lower bound on the worst-case $L_2$-approximation error for deterministic rank-1 lattice-based approximation proved by Byrenheid, Kämmerer, Ullrich, and Volkmer (2017). We also show a lower error bound of order $M^{-α/2-1/2}$ for our randomized algorithm, leaving a slight gap between the upper and lower bounds open for future research.

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arXiv2024

How sharp are error bounds? --lower bounds on quadrature worst-case errors for analytic functions

Numerical integration over the real line for analytic functions is studied. Our main focus is on the sharpness of the error bounds. We first derive two general lower estimates for the worst-case integration error, and then apply these to establish lower bounds for various quadrature rules. These bounds turn out to be either novel or improve upon existing results, leading to lower bounds that closely match upper bounds for various formulas. Specifically, for the suitably truncated trapezoidal rule, we improve upon general lower bounds on the worst-case error obtained by Sugihara [\textit{Numer. Math.}, 75 (1997), pp.~379--395] and provide exceptionally sharp lower bounds apart from a polynomial factor, and in particular show that the worst-case error for the trapezoidal rule by Sugihara is not improvable by more than a polynomial factor. Additionally, our research reveals a discrepancy between the error decay of the trapezoidal rule and Sugihara's lower bound for general numerical integration rules, introducing a new open problem. Moreover, Gauss--Hermite quadrature is proven sub-optimal under the decay conditions on integrands we consider, a result not deducible from upper-bound arguments alone. Furthermore, to establish the near-optimality of the suitably scaled Gauss--Legendre and Clenshaw--Curtis quadratures, we generalize a recent result of Trefethen [\textit{SIAM Rev.}, 64 (2022), pp.~132--150] for the upper error bounds in terms of the decay conditions.

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arXiv2024

Dynamical Low-Rank Approximation for Stochastic Differential Equations

In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations (SDEs). DLRA aims at approximating the solution as a linear combination of a small number of basis vectors with random coefficients (low rank format) with the peculiarity that both the basis vectors and the random coefficients vary in time. While the formulation and properties of DLRA are now well understood for random/parametric equations, the same cannot be said for SDEs and this work aims to fill this gap. We start by rigorously formulating a Dynamically Orthogonal (DO) approximation (an instance of DLRA successfully used in applications) for SDEs, which we then generalize to define a parametrization independent DLRA for SDEs. We show local well-posedness of the DO equations and their equivalence with the DLRA formulation. We also characterize the explosion time of the DO solution by a loss of linear independence of the random coefficients defining the solution expansion and give sufficient conditions for global existence.

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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.

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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.

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arXiv2022

Density estimation in RKHS with application to Korobov spaces in high dimensions

A kernel method for estimating a probability density function (pdf) from an i.i.d. sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate pdfs belonging to weighted Korobov spaces, for which a dimension independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.

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arXiv2021

Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a parabolic type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed by Lubich et al. [BIT Num. Math., 54:171-188, 2014; SIAM J. on Num. Anal., 53:917-941, 2015], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.

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arXiv2021

Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng, Kritzer, Hickernell (Constr. Approx., 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja, Kuo, Sloan (SIAM J. Numer. Anal., 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.

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arXiv2020

Existence of dynamical low rank approximations for random semi-linear evolutionary equations on the maximal interval

An existence result is presented for the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations. The DLR solution approximates the true solution at each time instant by a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time. A key to our proof is to find a suitable equivalent formulation of the original problem. The so-called Dual Dynamically Orthogonal formulation turns out to be convenient. Based on this formulation, the DLR approximation is recast to an abstract Cauchy problem in a suitable linear space, for which existence and uniqueness of the solution in the maximal interval are established.

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arXiv2019

Derandomised lattice rules for high dimensional integration

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider "half-shifted" rules, in which each component of the shift is an odd multiple of $1/(2N)$, where $N$ is the number of points in the lattice. We show, by applying the principle that \emph{there is always at least one choice as good as the average}, that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of order only ${1/N^2}$. Numerical experiments, in which the generating vector is chosen component-by-component (CBC) as for randomly shifted lattices and then the shift by a new "CBC for shift" algorithm, yield encouraging results.

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arXiv2018

Worst-case error for unshifted lattice rules without randomisation

An existence result is presented for the worst-case error of lattice rules for high dimensional integration over the unit cube, in an unanchored weighted space of functions with square-integrable mixed first derivatives. Existing studies rely on random shifting of the lattice to simplify the analysis, whereas in this paper neither shifting nor any other form of randomisation is considered. Given that a certain number-theoretic conjecture holds, it is shown that there exists an $N$-point rank-one lattice rule which gives a worst-case error of order $1/\sqrt{N}$ up to a (dimension-independent) logarithmic factor. Numerical results suggest that the conjecture is plausible.

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