arXiv subjects

Yuto Miyatake

Publications and source records attributed to Yuto Miyatake.

At least 19 recordsLinked to original sources

arXiv2026

Bayesian Inference of Discretization Error Means in ODEs via Ensemble Kalman Filtering

We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean-referred to as the discretization error mean-is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. The proposed prior depends on the step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh-Nagumo model demonstrate the effectiveness of the proposed approach.

math.NA↗

arXiv2026

Online TT-ALS for Streaming Tensor Decomposition with Incremental Orthogonalization

Tensor Train (TT) decomposition is a powerful technique for analyzing high-dimensional data. Existing algorithms for computing TT decompositions can be categorized into two main types: conventional batch-based approaches and recursive online methods. In the context of streaming data, batch methods typically achieve higher reconstruction accuracy but often suffer from memory exhaustion, while online methods provide greater computational efficiency. In this work, we introduce Online TT-ALS (Alternating Least Squares), an algorithm that sequentially enforces orthogonality constraints. This approach allows for efficient and exact updates of the core tensor while maintaining high reconstruction accuracy. Theoretically, we prove that enforcing these orthogonal gauge constraints guarantees monotonic decrease of the local objective function and temporal smoothness. Computationally, our deterministic single-sweep update reduces the rank dependence from quadratic to linear, achieving an overall complexity of $\mathcal{O}(I^{n-1} r)$. Experimental results demonstrate that the proposed method outperforms existing online techniques not only in terms of mathematical approximation accuracy but also in human perception-based video quality metrics. Furthermore, compared to recent deep learning-based paradigms, our algebraic approach achieves speedups of several orders of magnitude. Consequently, our method exhibits high computational efficiency and is suitable for low-latency real-time processing applications.

math.NA↗

arXiv2026

A polynomial moment approach to a rank condition for continuous-stage Runge--Kutta methods

In the study of energy-preserving methods for Hamiltonian systems, polynomial continuous-stage Runge--Kutta methods play an important role. Necessary and sufficient conditions for such methods to be energy-preserving have already been established. They are energy-preserving if the matrix $M\in \mathbb{R}^{s\times s}$ defining the method is symmetric, and the converse holds under the assumption that a certain $s\times \infty$ matrix $\Phi^\mathrm{CSRK}$ has full row rank. It was conjectured in Remark 3 in Miyatake and Butcher (SIAM J. Numer. Anal., 2016) that the full-rank assumption should always hold for every consistent polynomial continuous-stage Runge--Kutta method. In this paper, we prove the conjecture by showing that the matrix $\Phi^\mathrm{CSRK}$ has full row rank under the standard consistency condition. The proof is a direct application of the polynomial moment problem solved by Pakovich and Muzychuk (Proc. Lond. Math. Soc., 2009).

math.NA↗

arXiv2026

Accelerating SAV-based optimization via randomized low-rank Hessian approximation

We propose a new optimization method, the Nystr\"om-enhanced relaxed scalar auxiliary variable method (N-RSAV), which incorporates curvature information into the RSAV framework to accelerate convergence while preserving an unconditional modified energy dissipation law. Existing RSAV-based methods rely solely on first-order information and often suffer from slow convergence, particularly for ill-conditioned problems such as those arising in physics-informed neural networks (PINNs). To address this limitation, we design the linear operator in the RSAV scheme using approximate Hessian information obtained from a randomized low-rank Nystr\"om approximation. To preserve the dissipation structure, we enforce positive semidefiniteness through eigenvalue truncation. Furthermore, we introduce an adaptive strategy that reuses the approximate Hessian based on the deviation between the original and modified energies, significantly reducing computational cost. We also provide a convergence analysis of the RSAV scheme with a general positive semidefinite operator under the Polyak-Lojasiewicz (PL) condition and establish corresponding convergence guarantees for N-RSAV under the PL condition and an additional convexity assumption. Numerical experiments on ill-conditioned problems with effectively low-rank structure, including convex quadratic problems and training of PINNs, demonstrate that the proposed methods achieve substantially faster convergence than conventional RSAV-based approaches.

math.OC↗

arXiv2026

An error control framework for computing the exponential of matrices arising from the finite element discretization

Several methods for computing the action of the matrix exponential $\mathrm{e}^{\boldsymbol{A}} \boldsymbol{b}$ are expressed by substituting $\boldsymbol{A}$ into a rational approximation of the scalar exponential function. The error of such methods can be estimated using the numerical range of $\boldsymbol{A}$, which enables the computation of $\mathrm{e}^{\boldsymbol{A}}\boldsymbol{b}$ with a prescribed accuracy. However, when the input matrix has the structure $\boldsymbol{A} = \tau \boldsymbol{M}^{-1} \boldsymbol{K}$, this approach is challenging because computing the bounding box of numerical range is difficult and the numerical range may be too large to construct rational approximations on it. In this paper, focusing on the case where $\boldsymbol{M}$ is a well-conditioned symmetric positive definite matrix, we propose considering the numerical range of a similarity transformed matrix of $\boldsymbol{A}$. The numerical range of transformed matrix is not only numerically computable but can also be theoretically bounded depending on properties of $\boldsymbol{K}$. Numerical experiments confirm that the computations can be performed within the prescribed error tolerance.

math.NA↗

arXiv2025

Joint Bayesian Inference of Parameter and Discretization Error Uncertainties in ODE Models

We address the problem of Bayesian inference for parameters in ordinary differential equation (ODE) models based on observational data. Conventional approaches in this setting typically rely on numerical solvers such as the Euler or Runge-Kutta methods. However, these methods generally do not account for the discretization error induced by discretizing the ODE model. We propose a Bayesian inference framework for ODE models that explicitly quantifies discretization errors. Our method models discretization error as a random variable and performs Bayesian inference on both ODE parameters and variances of the randomized discretization errors, referred to as the discretization error variance. A key idea of our approach is the introduction of a Markov prior on the temporal evolution of the discretization error variances, enabling the inference problem to be formulated as a state-space model. Furthermore, we propose a specific form of the Markov prior that arises naturally from standard discretization error analysis. This prior depends on the step size in the numerical solver, and we discuss its asymptotic property in the limit as the step size approaches zero. Numerical experiments illustrate that the proposed method can simultaneously quantify uncertainties in both the ODE parameters and the discretization errors, and can produce posterior distributions over the parameters with broader support by accounting for discretization error.

stat.ME↗

arXiv2025

Low-loss Sb$_2$S$_3$ Optical Phase Shifter Enabled by Optimizing Sputtering Conditions

By quantitatively evaluating the atomic concentrations of sputtered Sb$_2$S$_3$ films with different sputtering powers and Ar flows, we reveal that a sputtered Sb$_2$S$_3$ film becomes close to the stoichiometric composition as the sputtering power and Ar flow decrease. We characterize the optical properties of Sb$_2$S$_3$ and show that the lower sputtering power leads to a better figure of merit of Sb$_2$S$_3$ as an optical phase shifter in the near infrared (NIR) range. Based on these results, we achieve a loss per phase shift of 0.33 dB/$π$ at a wavelength of 1.55 $μ$m, one of the lowest losses among Sb$_2$S$_3$ phase shifters in the NIR range.

physics.optics↗

arXiv2024

Quantifying uncertainty in the numerical integration of evolution equations based on Bayesian isotonic regression

This paper presents a new Bayesian framework for quantifying discretization errors in numerical solutions of ordinary differential equations. By modelling the errors as random variables, we impose a monotonicity constraint on the variances, referred to as discretization error variances. The key to our approach is the use of a shrinkage prior for the variances coupled with variable transformations. This methodology extends existing Bayesian isotonic regression techniques to tackle the challenge of estimating the variances of a normal distribution. An additional key feature is the use of a Gaussian mixture model for the $\log$-$χ^2_1$ distribution, enabling the development of an efficient Gibbs sampling algorithm for the corresponding posterior.

math.NA↗

arXiv2024

On the B-series composition theorem

The B-series composition theorem has been an important topic in numerical analysis of ordinary differential equations for the past-half century. Traditional proofs of this theorem rely on labelled trees, whereas recent developments in B-series analysis favour the use of unlabelled trees. In this paper, we present a new proof of the B-series composition theorem that does not depend on labelled trees. A key challenge in this approach is accurately counting combinations related to ``pruning.'' This challenge is overcome by introducing the concept of ``assignment.''

math.NA↗

arXiv2024

Adaptive projected SOR algorithms for nonnegative quadratic programming

The choice of relaxation parameter in the projected successive overrelaxation (PSOR) method for nonnegative quadratic programming problems is problem-dependent. We present novel adaptive PSOR algorithms that adaptively control the relaxation parameter using the Wolfe conditions. The method and its variants can be applied to various problems without requiring additional assumptions, barring the positive semidefiniteness concerning the matrix that defines the objective function, and the cost for updating the parameter is negligible in the whole iteration. Numerical experiments show that the proposed methods often perform comparably to (or sometimes superior to) the PSOR method with a nearly optimal relaxation parameter.

math.OC↗

arXiv2024

A new family of fourth-order energy-preserving integrators

For Hamiltonian systems with non-canonical structure matrices, a new family of fourth-order energy-preserving integrators is presented. The integrators take a form of a combination of Runge--Kutta methods and continuous-stage Runge--Kutta methods and feature a set of free parameters that offer greater flexibility and efficiency. Specifically, we demonstrate that by carefully choosing these free parameters a simplified Newton iteration applied to the integrators of order four can be parallelizable. This results in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators.

math.NA↗

arXiv2024

Lyapunov-Stable Deep Equilibrium Models

Deep equilibrium (DEQ) models have emerged as a promising class of implicit layer models, which abandon traditional depth by solving for the fixed points of a single nonlinear layer. Despite their success, the stability of the fixed points for these models remains poorly understood. By considering DEQ models as nonlinear dynamic systems, we propose a robust DEQ model named LyaDEQ with guaranteed provable stability via Lyapunov theory. The crux of our method is ensuring the Lyapunov stability of the DEQ model's fixed points, which enables the proposed model to resist minor initial perturbations. To avoid poor adversarial defense due to Lyapunov-stable fixed points being located near each other, we orthogonalize the layers after the Lyapunov stability module to separate different fixed points. We evaluate LyaDEQ models under well-known adversarial attacks, and experimental results demonstrate significant improvement in robustness. Furthermore, we show that the LyaDEQ model can be combined with other defense methods, such as adversarial training, to achieve even better adversarial robustness.

cs.LG↗

arXiv2024

Structure-Preserving Physics-Informed Neural Networks With Energy or Lyapunov Structure

Recently, there has been growing interest in using physics-informed neural networks (PINNs) to solve differential equations. However, the preservation of structure, such as energy and stability, in a suitable manner has yet to be established. This limitation could be a potential reason why the learning process for PINNs is not always efficient and the numerical results may suggest nonphysical behavior. Besides, there is little research on their applications on downstream tasks. To address these issues, we propose structure-preserving PINNs to improve their performance and broaden their applications for downstream tasks. Firstly, by leveraging prior knowledge about the physical system, a structure-preserving loss function is designed to assist the PINN in learning the underlying structure. Secondly, a framework that utilizes structure-preserving PINN for robust image recognition is proposed. Here, preserving the Lyapunov structure of the underlying system ensures the stability of the system. Experimental results demonstrate that the proposed method improves the numerical accuracy of PINNs for partial differential equations. Furthermore, the robustness of the model against adversarial perturbations in image data is enhanced.

cs.LG↗

arXiv2023

Piecewise monotone estimation in one-parameter exponential family

The problem of estimating a piecewise monotone sequence of normal means is called the nearly isotonic regression. For this problem, an efficient algorithm has been devised by modifying the pool adjacent violators algorithm (PAVA). In this study, we investigate estimation of a piecewise monotone parameter sequence for general one-parameter exponential families such as binomial, Poisson and chi-square. We develop an efficient algorithm based on the modified PAVA, which utilizes the duality between the natural and expectation parameters. We also provide a method for selecting the regularization parameter by using an information criterion. Simulation results demonstrate that the proposed method detects change-points in piecewise monotone parameter sequences in a data-driven manner. Applications to spectrum estimation, causal inference and discretization error quantification of ODE solvers are also presented.

stat.ME↗

arXiv2023

Non-volatile optical phase shift in ferroelectric hafnium zirconium oxide

A non-volatile optical phase shifter is a critical component for enabling large-scale, energy-efficient programmable photonic integrated circuits (PICs) on a silicon (Si) photonics platform. While ferroelectric materials like BaTiO3 offer non-volatile optical phase shift capabilities, their compatibility with complementary metal-oxide-semiconductor (CMOS) fabs is limited. Hence, the search for a novel CMOS-compatible ferroelectric material for non-volatile optical phase shifting in Si photonics is of utmost importance. Hafnium zirconium oxide (HZO) is an emerging ferroelectric material discovered in 2011, which exhibits CMOS compatibility due to the utilization of high-k dielectric HfO2 in CMOS transistors. Although extensively studied for ferroelectric transistors and memories, its application in photonics remains relatively unexplored. Here, we show the optical phase shift induced by ferroelectric HZO deposited on a SiN optical waveguide. We observed a negative change in refractive index at a 1.55 um wavelength in the pristine device regardless of the direction of an applied electric filed. We achieved approximately pi phase shift in a 4.5-mm-long device with negligible optical loss. The non-volatile multi-level optical phase shift was confirmed with a persistence of > 10000 s. This phase shift can be attributed to the spontaneous polarization within the HZO film along the external electric field. We anticipate that our results will stimulate further research on optical nonlinear effects, such as the Pockels effect, in ferroelectric HZO. This advancement will enable the development of various devices, including high-speed optical modulators. Consequently, HZO-based programmable PICs are poised to become indispensable in diverse applications, ranging from optical fiber communication and artificial intelligence to quantum computing and sensing.

physics.optics↗

arXiv2023

Modelling the discretization error of initial value problems using the Wishart distribution

This paper presents a new discretization error quantification method for the numerical integration of ordinary differential equations. The error is modelled by using the Wishart distribution, which enables us to capture the correlation between variables. Error quantification is achieved by solving an optimization problem under the order constraints for the covariance matrices. An algorithm for the optimization problem is also established in a slightly broader context.

stat.ME↗

arXiv2023

A modified model for topic detection from a corpus and a new metric evaluating the understandability of topics

This paper presents a modified neural model for topic detection from a corpus and proposes a new metric to evaluate the detected topics. The new model builds upon the embedded topic model incorporating some modifications such as document clustering. Numerical experiments suggest that the new model performs favourably regardless of the document's length. The new metric, which can be computed more efficiently than widely-used metrics such as topic coherence, provides variable information regarding the understandability of the detected topics.

cs.CL↗

arXiv2022

Structure-preserving numerical methods for constrained gradient flows of planar closed curves with explicit tangential velocities

In this paper, we consider numerical approximation of constrained gradient flows of planar closed curves, including the Willmore and the Helfrich flows. These equations have energy dissipation and the latter has conservation properties due to the constraints. We will develop structure-preserving methods for these equations that preserve both the dissipation and the constraints. To preserve the energy structures, we introduce the discrete version of gradients according to the discrete gradient method and determine the Lagrange multipliers appropriately. We directly address higher order derivatives by using the Galerkin method with B-spline curves to discretize curves. Moreover, we will consider stabilization of the schemes by adding tangential velocities. We introduce a new Lagrange multiplier to obtain both the energy structures and the stability. Several numerical examples are presented to verify that the proposed schemes preserve the energy structures with good distribution of control points.

math.NA↗