arXiv subjects

Kota Takeda

Publications and source records attributed to Kota Takeda.

3 recordsLinked to original sources

arXiv2026

Error control technique of quadrature-based algorithms for the action of real powers of a Hermitian positive-definite matrix

This study considers quadrature-based algorithms to compute $A^\alpha \boldsymbol{b}$, the action of a real power of a Hermitian positive-definite matrix $A$ on a vector $ \boldsymbol{b}$. In these algorithms, the computation of an integral representation of $A^{\alpha} \boldsymbol{b}$ is reduced to solving several tens or hundreds of shifted linear systems. Current approaches usually analyze the quadrature discretization error, but rarely take into account the additional error introduced by solving these shifted linear systems with iterative solvers. Here, we bound this error with the residual of the approximated solution of these linear systems. This allows the derivation of a stopping criterion for iterative solvers to keep the error of $A^\alpha \boldsymbol{b}$ below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of $A^\alpha \boldsymbol{b}$ within prescribed tolerance limits.

math.NA↗

arXiv2025

Error analysis of the projected PO method with additive inflation for the partially observed Lorenz 96 model

We consider the filtering problem with the partially observed Lorenz 96 model. Although the accuracy of the 3DVar filter in this problem has been established, the theoretical guarantee for the ensemble Kalman filter (EnKF) remains limited due to the analytical difficulty of handling non-symmetric matrices that emerge in the partial observation setting. This study establishes uniform-in-time error bounds of a stochastic variant of the EnKF, known as the perturbed observation (PO) method. By utilizing additive covariance inflation, we successfully obtain the bounds both with and without projecting the background covariance onto the observation space. Our analysis with the projection complements existing results for the deterministic variant of the EnKF, while our approach without the projection offers an extended mathematical framework to handle the non-symmetric matrix products directly. A numerical example validates the theoretical findings and shows comparable accuracies between the two settings.

math.NA↗

arXiv2024

Uniform error bounds of the ensemble transform Kalman filter for infinite-dimensional dynamics with multiplicative covariance inflation

Data assimilation is a method of uncertainty quantification to estimate the hidden true state by updating the prediction owing to model dynamics with observation data. As a prediction model, we consider a class of nonlinear dynamical systems on Hilbert spaces including the two-dimensional Navier-Stokes equations and the Lorenz '63 and '96 equations. For nonlinear model dynamics, the ensemble Kalman filter (EnKF) is often used to approximate the mean and covariance of the probability distribution with a set of particles called an ensemble. In this paper, we consider a deterministic version of the EnKF known as the ensemble transform Kalman filter (ETKF), performing well even with limited ensemble sizes in comparison to other stochastic implementations of the EnKF. When the ETKF is applied to large-scale systems, an ad-hoc numerical technique called a covariance inflation is often employed to reduce approximation errors. Despite the practical effectiveness of the ETKF, little is theoretically known. The present study aims to establish the theoretical analysis of the ETKF. We obtain that the estimation error of the ETKF with and without the covariance inflation is bounded for any finite time. In particular, the uniform-in-time error bound is obtained when an inflation parameter is chosen appropriately, justifying the effectiveness of the covariance inflation in the ETKF.

math.ST↗