arXiv subjects

Tomoya Kemmochi

Publications and source records attributed to Tomoya Kemmochi.

18 recordsLinked to original sources

arXiv2025

A preconditioning technique of Gauss--Legendre quadrature for the logarithm of symmetric positive definite matrices

This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss--Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that of the original matrix. Although the matrix logarithm needs to be computed twice, each computation can be performed more efficiently, and it potentially reduces the overall computational cost. It is shown that the proposed technique is effective when the condition number of the given matrix is approximately between $130$ and $3.0\times 10^5$.

math.NA↗

arXiv2024

Discrete maximal regularity for the discontinuous Galerkin time-stepping method without logarithmic factor

Maximal regularity is a kind of a priori estimates for parabolic-type equations and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green's function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.

math.NA↗

arXiv2024

Migrating elastic flows

Huisken's problem asks whether there is an elastic flow of closed planar curves that is initially contained in the upper half-plane but `migrates' to the lower half-plane at a positive time. Here we consider variants of Huisken's problem for open curves under the natural boundary condition, and construct various migrating elastic flows both analytically and numerically.

math.AP↗

arXiv2023

Matrix equation representation of the convolution equation and its unique solvability

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.

math.NA↗

arXiv2023

Higher order discrete gradient method by the discontinuous Galerkin time-stepping method

Many differential equations with physical backgrounds are described as gradient systems, which are evolution equations driven by the gradient of some functionals, and such problems have energy conservation or dissipation properties. For numerical computation of gradient systems, numerical schemes that inherit the energy structure of the equation play important roles, which are called structure-preserving. The discrete gradient method is one of the most classical framework of structure-preserving methods, which is at most second order accurate. In this paper, we develop a higher-order structure-preserving numerical method for gradient systems, which includes the discrete gradient method. We reformulate the gradient system as a coupled system and then apply the discontinuous Galerkin time-stepping method. Numerical examples suggests that the order of accuracy of our scheme is $(k+1)$ in general and $(2k+1)$ at nodal times, where $k$ is the degree of polynomials.

math.NA↗

arXiv2023

Computing the matrix exponential with the double exponential formula

This paper considers the computation of the matrix exponential $\mathrm{e}^A$ with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore it is necessary to truncate the infinite interval. In this paper, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates $\mathrm{e}^A$ with the fixed mesh size which is a parameter in the DE formula affecting the accuracy. Second one computes $\mathrm{e}^A$ based on the first one with automatic selection of the mesh size depending on the given error tolerance.

math.NA↗

arXiv2023

$L^p$-resolvent estimate for finite element approximation of the Stokes operator

In this paper, we will show the $L^p$-resolvent estimate for the finite element approximation of the Stokes operator for $p \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$, where $N \ge 2$ is the dimension of the domain. It is expected that this estimate can be applied to error estimates for finite element approximation of the non-stationary Navier--Stokes equations, since studies in this direction are successful in numerical analysis of nonlinear parabolic equations. To derive the resolvent estimate, we introduce the solution of the Stokes resolvent problem with a discrete external force. We then obtain local energy error estimate according to a novel localization technique and establish global $L^p$-type error estimates. The restriction for $p$ is caused by the treatment of lower-order terms appearing in the local energy error estimate. Our result may be a breakthrough in the $L^p$-theory of finite element methods for the non-stationary Navier--Stokes equations.

math.NA↗

arXiv2023

Discrete maximal regularity for the finite element approximation of the Stokes operator and its application

Maximal regularity for the Stokes operator plays a crucial role in the theory of the non-stationary Navier--Stokes equations. In this paper, we consider the finite element semi-discretization of the non-stationary Stokes problem and establish the discrete counterpart of maximal regularity in $L^q$ for $q \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$. For the proof of discrete maximal regularity, we introduce the temporally regularized Green's function. With the aid of this notion, we prove discrete maximal regularity without the Gaussian estimate. As an application, we present $L^p(0,T;L^q(Ω))$-type error estimates for the approximation of the non-stationary Stokes problem.

math.NA↗

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↗

arXiv2021

Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy

In this paper, we present a novel investigation of the so-called SAV approach, which is a framework to construct linearly implicit geometric numerical integrators for partial differential equations with variational structure. SAV approach was originally proposed for the gradient flows that have lower-bounded nonlinear potentials such as the Allen-Cahn and Cahn-Hilliard equations, and this assumption on the energy was essential. In this paper, we propose a novel approach to address gradient flows with unbounded energy such as the KdV equation by a decomposition of energy functionals. Further, we will show that the equation of the SAV approach, which is a system of equations with scalar auxiliary variables, is expressed as another gradient system that inherits the variational structure of the original system. This expression allows us to construct novel higher-order integrators by a certain class of Runge-Kutta methods. We will propose second and fourth order schemes for conservative systems in our framework and present several numerical examples.

math.NA↗

arXiv2021

Computing the matrix fractional power with the double exponential formula

Two quadrature-based algorithms for computing the matrix fractional power $A^α$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^α$, is proposed. Then, two algorithms are presented -- one computes $A^α$ with a fixed number of abscissas, and the other computes $A^α$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $α$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.

math.NA↗

arXiv2019

Modified Strang splitting for semilinear parabolic problems

We consider applying the Strang splitting to semilinear parabolic problems. The key ingredients of the Strang splitting are the decomposition of the equation into several parts and the computation of approximate solutions by combining the time evolution of each split equation. However, when the Dirichlet boundary condition is imposed, order reduction could occur due to the incompatibility of the split equations with the boundary condition. In this paper, to overcome the order reduction, a modified Strang splitting procedure is presented for the one-dimensional semilinear parabolic equation with first-order spatial derivatives, like the Burgers equation.

math.NA↗

arXiv2019

Relation between the T-congruence Sylvester equation and the generalized Sylvester equation

The T-congruence Sylvester equation is the matrix equation $AX+X^{\mathrm{T}}B=C$, where $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{m\times m}$ are given, and $X\in\mathbb{R}^{n\times m}$ is to be determined. Recently, Oozawa et al. discovered a transformation that the matrix equation is equivalent to one of the well-studied matrix equations (the Lyapunov equation); however, the condition of the transformation seems to be too limited because matrices $A$ and $B$ are assumed to be square matrices ($m=n$). In this paper, two transformations are provided for rectangular matrices $A$ and $B$. One of them is an extension of the result of Oozawa et al. for the case $m\ge n$, and the other is a novel transformation for the case $m\le n$.

math.NA↗

arXiv2018

Maximum norm error estimates for the finite element approximation of parabolic problems on smooth domains

In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $Ω\subset \mathbb{R}^N$ with the inhomogeneous Neumann boundary condition. We emphasize that the domain can be non-convex in general. We implement the finite element method for this problem by constructing a family of polygonal or polyhedral domains $\{ Ω_h \}_h$ that approximate the original domain $Ω$. The main result of this study is the $L^\infty$-error estimate for this approximation. We shall show that the convergence rate is not optimal for higher order elements since the symmetric difference $Ω\bigtriangleup Ω_h$ is not empty in general. In order to address the effect of the symmetric difference of domains, we introduce the tubular neighborhood of the original boundary $\partialΩ$. We will also present a slightly new approach to establish the $L^\infty$-error estimate. Moreover, we present the smoothing property for the discrete parabolic semigroup and the spatially discretized maximal regularity as corollaries of the main result.

math.NA↗

arXiv2018

$L^\infty$- and $W^{1,\infty}$-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Pointwise error analysis of the linear finite element approximation for $-Δu + u = f$ in $Ω$, $\partial_n u = τ$ on $\partialΩ$, where $Ω$ is a bounded smooth domain in $\mathbb R^N$, is presented. We establish $O(h^2|\log h|)$ and $O(h)$ error bounds in the $L^\infty$- and $W^{1,\infty}$-norms respectively, by adopting the technique of regularized Green's functions combined with local $H^1$- and $L^2$-estimates in dyadic annuli. Since the computational domain $Ω_h$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $Ω_h \neq Ω$. In particular, the so-called Galerkin orthogonality relation, utilized three times in the proof, does not exactly hold and involves domain perturbation terms (or boundary-skin terms), which need to be addressed carefully. A numerical example is provided to confirm the theoretical result.

math.NA↗

arXiv2017

On the finite element approximation for non-stationary saddle-point problems

In this paper, we present a numerical analysis of the hydrostatic Stokes equations, which are linearization of the primitive equations describing the geophysical flows of the ocean and the atmosphere. The hydrostatic Stokes equations can be formulated as an abstract non-stationary saddle-point problem, which also includes the non-stationary Stokes equations. We first consider the finite element approximation for the abstract equations with a pair of spaces under the discrete inf-sup condition. The aim of this paper is to establish error estimates for the approximated solutions in various norms, in the framework of analytic semigroup theory. Our main contribution is an error estimate for the pressure with a natural singularity term $t^{-1}$, which is induced by the analyticity of the semigroup. We also present applications of the error estimates for the finite element approximations of the non-stationary Stokes and the hydrostatic Stokes equations.

math.NA↗

arXiv2016

Energy dissipative numerical scheme for gradient flows of planar curves using discrete partial derivatives and B-spline curves

In this paper, we develop an energy dissipative numerical scheme for gradient flows of planar curves, such as the curvature flow and the elastic flow. Our study presents a general framework for solving such equations. To discretize time, we use a similar approach to the discrete partial derivative method, which is a structure-preserving method for the gradient flows of graphs. For the approximation of curves, we use B-spline curves. Owing to the smoothness of B-spline functions, we can directly address higher order derivatives. In the last part of the paper, we consider some numerical examples of the elastic flow, which exhibit topology-changing solutions and more complicated evolution. Videos illustrating our method are available on YouTube.

math.NA↗

arXiv2016

Discrete maximal regularity and the finite element method for parabolic equations

Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in $L^p(0,T;L^q(Ω))$ norm, $p,q\in (1,\infty)$ for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of G.~Dore and A.~Venni (On the closedness of the sum of two closed operators. \emph{Math.\ Z.}, 196(2):189--201, 1987). As an application, optimal order error estimates in that norm are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinearity and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo-Nirenberg and Sobolev inequalities are also presented.

math.NA↗