arXiv subjects

Hirofumi Notsu

Publications and source records attributed to Hirofumi Notsu.

At least 19 recordsLinked to original sources

arXiv2026

Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement

This paper proposes a unified shape and coefficient optimization approach for inverse problems governed by diffusion equations. The associated forward problem is considered with a Robin boundary condition, physically motivated in diffuse optical tomography to model partial reflection of light at tissue boundaries. The main objective is the recovery of the piecewise-defined absorption coefficient together with its underlying interface from a single boundary measurement. To this end, a shape-based reconstruction approach is formulated in which the interface is introduced as a geometric unknown governing the piecewise structure of the absorption coefficient. While classical approaches rely on the Fréchet derivative with respect to spatially varying parameters, the Eulerian derivative with respect to the interface is additionally exploited. This leads to a unified framework for the simultaneous recovery of the coefficient and the geometry under the single-measurement setting. Numerical experiments demonstrate the effectiveness of the proposed method, even for complex and non-convex interfaces.

math.NA↗

arXiv2026

Numerical methods for diffusion coefficient recovery

We revisit the inverse problem of reconstructing a spatially varying diffusion coefficient in stationary elliptic equations from boundary Cauchy data. From a theoretical perspective, we introduce a gradient-weighted modification of the coupled complex-boundary method (CCBM) incorporating an $H^1$-type term, and formulate the reconstruction as a regularized optimization problem over bounded admissible coefficients. We establish continuity and differentiability of the forward map, Lipschitz continuity of the modified cost functional, existence of minimizers, stability with respect to noisy data, and convergence under vanishing noise. From a numerical perspective, reconstructions are computed using a Sobolev-gradient descent scheme and evaluated through extensive numerical experiments across a range of noise levels, boundary inputs, and coefficient structures. In the reported tests, for sufficiently large but not excessive $H^1$-weights, the modified CCBM is observed to yield more stable reconstructions and to reduce certain high-frequency artifacts. Across the numerical scenarios considered in this study, the method often demonstrates favorable stability and robustness properties relative to several classical boundary-based formulations, although performance remains problem- and parameter-dependent. A projection-based extension further supports stable recovery of piecewise-constant diffusion coefficients in multi-subregion test cases. Our results indicate that, as long as all subdomains share a portion of the boundary, the proposed CCBM-based Tikhonov regularization approach with a pick-a-point strategy enables stable and reliable reconstruction of diffusion parameters.

math.NA↗

arXiv2026

Ensemble Reservoir Computing for Physical Systems

Physical computing exploits unconventional physical substrates to overcome limitations such as the high energy consumption inherent in digital computation. However, intrinsic noise and temporal fluctuations (e.g., oscillations) generally deteriorate computational performance. Here, we propose ensemble reservoir computing (ERC), a novel framework that employs ensemble averaging of spatially multiplexed systems to achieve robust information processing despite noise and temporal fluctuations. First, we prove that ensemble averaging in ERC eliminates temporal fluctuations and noise from dynamical states under certain conditions, thereby restoring computational performance to its noise-free level. Next, we show that ERC not only removes the noise and fluctuations but also actively exploits the computational capabilities that conventional reservoir computing (RC) leaves unutilized. This computational enhancement is demonstrated across diverse dynamical systems (e.g., periodic, chaotic, and strange-nonchaotic systems), in which ERC outperforms conventional RC. Finally, using energy-efficient spin-torque oscillators (STOs), we demonstrate that ERC maintains high performance even under realistic conditions, in which noise and temporal fluctuations coexist: STOs with ERC achieved 99\% accuracy on an error detection test, where conventional STO reservoir with linear regression only shows a chance level performance, highlighting ERC's robustness and performance gains for physical systems.

math.DS↗

arXiv2025

Mathematical and numerical study of symmetry and positivity of the tensor-valued spring constant defined from P1-FEM for two- and three-dimensional linear elasticity

In this study, we consider a spring-block system that approximates a $d$-dimensional linear elastic body, where $d=2$ or $d=3$. We derive a $d\times d$ matrix as the spring constant using the P1 finite element method with a triangular mesh for the linear elasticity equations. We mathematically analyze the symmetry and positive-definiteness of the spring constant. Even if we assume full symmetry of the elasticity tensor, the symmetry of the matrix obtained as the spring constant is not trivial. However, we have succeeded in proving this in a unified manner for both 2D and 3D cases. This is an alternative proof for the 2D case in Notsu-Kimura (2014) and is a new result for the 3D case. We provide a necessary and sufficient condition for the spring constant to be positive-definite in the case of an isotropic elasticity tensor, along with a sufficient condition in terms of mesh regularity and the Poisson ratio. These theoretical results are supported by several numerical experiments. The positive-definiteness of the spring constant derived from the finite element method plays a vital role in fracture simulations of elastic bodies using the spring-block system.

math.NA↗

arXiv2025

Energy Transfer Dynamics Generated by Non-Axisymmetric Tornado-Type Flows

The energy cascade in turbulence, first statistically described by Richardson (1922) and Kolmogorov (1941), lacked connection to the underlying fluid dynamics. Recent numerical studies of Goto et al. (2017) and Yoneda et al. (2022) revealed scale-local energy transfer via vortex stretching but remained within spatial statistics. This study aims to uncover the time-dependent elementary process behind the energy cascade by constructing a tornado-type flow in a non-axisymmetric curved cylindrical domain. Our approach reveals specific vortex dynamics responsible for energy transfer, offering new insight into the physical mechanisms of turbulence.

math.AP↗

arXiv2025

Harnessing omnipresent oscillator networks as computational resource

Nature is pervaded with oscillatory dynamics. In networks of coupled oscillators patterns can arise when the system synchronizes to an external input. Hence, these networks provide processing and memory of input. We present a universal framework for harnessing oscillator networks as computational resource. This computing framework is introduced by the ubiquitous model for phase-locking, the Kuramoto model. We force the Kuramoto model by a nonlinear target-system, then after substituting the target-system with a trained feedback-loop it emulates the target-system. Our results are two-fold. Firstly, the trained network inherits performance properties of the Kuramoto model, where all-to-all coupling is performed in linear time with respect to the number of nodes and parameters for synchronization are abundant. The latter implies that the network is generically successful since the system learns via sychronization. Secondly, the learning capabilities of the oscillator network, which describe a type of collective intelligence, can be explained using Kuramoto model's order parameter. In summary, this work provides the foundation for utilizing nature's oscillator networks as a new class of information processing systems.

cs.LG↗

arXiv2024

Detecting immersed obstacle in Stokes fluid flow using the coupled complex boundary method

A non-conventional shape optimization approach is introduced to address the identification of an obstacle immersed in a fluid described by the Stokes equation within a larger bounded domain, relying on boundary measurements on the accessible surface. The approach employs tools from shape optimization, utilizing the coupled complex boundary method to transform the over-specified problem into a complex boundary value problem by incorporating a complex Robin boundary condition. This condition is derived by coupling the Dirichlet and Neumann boundary conditions along the accessible boundary. The identification of the obstacle involves optimizing a cost function constructed based on the imaginary part of the solution across the entire domain. The subsequent calculation of the shape gradient of this cost function, rigorously performed via the rearrangement method, enables the iterative solution of the optimization problem using a Sobolev gradient descent algorithm. The feasibility of the method is illustrated through numerical experiments in both two and three spatial dimensions, demonstrating its effectiveness in reconstructing obstacles with pronounced concavities under high-level noise-contaminated data, all without perimeter or volume functional penalization.

math.OC↗

arXiv2024

Error Estimates for First- and Second-Order Lagrange-Galerkin Moving Mesh Schemes for the One-Dimensional Convection-Diffusion Equation

A new moving mesh scheme based on the Lagrange-Galerkin method for the approximation of the one-dimensional convection-diffusion equation is studied. The mesh movement, which is prescribed by a discretized dynamical system for the nodal points, follows the direction of convection. It is shown that under a restriction of the time increment the mesh movement cannot lead to an overlap of the elements and therefore an invalid mesh. For the linear element, optimal error estimates in the $\ell^\infty(L^2) \cap \ell^2(H_0^1)$ norm are proved in case of both, a first-order backward Euler method and a second-order two-step method in time. These results are based on new estimates of the time dependent interpolation operator derived in this work. Preservation of the total mass is verified for both choices of the time discretization. Numerical experiments are presented that confirm the error estimates and demonstrate that the proposed moving mesh scheme can circumvent limitations that the Lagrange-Galerkin method on a fixed mesh exhibits.

math.NA↗

arXiv2023

Virtual reservoir acceleration for CPU and GPU: Case study for coupled spin-torque oscillator reservoir

We provide high-speed implementations for simulating reservoirs described by $N$-coupled spin-torque oscillators. Here $N$ also corresponds to the number of reservoir nodes. We benchmark a variety of implementations based on CPU and GPU. Our new methods are at least 2.6 times quicker than the baseline for $N$ in range $1$ to $10^4$. More specifically, over all implementations the best factor is 78.9 for $N=1$ which decreases to 2.6 for $N=10^3$ and finally increases to 23.8 for $N=10^4$. GPU outperforms CPU significantly at $N=2500$. Our results show that GPU implementations should be tested for reservoir simulations. The implementations considered here can be used for any reservoir with evolution that can be approximated using an explicit method.

cs.DC↗

arXiv2023

Numerical solution to a free boundary problem for the Stokes equation using the coupled complex boundary method in shape optimization settings

A new reformulation of a free boundary problem for the Stokes equations governing a viscous flow with overdetermined condition on the free boundary is proposed. The idea of the method is to transform the governing equations to a boundary value problem with a complex Robin boundary condition coupling the two boundary conditions on the free boundary. The proposed formulation give rise to a new cost functional that apparently has not been exploited yet in the literature, specifically, and at least, in the context of free surface problems. The shape derivatives of the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary is explicitly determined. Using the computed shape gradient information, a domain variation method from a preconditioned steepest descent algorithm is applied to solve the shape optimization problem. Numerical results illustrating the applicability of the method is then provided both in two and three spatial dimensions. For validation and evaluation of the method, the numerical results are compared with the ones obtained via the classical tracking Dirichlet data.

math.OC↗

arXiv2023

A two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition and its application to the Bay of Bengal region. Part I: Flat bottom topography

This study presents a two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC). Firstly, the experimental order of convergence of the scheme is shown to see the second-order accuracy in time. Secondly, the effect of the TBC on a simple domain is discussed; the artificial reflections are kept from the Dirichlet boundaries and removed significantly from the transmission boundaries. Thirdly, the scheme is applied to a complex practical domain, i.e., the Bay of Bengal region, which is non-convex and includes islands. The effect of the TBC is discussed again for the complex domain; the artificial reflections are removed significantly from transmission boundaries, which are set on open sea boundaries. Based on the numerical results, it is revealed that the scheme has the following properties; (i) the same advantages of Lagrange-Galerkin methods (the CFL-free robustness for convection-dominated problems and the symmetry of the matrices for the system of linear equations); (ii) second-order accuracy in time; (iii) mass preservation of the function for the water level from the reference height (until the contact with the transmission boundaries of the wave); and (iv) no significant artificial reflection from the transmission boundaries. The numerical results by the scheme are presented in this paper for the flat bottom topography of the domain. In the next part of this work, Part II, the scheme will be applied to rapidly varying bottom surfaces and a real bottom topography of the Bay of Bengal region.

math.NA↗

arXiv2022

A mass-preserving two-step Lagrange-Galerkin scheme for convection-diffusion problems

A mass-preserving two-step Lagrange-Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $L^2$-theory. The introduced scheme maintains the advantages of the Lagrange-Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the $\ell^\infty(L^2)$- and $\ell^2(H^1_0)$-norms. For the time increment $Δt$, the mesh size $h$ and a conforming finite element space of polynomial degree $k$, the convergence order is of $O(Δt^2 + h^k)$ in the $\ell^\infty(L^2)\cap \ell^2(H^1_0)$-norm and of $O(Δt^2 + h^{k+1})$ in the $\ell^\infty(L^2)$-norm if the duality argument can be employed. Error estimates of $O(Δt^{3/2}+h^k)$ in discrete versions of the $L^\infty(H^1_0)$- and $H^1(L^2)$-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.

math.NA↗

arXiv2021

A Convective Boundary Condition for the Navier-Stokes Equations: Existence Analysis and Numerical Implementations

Due to computational complexity, fluid flow problems are mostly defined on a bounded domain. Hence, capturing fluid outflow calls for imposing an appropriate condition on the boundary where the said outflow is prescribed. Usually, the Neumann-type boundary condition called do-nothing condition is the go-to description for such outflow phenomenon However, such condition does not ensure an energy estimate for the Navier--Stokes equations - let alone establish the existence of solutions. In this paper, we analyze a convective boundary condition that will capture outflow and establish the existence of solutions to the governing equation. We shall show existence and uniqueness results for systems with mixed boundary conditions - Dirichlet condition and the convective boundary condition. The first system is a stationary equation where the Dirichlet condition is purely homogeneous, the other is where an input function is prescribed, and lastly a dynamic system with prescribed input function. We end by showing numerical examples to illustrate the difference between the current outflow condition and the usual do-nothing condition.

math.AP↗

arXiv2021

A Shape Optimization Problem Constrained with the Stokes Equations to Address Maximization of Vortices

We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the $L^2$-norm of the curl and the {\it det-grad} measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.

math.NA↗

arXiv2021

Maximizing Vortex for the Navier--Stokes Flow with a Convective Boundary Condition: A Shape Design Problem

In this study, a shape optimization problem for the two-dimensional stationary Navier--Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the artificial boundary condition is formulated so as to take into account the possibility of ill-posedness caused by the usual do-nothing boundary condition. The goal of the optimization problem is to maximize the vorticity of the said fluid by determining the shape of an obstacle inside the channel. Meanwhile, the shape variation is limited by a perimeter functional and a volume constraint. The perimeter functional was considered to act as a Tikhonov regularizer and the volume constraint is added to exempt us from topological changes in the domain. The shape derivative of the objective functional was formulated using the rearrangement method, and this derivative was later on used for gradient descent methods. Additionally, an augmented Lagrangian method and a class of solenoidal deformation fields were considered to take into account the goal of volume preservation. Lastly, numerical examples based on the gradient descent and the volume preservation methods are presented.

math.OC↗

arXiv2021

Second-order finite difference approximations of the upper-convected time derivative

In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation of the popular viscoelastic models, is reformulated in order to obtain approximations of second-order in time for solving a simplified constitutive equation in one and two dimensions. The theoretical analysis of the truncation errors of the methods takes into account the linear and quadratic interpolation operators based on a Lagrangian framework. Numerical experiments illustrating the theoretical results for the model equation defined in one and two dimensions are included. Finally, the finite difference approximations of second-order in time are also applied for solving a two-dimensional Oldroyd-B constitutive equation subjected to a prescribed velocity field at different Weissenberg numbers.

math.NA↗

arXiv2020

Computing with vortices: Bridging fluid dynamics and its information-processing capability

Herein, the Karman vortex system is considered to be a large recurrent neural network, and the computational capability is numerically evaluated by emulating nonlinear dynamical systems and the memory capacity. Therefore, the Reynolds number dependence of the Karman vortex system computational performance is revealed and the optimal computational performance is achieved near the critical Reynolds number at the onset of Karman vortex shedding, which is associated with a Hopf bifurcation. Our finding advances the understanding of the relationship between the physical properties of fluid dynamics and its computational capability as well as provides an alternative to the widely believed viewpoint that the information processing capability becomes optimal at the edge of chaos.

physics.flu-dyn↗

arXiv2019

Chaos in nanomagnet via feedback current

Nonlinear magnetization dynamics excited by spin-transfer effect with feedback current is studied both numerically and analytically. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates the positive Lyapunov exponent for a certain range of the feedback rate, which identifies the existence of chaos in a nanostructured ferromagnet. Transient behavior from chaotic to steady oscillation is also observed in another range of the feedback parameter. An analytical theory is also developed, which indicates the appearance of multiple attractors in a phase space due to the feedback current. An instantaneous imbalance between the spin-transfer torque and damping torque causes a transition between the attractors, and results in the complex dynamics.

cond-mat.mes-hall↗