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Kazufumi Ito

Publications and source records attributed to Kazufumi Ito.

At least 19 recordsLinked to original sources
arXiv2026

Direct Sampling Methods for Inverse Interface Problems

This work investigates two types of inverse interface problems in scenarios where only very limited Cauchy data is available. These problems are associated with the Laplace equation featuring a Robin-type flux jump across an internal interface. The first problem focuses on reconstructing the location of cracks along a known interface using Cauchy data measured on the outer boundary. The second problem involves determining the location of an unknown interface based on Cauchy data from the outer boundary. To address these challenges, we adopt an efficient Direct Sampling Method (DSM) and introduce innovative enhancements to the boundary conditions in the reference system, thereby maximizing the utility of the available Cauchy data. Additionally, we propose a novel refinement to further improve the robustness of the DSM against noise. We provide a detailed exposition of the general principles underlying the DSM and systematically present its computational implementation steps. Through detailed Fourier analysis and computations, we illustrate the theoretical background of the DSM as well as the effectiveness of our refinement approach. A series of numerical experiments demonstrates that our method yields highly satisfactory results, even when processing incomplete and noisy Cauchy data on the outer boundary. We introduce quantitative metrics, such as Mean Localization Error (MLE) and Contrast-to-Noise Ratio (CNR), to rigorously evaluate the performance of our method. These findings underscore the exceptional effectiveness and broad applicability of the proposed approach.

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arXiv2025

Anderson-type acceleration method for Deep Neural Network optimization

In this paper we consider the neural network optimization. We develop Anderson-type acceleration method for the stochastic gradient decent method and it improves the network permanence very much. We demonstrate the applicability of the method for Deep Neural Network (DNN) and Convolution Neural Network (CNN).

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arXiv2025

Reduced order method based Anderson-type acceleration method for nonlinear least square problems and large scale ill-posed problems

In this paper, we propose an acceleration framework for a class of iterative methods using the Reduced Order Method (ROM). Assuming that the underlying iterative scheme generates a rich basis for the solution space, we construct the next iterate by minimizing the equation error over the linear manifold spanned by this basis. The resulting optimal linear combination yields a more accurate approximation of the solution and significantly enhances convergence. In essence, the method can be seen as a history-based acceleration technique, akin to a delayed or memory-enhanced iterative scheme. This approach effectively remedies semi-ill-posed problems, enabling convergence where standard methods may fail, and also acts as a stabilizing and regularizing mechanism for the original iteration.

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arXiv2025

A Weighted Sampling Method for Inverse Medium Problem with Limited Aperture

Inverse medium scattering problems arise in many applications, but in practice, the measurement data are often restricted to a limited aperture by physical or experimental constraints. Classical sampling methods, such as MUSIC and the linear sampling method, are well understood for full-aperture data, yet their performance deteriorates severely under limited-aperture conditions, especially in the presence of noise. We propose a new sampling method tailored to the inverse medium problem with limited-aperture data. The method is motivated by the linear sampling framework and incorporates a weight function into the index function. The weight is designed so that the modified kernel reproduces the full-aperture behavior using only limited data, which both localizes oscillations and improves the conditioning of the far-field system, thereby yielding more accurate and stable reconstructions. We provide a theoretical justification of the method under the Born approximation and an efficient algorithm for computing the weight. Numerical experiments in two and three dimensions demonstrate that the proposed method achieves greater accuracy and robustness than existing sampling-type methods, particularly for noisy, limited-aperture data.

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arXiv2025

Iterative Direct Sampling Method for Elliptic Inverse Problems with Limited Cauchy Data

In this work, we propose an innovative iterative direct sampling method to solve nonlinear elliptic inverse problems from a limited number of pairs of Cauchy data. It extends the original direct sampling method (DSM) by incorporating an iterative mechanism, enhancing its performance with a modest increase in computational effort but a clear improvement in its stability against data noise. The method is formulated in an abstract framework of operator equations and is applicable to a broad range of elliptic inverse problems. Numerical results on electrical impedance tomography, optical tomography and cardiac electrophysiology etc. demonstrate its effectiveness and robustness, especially with an improved accuracy for identifying the locations and geometric shapes of inhomogeneities in the presence of large noise, when compared with the standard DSM.

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arXiv2024

Inverse medium problems, saddle point formulation

In this paper we discuss inverse medium problems. We develop the direct sampling method based on probing indices using the saddle point formulation. The medium is constructed by solutions of saddle point problems. The method improves the probing functions for the direct sampling method and directly images the medium. The method is very efficient and can be applied to a general class of inverse medium problems.

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arXiv2023

A Neumann interface optimal control problem with elliptic PDE constraints and its discretization and numerical analysis

We study an optimal control problem governed by elliptic PDEs with interface, which the control acts on the interface. Due to the jump of the coefficient across the interface and the control acting on the interface, the regularity of solution of the control problem is limited on the whole domain, but smoother on subdomains. The control function with pointwise inequality constraints is served as the flux jump condition which we called Neumann interface control. We use a simple uniform mesh that is independent of the interface. The standard linear finite element method can not achieve optimal convergence when the uniform mesh is used. Therefore the state and adjoint state equations are discretized by piecewise linear immersed finite element method (IFEM). While the accuracy of the piecewise constant approximation of the optimal control on the interface is improved by a postprocessing step which possesses superconvergence properties; as well as the variational discretization concept for the optimal control is used to improve the error estimates. Optimal error estimates for the control, suboptimal error estimates for state and adjoint state are derived. Numerical examples with and without constraints are provided to illustrate the effectiveness of the proposed scheme and correctness of the theoretical analysis.

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arXiv2023

A direct probing method of an inverse problem for the Eikonal equation

In this paper, we propose a direct probing method for the inverse problem based on the Eikonal equation. For the Eikonal equation with a point source, the viscosity solution represents the least travel time of wave fields from the source to the point at the high-frequency limit. The corresponding inverse problem is to determine the inhomogeneous wave-speed distribution from the first-arrival time data at the measurement surfaces corresponding to distributed point sources, which is called transmission travel-time tomography. At the low-frequency regime, the reconstruction approximates the frequency-depend wave-speed distribution. We analyze the Eikonal inverse problem and show that it is highly ill-posed. Then we developed a direct probing method that incorporates the solution analysis of the Eikonal equation and several aspects of the velocity models. When the wave-speed distribution has a small variation from the homogeneous medium, we reconstruct the inhomogeneous media using the filtered back projection method. For the high-contrast media, we assume a background medium and develop the adjoint-based back projection method to identify the variations of the medium from the assumed background.

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arXiv2022

Least-Squares Method for Inverse Medium Problems

We present a two-stage least-squares method to inverse medium problems of reconstructing multiple unknown coefficients simultaneously from noisy data. A direct sampling method is applied to detect the location of the inhomogeneity in the first stage, while a total least-squares method with mixed regularization is used to recover the medium profile in the second stage. The total least-squares method is designed to minimize the residual of the model equation and the data fitting, along with an appropriate regularization, in an attempt to significantly improve the accuracy of the approximation obtained from the first stage. We shall also present an analysis on the well-posedness and convergence of this algorithm. Numerical experiments are carried out to verify the accuracies and robustness of this novel two-stage least-squares algorithm, with great tolerance of noise.

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arXiv2020

A neural network based policy iteration algorithm with global $H^2$-superlinear convergence for stochastic games on domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the $H^2$-norm, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to HJBI boundary value problems corresponding to controlled diffusion processes with oblique boundary reflection. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

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arXiv2019

Regularized Linear Inversion with Randomized Singular Value Decomposition

In this work, we develop efficient solvers for linear inverse problems based on randomized singular value decomposition (RSVD). This is achieved by combining RSVD with classical regularization methods, e.g., truncated singular value decomposition, Tikhonov regularization, and general Tikhonov regularization with a smoothness penalty. One distinct feature of the proposed approach is that it explicitly preserves the structure of the regularized solution in the sense that it always lies in the range of a certain adjoint operator. We provide error estimates between the approximation and the exact solution under canonical source condition, and interpret the approach in the lens of convex duality. Extensive numerical experiments are provided to illustrate the efficiency and accuracy of the approach.

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arXiv2018

Fully Discrete Schemes and Their Analyses for Forward-Backward Stochastic Differential Equations

We propose some numerical schemes for forward-backward stochastic differential equations (FBSDEs) based on a new fundamental concept of transposition solutions. These schemes exploit time-splitting methods for the variation of constants formula of the associated partial differential equations and a discrete representation of the transition semigroups. The convergence of the schemes is established for FBSDEs with uniformly Lipschitz drivers, locally Lipschitz and maximal monotone drivers. Numerical experiments are presented for several nonlinear financial derivative pricing problems to demonstrate the adaptivity and effectiveness of the new schemes. The ideas here can be applied to construct high-order schemes for FBSDEs with general Markov forward processes.

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arXiv2017

Variational Gaussian Approximation for Poisson Data

The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian approximation to the posterior distribution arising from the Poisson model with a Gaussian prior. This is achieved by seeking an optimal Gaussian distribution minimizing the Kullback-Leibler divergence from the posterior distribution to the approximation, or equivalently maximizing the lower bound for the model evidence. We derive an explicit expression for the lower bound, and show the existence and uniqueness of the optimal Gaussian approximation. The lower bound functional can be viewed as a variant of classical Tikhonov regularization that penalizes also the covariance. Then we develop an efficient alternating direction maximization algorithm for solving the optimization problem, and analyze its convergence. We discuss strategies for reducing the computational complexity via low rank structure of the forward operator and the sparsity of the covariance. Further, as an application of the lower bound, we discuss hierarchical Bayesian modeling for selecting the hyperparameter in the prior distribution, and propose a monotonically convergent algorithm for determining the hyperparameter. We present extensive numerical experiments to illustrate the Gaussian approximation and the algorithms.

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arXiv2017

A convex analysis approach to optimal controls with switching structure for partial differential equations

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.

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arXiv2015

Analysis on Non-negative Factorizations and Applications

In this work we perform some mathematical analysis on non-negative matrix factorizations (NMF) and apply NMF to some imaging and inverse problems. We will propose a sparse low-rank approximation of big positive data and images in terms of tensor products of positive vectors, and investigate its effectiveness in terms of the number of tensor products to be used in the approximation. A new concept of multi-level analysis (MLA) framework is also suggested to extract major components in the matrix representing structures of different resolutions, but still preserving the positivity of the basis and sparsity of the approximation. We will also propose a semi-smooth Newton method based on primal-dual active sets for the non-negative factorization. Numerical results are given to demonstrate the effectiveness of the proposed method to capture features in images and structures of inverse problems under no a-priori assumption on the data structure, as well as to provide a sparse low-rank representation of the data.

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arXiv2014

Direct Sampling Method for Diffusive Optical Tomography

In this work, we are concerned with the diffusive optical tomography (DOT) problem in the case when only one or two pairs of Cauchy data is available. We propose a simple and efficient direct sampling method (DSM) to locate inhomogeneities inside a homogeneous background and solve the DOT problem in both full and limited aperture cases. This new method is easy to implement and less expensive computationally. Numerical experiments demonstrate its effectiveness and robustness against noise in the data. This provides a new promising numerical strategy for the DOT problem.

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arXiv2014

An Inexact Uzawa Algorithm for Generalized Saddle-Point Problems and Its Convergence

We propose an inexact Uzawa algorithm with two variable relaxation parameters for solving the generalized saddle-point system. The saddle-point problems can be found in a wide class of applications, such as the augmented Lagrangian formulation of the constrained minimization, the mixed finite element method, the mortar domain decomposition method and the discretization of elliptic and parabolic interface problems. The two variable parameters can be updated at each iteration, requiring no a priori estimates on the spectrum of two preconditioned subsystems involved. The convergence and convergence rate of the algorithm are analysed. Both symmetric and nonsymmetric saddle-point systems are discussed, and numerical experiments are presented to demonstrate the robustness and effectiveness of the algorithm.

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