arXiv subjects

Keji Liu

Publications and source records attributed to Keji Liu.

9 recordsLinked to original sources

arXiv2026

Derivative-Free Recovery of a Nonlinearity in a Free-Boundary DCIS Model

We investigate an inverse coefficient problem for a multidimensional free-boundary model of ductal carcinoma in situ (DCIS), in which the tumor interface is governed by the nonlinear coupling of nutrient concentration, tissue pressure and curvature, and the unknown nutrient consumption function is recovered from a temporal trace of the nutrient concentration obtained by needle aspiration biopsy. For the forward problem, we establish uniform local well-posedness over an admissible class of consumption functions. The inverse problem is recast as a fixed-point problem: approximating the admissible set by finite-dimensional spaces yields discrete iteration operators, for which we prove the existence of fixed points, and the strong convergence of a subsequence of discrete fixed points to a fixed point of the continuous operator, which solves the inverse problem under a consistency condition. To approximate these fixed points, we develop a homotopy-continuation method combining a linearly convergent Picard iteration with a cubical Sperner search, without differentiating an objective functional or computing an adjoint state. Several numerical experiments on radially symmetric and non-symmetric DCIS models corroborate the theoretical findings.

math.NA↗

arXiv2026

ATLAS-NN: Adaptive Transfer Learnable Symplectic-aware Neural Network for Long-Time Hamiltonian Dynamics

Modeling Hamiltonian systems over long temporal intervals remains a significant challenge due to intrinsic multiscale structures and rapid nonlinear transitions. While Hamiltonian Neural Networks (HNNs) incorporate geometric invariants to improve stability, they typically rely on a fixed, externally prescribed temporal structure. This lack of adaptability often leads to accumulated phase errors and degraded accuracy in systems with heterogeneous temporal scales. To address these limitations, we put forward the Adaptive Transfer Learnable Symplectic-aware Neural Network (ATLAS-NN). Our framework augments the HNN architecture with a learnable temporal scaling mechanism that parametrize a nonlinear mapping of time, automatically adapting to the system's intrinsic complexity. We propose a two-stage transfer learning strategy: the model is first trained on a short-time \textit{source} interval to identify the Hamiltonian structure and optimal temporal reparameterization; the learned scaling function is then frozen and transferred to an extended \textit{target} interval for fine-tuning. Numerical experiments on nonlinear oscillators and the chaotic H\'enon--Heiles system demonstrate that ATLAS-NN provides a more efficient alternative to standard HNNs and traditional symplectic integrators, yielding nearly an order of magnitude reduction in long-time prediction error.

physics.comp-ph↗

arXiv2020

A Time Delay Dynamic System with External Source for the Local Outbreak of 2019-nCoV

How to model the 2019 CoronaVirus (2019-nCov) spread in China is one of the most urgent and interesting problems in applied mathematics. In this paper, we propose a novel time delay dynamic system with external source to describe the trend of local outbreak for the 2019-nCoV. The external source is introduced in the newly proposed dynamic system, which can be considered as the suspected people travel to different areas. The numerical simulations exhibit the dynamic system with the external source is more reliable than the one without it, and the rate of isolation is extremely important for controlling the increase of cumulative confirmed people of 2019-nCoV. Based on our numerical simulation results with the public data, we suggest that the local government should have some more strict measures to maintain the rate of isolation. Otherwise the local cumulative confirmed people of 2019-nCoV might be out of control.

physics.soc-ph↗

arXiv2020

A Time Delay Dynamical Model for Outbreak of 2019-nCoV and the Parameter Identification

In this paper, we propose a novel dynamical system with time delay to describe the outbreak of 2019-nCoV in China. One typical feature of this epidemic is that it can spread in latent period, which is therefore described by the time delay process in the differential equations. The accumulated numbers of classified populations are employed as variables, which is consistent with the official data and facilitates the parameter identification. The numerical methods for the prediction of outbreak of 2019-nCoV and parameter identification are provided, and the numerical results show that the novel dynamic system can well predict the outbreak trend so far. Based on the numerical simulations, we suggest that the transmission of individuals should be greatly controlled with high isolation rate by the government.

q-bio.PE↗

arXiv2018

Optimal Mesh Size for Inverse Medium Scattering Problems

An optimal mesh size of the sampling region can help to reduce computational burden in practical applications. In this work, we investigate optimal choices of mesh sizes for the identifications of medium obstacles from either the far-field or near-field data in two and three dimensions. The results would have applications in the reconstruction process of inverse scattering problems.

math.NA↗

arXiv2016

A Multilevel Sampling Method for Detecting Sources in a Stratified Ocean Waveguide

In the reconstruction process of sound waves in a 3D stratified waveguide, a key technique is to effectively reduce the huge computational demand. In this work, we propose an efficient and simple multilevel reconstruction method to help locate the accurate position of a point source in a stratified ocean. The proposed method can be viewed as a direct sampling method since no solutions of optimizations or linear systems are involved. The novel method exhibits several strengths: fast convergence, robustness against noise, advantages in computational complexity and applicability for a very small number of receivers.

math.NA↗

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.

math-ph↗

arXiv2013

Optimal Shape Design by Partial Spectral Data

In this paper, we are concerned with a shape design problem, in which our target is to design, up to rigid transformations and scaling, the shape of an object given either its polarization tensor at multiple contrasts or the partial eigenvalues of its Neumann-Poincar\'e operator, which are known as the Fredholm eigenvalues. We begin by proposing to recover the eigenvalues of the Neumann-Poincar\'e operator from the polarization tensor by means of the holomorphic functional calculus. Then we develop a regularized Gauss-Newton optimization method for the shape reconstruction process. We present numerical results to demonstrate the effectiveness of the proposed methods and to illustrate important properties of the Fredholm eigenvalues and their associated eigenfunctions. Our results are expected to have important applications in the design of plasmon resonances in nanoparticles as well as in the multifrequency or pulsed imaging of small anomalies.

math.OC↗

arXiv2012

A Multilevel Sampling Algorithm for Locating Inhomogeneous Media

In the reconstruction process of unknown multiple scattering objects in inverse medium scattering problems, the first important step is to effectively locate some approximate domains that contain all inhomogeneous media. Without such an effective step, one may have to take a much larger computational domain than actually needed in the reconstruction of all scattering objects, thus resulting in a huge additional computational efforts. In this work we propose a simple and efficient multilevel reconstruction algorithm to help locate an accurate position and shape of each inhomogeneous medium. Then other existing effective but computationally more demanding reconstruction algorithms may be applied in these initially located computational domains to achieve more accurate shapes of the scatter and the contrast values over each medium domain. The new algorithm exhibits several strengths: robustness against noise, requiring less incidences, fast convergence, flexibility to deal with scatterers of special shapes, and advantages in computational complexity.

math.NA↗