Jiguang Sun

Ruihao Huang, Allan A. Struthers, Jiguang Sun et al.

Recently, a new eigenvalue problem, called the transmission eigenvalue problem, has attracted many researchers. The problem arose in inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing. The problem is numerically challenging because it is non-selfadjoint and nonlinear. In this paper, we propose a recursive integral method for computing transmission eigenvalues from a finite element discretization of the continuous problem. The method, which overcomes some difficulties of existing methods, is based on eigenprojectors of compact operators. It is self-correcting, can separate nearby eigenvalues, and does not require an initial approximation based on some a priori spectral information. These features make the method well suited for the transmission eigenvalue problem whose spectrum is complicated. Numerical examples show that the method is effective and robust.

Juan Liu, Jiguang Sun, Tiara Turner

We propose an efficient numerical method for a non-selfadjoint Steklov eigenvalue problem. The Lagrange finite element is used for discretization. The convergence is proved using the spectral perturbation theory for compact operators. The non-sefadjointness of the problem leads to non-Hermitian matrix eigenvalue problem. Due to the existence of complex eigenvalues and lack of a priori spectral information, we propose a modified version of the recently developed spectral indicator method to compute (complex) eigenvalues in a given region on the complex plane. In particular, to reduce computational cost, the problem is transformed into a much smaller matrix eigenvalue problem involving the unknowns only on the boundary of the domain. Numerical examples are presented to validate the effectiveness of the proposed method.

Fang Zeng, Jiguang Sun, Liwei Xu

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.

Xia Ji, Peijun Li, Jiguang Sun

The goal of this paper is to develop numerical methods computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. We construct a nonlinear function whose values are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. Using an $H^2$-conforming finite element for the self-adjoint fourth order eigenvalue problems, we employ a secant method to compute the roots of the nonlinear function. The convergence of the proposed method is proved. In addition, a mixed finite element method is developed for the purpose of verification. Numerical examples are presented to verify the theory and demonstrate the effectiveness of the two methods.

Xiaodong Liu, Jiguang Sun

The inverse scattering problems have been popular for the past thirty years. While very successful in many cases, progress has lagged when only {\em limited-aperture} measurement is available. In this paper, we perform some elementary study to recover data that can not be measured directly. To be precise, we aim at recovering the {\em full-aperture} far field data from {\em limited-aperture} measurement. Due to the reciprocity relation, the multi-static response matrix (MSR) has a symmetric structure. Using the Green's formula and single layer potential, we propose two schemes to recover {\em full-aperture} MSR. The recovered data is tested by a recently proposed direct sampling method and the factorization method. The numerical results show the possibility to recover, at least partially, the missing data and consequently improve the reconstruction of the scatterer.

Bo Gong, Jiguang Sun

Scattering resonances arise in wave phenomena and play an important role in many applications. While extensive theoretical studies have been conducted, effective numerical computation remains limited, and most existing methods suffer from spurious modes. In this paper, we propose a spurious-mode-free method for computing scattering resonances in transmission problems. The unbounded domain is truncated using a Dirichlet-to-Neumann (DtN) map. The resonances are formulated as eigenvalues of a holomorphic Fredholm operator function, which is discretized by the finite element method. The spectrum indicator method is then used to compute the eigenvalues of the nonlinear matrix eigenvalue problems. We establish optimal order convergence and present extensive examples that demonstrate the effectiveness of the proposed method. The results are consistent with existing theoretical findings in the literature and offer new insights that may inform further theoretical developments.

Bo Gong, Takumi Sato, Jiguang Sun et al.

Meromorphic continuation of the scattering operator leads to scattering poles (resonances) in the complex plane. Despite their significance, numerical investigation of scattering poles remains limited. In this paper, we propose and analyze a numerical method to compute electromagnetic poles of perfectly conducting obstacles. The unbounded domain for the scattering problem is truncated using the DtN mapping and the poles are shown to be the eigenvalues of a holomorphic Fredholm operator function related to Maxwell's equations. Edge elements are used for discretization. The convergence is proved using the abstract approximation theory for eigenvalue problems of holomorphic Fredholm operator functions. The proposed finite element DtN approach is free of non-physical poles. A spectral indicator method is then employed to compute the resulting nonlinear matrix eigenvalue problem. Numerical examples are presented to demonstrate the effectiveness of the method.

Tanxin Zhu, Emran Hossen, Chen Zhao et al.

Purpose of Review Imaging derived fractional flow reserve (FFR) is rapidly evolving beyond conventional computational fluid dynamics (CFD) based pipelines toward machine learning (ML), deep learning (DL), and physics informed approaches that enable fast, wire free, and scalable functional assessment of coronary stenosis. This review synthesizes recent advances in CT and angiography based FFR, with particular emphasis on emerging physics informed neural networks and neural operators (PINNs and PINOs) and key considerations for their clinical translation. Recent Findings ML/DL approaches have markedly improved automation and computational speed, enabling prediction of pressure and FFR from anatomical descriptors or angiographic contrast dynamics. However, their real-world performance and generalizability can remain variable and sensitive to domain shift, due to multi-center heterogeneity, interpretability challenges, and differences in acquisition protocols and image quality. Physics informed learning introduces conservation structure and boundary condition consistency into model training, improving generalizability and reducing dependence on dense supervision while maintaining rapid inference. Recent evaluation trends increasingly highlight deployment oriented metrics, including calibration, uncertainty quantification, and quality control gatekeeping, as essential for safe clinical use. Summary The field is converging toward imaging derived FFR methods that are faster, more automated, and more reliable. While ML/DL offers substantial efficiency gains, physics informed frameworks such as PINNs and PINOs may provide a more robust balance between speed and physical consistency. Prospective multi center validation and standardized evaluation will be critical to support broad and safe clinical adoption.