Xian‐Ming Gu

Xian-Ming Gu, Bruno Carpentieri, Ting-Zhu Huang et al.

In the present study, we establish two new block variants of the Conjugate Orthogonal Conjugate Gradient (COCG) and the Conjugate A-Orthogonal Conjugate Residual (COCR) Krylov subspace methods for solving complex symmetric linear systems with multiple right hand sides. The proposed Block iterative solvers can fully exploit the complex symmetry property of coefficient matrix of the linear system. We report on extensive numerical experiments to show the favourable convergence properties of our newly developed Block algorithms for solving realistic electromagnetic simulations.

Hou-Biao Li, Ming-Yan Song, Er-Jie Zhong et al.

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(τ^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. Moreover, the convergence order of this kind of method is also $O(τ^2+h^4)$ under the discrete maximum norm when the space step size is just twice the one of H-OCD method, which accelerates the computational process and makes the result much smoother to some extent. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in time direction. The results of numerical experiments are also consistent with these theoretical analysis.

Pu Yuan, Paul Zegeling, Xian-Ming Gu

In this paper, we study an evolution equation that involves a half-Laplacian operator derived from the Riesz fractional Laplacian, combined with a differential operator \(\mathcal{L}\). Using the identity $(-Δ)^{1/2}=\mathcal H(\partial_x)$, we introduce a Spectrum Doubling (SD) reformulation that transforms the original half-diffusion equation into a first-order doubled system. The reformulated system exhibits stable and unstable spectral branches, and the original half-diffusion dynamics is recovered on a suitable stable invariant subspace characterized by a compatibility condition on the initial condition. The SD reformulation provides a practical numerical advantage: the half-Laplacian is applied only to the initial condition and source term, avoiding repeated evaluation of singular integrals during time marching. For the resulting integer-order system, we develop a Boundary Value Method (BVM) and study a second-order generalized midpoint scheme. We establish its stability and second-order temporal convergence. The fully discrete scheme leads to a large Kronecker-structured linear system, which is solved efficiently by GMRES with a block $ω$-circulant preconditioner. Under simultaneous diagonalizability of the spatial discretization matrices, the preconditioner can be implemented efficiently through fast discrete transforms. Numerical experiments for three evelutionary models confirm the theoretical convergence results and demonstrate the robustness and efficiency of the proposed method, including in strongly advective regimes. The experiments also show that the approach remains effective when the Hilbert transform is evaluated numerically, and illustrate the applicability of the SD framework to a nonlocal Schrödinger-type example.

Xianming Gu, Lihui Wang, Zeyu Deng et al.

Multi-modal medical image fusion is essential for the precise clinical diagnosis and surgical navigation since it can merge the complementary information in multi-modalities into a single image. The quality of the fused image depends on the extracted single modality features as well as the fusion rules for multi-modal information. Existing deep learning-based fusion methods can fully exploit the semantic features of each modality, they cannot distinguish the effective low and high frequency information of each modality and fuse them adaptively. To address this issue, we propose AdaFuse, in which multimodal image information is fused adaptively through frequency-guided attention mechanism based on Fourier transform. Specifically, we propose the cross-attention fusion (CAF) block, which adaptively fuses features of two modalities in the spatial and frequency domains by exchanging key and query values, and then calculates the cross-attention scores between the spatial and frequency features to further guide the spatial-frequential information fusion. The CAF block enhances the high-frequency features of the different modalities so that the details in the fused images can be retained. Moreover, we design a novel loss function composed of structure loss and content loss to preserve both low and high frequency information. Extensive comparison experiments on several datasets demonstrate that the proposed method outperforms state-of-the-art methods in terms of both visual quality and quantitative metrics. The ablation experiments also validate the effectiveness of the proposed loss and fusion strategy.

Xian-Ming Gu, Ting-Zhu Huang, Guojian Yin et al.

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method [M. Heyouni, Méthode de Hessenberg Généralisée et Applications, Ph.D. Thesis, Université des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough CPU time elapsed to converge than the earlier established restarted shifted FOM, weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recent popular applications of handling the time fractional differential equations.