Scattered data interpolation aims to reconstruct a continuous (smooth) function that approximates the underlying function by fitting (meshless) data points. There are extensive applications of scattered data interpolation in computer graphics, fluid dynamics, inverse kinematics, machine learning, etc. In this paper, we consider a novel generalized Mercel kernel in the reproducing kernel Banach space for scattered data interpolation. The system of interpolation equations is formulated as a multilinear system with a structural tensor, which is an absolutely and uniformly convergent infinite series of symmetric rank-one tensors. Then we design a fast numerical method for computing the product of the structural tensor and any vector in arbitrary precision. Whereafter, a scalable optimization approach equipped with limited-memory BFGS and Wolfe line-search techniques is customized for solving these multilinear systems. Using the Łojasiewicz inequality, we prove that the proposed scalable optimization approach is a globally convergent algorithm and possesses a linear or sublinear convergence rate. Numerical experiments illustrate that the proposed scalable optimization approach can improve the accuracy of interpolation fitting and computational efficiency.

In this paper, we propose a novel Lagrange multiplier approach, named zero-factor (ZF) approach to solve a series of gradient flow problems. The numerical schemes based on the new algorithm are unconditionally energy stable with the original energy and do not require any extra assumption conditions. We also prove that the ZF schemes with specific zero factors lead to the popular SAV-type method. To reduce the computation cost and improve the accuracy and consistency, we propose a zero-factor approach with relaxation, which we named the relaxed zero-factor (RZF) method, to design unconditional energy stable schemes for gradient flows. The RZF schemes can be proved to be unconditionally energy stable with respect to a modified energy that is closer to the original energy, and provide a very simple calculation process. The variation of the introduced zero factor is highly consistent with the nonlinear free energy which implies that the introduced ZF method is a very efficient way to capture the sharp dissipation of nonlinear free energy. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

This paper is concerned with a $C^1$-conforming Gauss collocation approximation to the solution of a model two-dimensional elliptic boundary problem. Superconvergence phenomena for the numerical solution at mesh nodes, at roots of a special Jacobi polynomial, and at the Lobatto and Gauss lines are identified with rigorous mathematical proof, when tensor products of $C^1$ piecewise polynomials of degree not more than $k,$ $k≥3$ are used. This method is shown to be superconvergent with $(2k−2)$-th order accuracy in both the function value and its gradient at mesh nodes, $(k+2)$-th order accuracy at all interior roots of a special Jacobi polynomial, $(k+1)$-th order accuracy in the gradient along the Lobatto lines, and $k$-th order accuracy in the second-order derivative along the Gauss lines. Numerical experiments are presented to indicate that all the superconvergence rates are sharp.

Computed tomography (CT) reconstruction with sparse-view projections is a challenging problem in medical imaging. The learning-based methods lack generalization ability and mathematical interpretability. Since the model-based iterative reconstruction (IR) methods need inner gradient-based iterations to deal with the CT system matrix, the algorithms may not be efficient enough, and IR methods with deep networks have no convergence guarantees. In this paper, we propose an efficient deep semi-proximal iterative method (DeepSPIM) to reconstruct CT images from sparseview projections. Unlike the existing IR methods, a carefully designed semi-proximal term is introduced to make the system matrix-related subproblem solvable. Theoretically, we give some useful mathematical analysis, including the existence of the solutions to the reconstruction model with an implicit image prior, the global convergence of the proposed method under gradient step denoiser assumption. Experimental results show that DeepSPIM is efficient and outperforms the closely related state-ofthe-art methods regarding quantitative image quality values, details preservation, and structure recovery.

The fluctuation of mRNA molecule numbers within an isogenic cell population is primarily attributed to randomly switching between active (ON) and inactive (OFF) periods of gene transcription. In most studies the waiting-times for ON or OFF states are modeled as exponential distributions. However, increasing data suggest that the residence durations at ON or OFF are non-exponential distributed for which the traditional master equations cannot be presented. By combining Kolmogorov forward equations with alternating renewal processes, we present a novel method to compute the average transcription level and its noise by circumventing the bottleneck of master equations under gene ON and OFF switch. As an application, we consider lifetimes of OFF and ON states having Erlang distributions. We show that: (i) multiple steps from OFF to ON force the oscillating transcription while multiple steps from ON to OFF accelerate the transcription, (ii) the increase of steps between ON and OFF rapidly reduces the transcription noise to approach its minimum value. This suggests that a large number of steps between ON and OFF are not needed in the model to capture the stochastic transcription data. Our computation approach can be further used to treat a series of transcription cycles which are non-lattice distributed.

Body attitude coordination plays an important role in multi-airplane synchronization. In this paper, we study the flocking dynamics of a modified model for body attitude coordination. In contrast to the original body attitude alignment models in Degond et al. (Math. Models Methods Appl. Sci., 27(6): 1005–1049, 2017) and Ha et al. (Discrete Contin. Dyn. Syst., 40(4): 2037–2060, 2020), we introduce the velocity alignment term and assume the velocity of each agent is variable. More precisely, the adjoint coefficient will vary with the linked individual changes. In this case, synchronization would include the body attitude alignment and velocity alignment. It will generate a new collective behaviour which is called body attitude flocking. As results, we present two sufficient frameworks leading to the body attitude flocking by technique estimates. Also, we show the finite-in-time stability of the system which is valid on any finite time interval. In addition, we formally derive a kinetic model of the model for body attitude coordination using the BBGKY hierarchy. We prove the well-posedness of the kinetic equation and show a rigorous justification for the mean-field limit of our model. Moreover, we present a sufficient condition for asymptotic flocking in the kinetic model. Finally, we also give the numerical simulations to verify our analysis results.

This paper presents a Fourier matching method to rigorously study resonances in a sound-hard slab with a finite number of narrow cylindrical holes. The cross sections of the holes, of diameters O(h) for h ≪ 1, can be arbitrarily shaped. Outside the slab, a sound field can be represented in terms of its normal derivatives on the apertures of the holes. Inside each hole, the field can be represented in terms of a countable Fourier basis due to the zero Neumann boundary condition on the side surface. The countably infinite Fourier coefficients for all the holes constitute the unknowns. Matching the two field representatives leads to a countable-dimensional linear system governing the unknowns. Due to the invertibility of a principal submatrix of the infinite-dimensional coefficient matrix, we reduce the linear system to a finitedimensional one. Resonances are those when the finite-dimensional linear system becomes singular. We derive asymptotic formulae for the resonances in the subwavelength structure for h ≪ 1. They reveal that a sound field with its real frequency close to a resonance frequency can be enhanced by a magnitude O(h −2 ). Numerical experiments are carried out to validate the proposed resonance formulae.

In this paper, we present a novel adaptive sampling strategy for enhancing the performance of physics-informed neural networks (PINNs) in addressing inverse problems with low regularity and high dimensionality. The framework is based on failure-informed PINNs, which was recently developed in [Gao et al., SIAM J. Sci. Comput., 45(4), 2023]. Specifically, we employ a truncated Gaussian mixture model to estimate the failure probability; this model additionally serves as an error indicator in our adaptive strategy. New samples for further computation are also produced using the truncated Gaussian mixture model. To describe the new framework, we consider two important classes of inverse problems: the inverse conductivity problem in electrical impedance tomography and the inverse source problem in a parabolic system. The effectiveness of our method is demonstrated through a series of numerical examples.