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This paper presents a fast and robust numerical method for reconstructing point-like sources in the time-harmonic Maxwell's equations given Cauchy data at a fixed frequency. This is an electromagnetic inverse source problem with broad applications, such as antenna synthesis and design, medical imaging, and pollution source tracing. We introduce new imaging functions and a computational algorithm to determine the number of point sources, their locations, and associated moment vectors, even when these vectors have notably different magnitudes. The number of sources and locations are estimated using significant peaks of the imaging functions, and the moment vectors are computed via explicitly simple formulas. The theoretical analysis and stability of the imaging functions are investigated, where the main challenge lies in analyzing the behavior of the dot products between the columns of the imaginary part of the Green's tensor and the unknown moment vectors. Additionally, we extend our method to reconstruct small-volume sources using an asymptotic expansion of their radiated electric field. We provide numerical examples in three dimensions to demonstrate the performance of our method.

This paper addresses the inverse scattering problem for Maxwell's equations. We first show that a bianisotropic scatterer can be uniquely determined from multi-static far-field data through the factorization analysis of the far-field operator. Next, we investigate a modified version of the orthogonality sampling method, as proposed in \cite{Le2022}, for the numerical reconstruction of the scatterer. Finally, we apply this sampling method to invert unprocessed 3D experimental data obtained from the Fresnel Institute \cite{Geffrin2009}. Numerical examples with synthetic scattering data for bianisotropic targets are also presented to demonstrate the effectiveness of the method.

Detection of abrupt spatial changes in physical properties representing unique geometric features such as buried objects, cavities, and fractures is an important problem in geophysics and many engineering disciplines. In this context, simultaneous spatial field and geometry estimation methods that explicitly parameterize the background spatial field and the geometry of the embedded anomalies are of great interest. This paper introduces an advanced inversion procedure for simultaneous estimation using the domain independence property of the Karhunen-Lo\`eve (K-L) expansion. Previous methods pursuing this strategy face significant computational challenges. The associated integral eigenvalue problem (IEVP) needs to be solved repeatedly on evolving domains, and the shape derivatives in gradient-based algorithms require costly computations of the Moore-Penrose inverse. Leveraging the domain independence property of the K-L expansion, the proposed method avoids both of these bottlenecks, and the IEVP is solved only once on a fixed bounding domain. Comparative studies demonstrate that our approach yields two orders of magnitude improvement in K-L expansion gradient computation time. Inversion studies on one-dimensional and two-dimensional seepage flow problems highlight the benefits of incorporating geometry parameters along with spatial field parameters. The proposed method captures abrupt changes in hydraulic conductivity with a lower number of parameters and provides accurate estimates of boundary and spatial-field uncertainties, outperforming spatial-field-only estimation methods.

In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method.

Studying unified model averaging estimation for situations with complicated data structures, we propose a novel model averaging method based on cross-validation (MACV). MACV unifies a large class of new and existing model averaging estimators and covers a very general class of loss functions. Furthermore, to reduce the computational burden caused by the conventional leave-subject/one-out cross validation, we propose a SEcond-order-Approximated Leave-one/subject-out (SEAL) cross validation, which largely improves the computation efficiency. In the context of non-independent and non-identically distributed random variables, we establish the unified theory for analyzing the asymptotic behaviors of the proposed MACV and SEAL methods, where the number of candidate models is allowed to diverge with sample size. To demonstrate the breadth of the proposed methodology, we exemplify four optimal model averaging estimators under four important situations, i.e., longitudinal data with discrete responses, within-cluster correlation structure modeling, conditional prediction in spatial data, and quantile regression with a potential correlation structure. We conduct extensive simulation studies and analyze real-data examples to illustrate the advantages of the proposed methods.

In this paper we obtain the Wedderburn-Artin decomposition of a semisimple group algebra associated to a direct product of finite groups. We also provide formulae for the number of all possible group codes, and their dimensions, that can be constructed in a group algebra. As particular cases, we present the complete algebraic description of the group algebra of any direct product of groups whose direct factors are cyclic, dihedral, or generalised quaternion groups. Finally, in the specific case of semisimple dihedral group algebras, we give a method to build quantum error-correcting codes, based on the CSS construction.

In this paper we consider a nonconvex unconstrained optimization problem minimizing a twice differentiable objective function with H\"older continuous Hessian. Specifically, we first propose a Newton-conjugate gradient (Newton-CG) method for finding an approximate first- and second-order stationary point of this problem, assuming the associated the H\"older parameters are explicitly known. Then we develop a parameter-free Newton-CG method without requiring any prior knowledge of these parameters. To the best of our knowledge, this method is the first parameter-free second-order method achieving the best-known iteration and operation complexity for finding an approximate first- and second-order stationary point of this problem. Finally, we present preliminary numerical results to demonstrate the superior practical performance of our parameter-free Newton-CG method over a well-known regularized Newton method.

In this paper, we consider a class of non-convex and non-smooth sparse optimization problems, which encompass most existing nonconvex sparsity-inducing terms. We show the second-order optimality conditions only depend on the nonzeros of the stationary points. We propose two damped iterative reweighted algorithms including the iteratively reweighted $\ell_1$ algorithm (DIRL$_1$) and the iteratively reweighted $\ell_2$ (DIRL$_2$) algorithm, to solve these problems. For DIRL$_1$, we show the reweighted $\ell_1$ subproblem has support identification property so that DIRL$_1$ locally reverts to a gradient descent algorithm around a stationary point. For DIRL$_2$, we show the solution map of the reweighted $\ell_2$ subproblem is differentiable and Lipschitz continuous everywhere. Therefore, the map of DIRL$_1$ and DIRL$_2$ and their inverse are Lipschitz continuous, and the strict saddle points are their unstable fixed points. By applying the stable manifold theorem, these algorithms are shown to converge only to local minimizers with randomly initialization when the strictly saddle point property is assumed.

In this contribution we study the formal ability of a multi-resolution-times lattice Boltzmann scheme to approximate isothermal and thermal compressible Navier Stokes equations with a single particle distribution. More precisely, we consider a total of 12 classical square lattice Boltzmann schemes with prescribed sets of conserved and nonconserved moments. The question is to determine the algebraic expressions of the equilibrium functions for the nonconserved moments and the relaxation parameters associated to each scheme. We compare the fluid equations and the result of the Taylor expansion method at second order accuracy for bidimensional examples with a maximum of 17 velocities and three-dimensional schemes with at most 33 velocities. In some cases, it is not possible to fit exactly the physical model. For several examples, we adjust the Navier Stokes equations and propose nontrivial expressions for the equilibria.

This paper is concerned with a Bayesian approach to testing hypotheses in statistical inverse problems. Based on the posterior distribution $\Pi \left(\cdot |Y = y\right)$, we want to infer whether a feature $\langle\varphi, u^\dagger\rangle$ of the unknown quantity of interest $u^\dagger$ is positive. This can be done by the so-called maximum a posteriori test. We provide a frequentistic analysis of this test's properties such as level and power, and prove that it is a regularized test in the sense of Kretschmann et al. (2024). Furthermore we provide lower bounds for its power under classical spectral source conditions in case of Gaussian priors. Numerical simulations illustrate its superior performance both in moderately and severely ill-posed situations.