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1. Fast numerical solutions to direct and inverse scattering for bi-anisotropic periodic Maxwell's equations

   Nguyen, Dinh-Liem; Truong, Trung (January 2023, AMS Contemporary Mathematics)

   Nguyen, Dinh-Liem; Nguyen, Loc; Nguyen, Thi-Phong (Ed.)

   This paper is concerned with the numerical solution to the direct and inverse electromagnetic scattering problem for bi-anisotropic periodic structures. The direct problem can be reformulated as an integro-di erential equation. We study the existence and uniqueness of solution to the latter equation and analyze a spectral Galerkin method to solve it. This spectral method is based on a periodization technique which allows us to avoid the evaluation of the quasiperiodic Green's tensor and to use the fast Fourier transform in the numerical implementation of the method. For the inverse problem, we study the orthogonality sampling method to reconstruct the periodic structures from scattering data generated by only two incident fields. The sampling method is fast, simple to implement, regularization free, and very robust against noise in the data. Numerical examples for both direct and inverse problems are presented to examine the efficiency of the numerical solvers. 

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2. Reconstructing a space-dependent source term via the quasi-reversibility method

   https://doi.org/10.1090/conm/784/15755  

   Nguyen, L. H.; Vu, H. (April 2023, Contemporary mathematics)

   The aim of this paper is to solve an important inverse source problem which arises from the well-known inverse scattering problem. We propose to truncate the Fourier series of the solution to the governing equation with respect to a special basis of L2. By this, we obtain a system of linear elliptic equations. Solutions to this system are the Fourier coefficients of the solution to the governing equation. After computing these Fourier coefficients, we can directly find the desired source function. Numerical examples are presented. 

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3. Model order reduction of layered waveguides via rational Krylov fitting

   https://doi.org/10.1007/s10543-022-00922-2  

   Druskin, Vladimir; Güttel, Stefan; Knizhnerman, Leonid (April 2022, BIT Numerical Mathematics)

   Abstract Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments indicate that RKFIT computes more accurate grids than previous analytic approaches and even works in the presence of pronounced scattering resonances. Spectral adaptation effects allow for finite difference grids with dimensions near or even below the Nyquist limit. 

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4. Reconstruction techniques for quantum trees

   https://doi.org/10.1002/mma.9963  

   Avdonin, Sergei A.; Khmelnytskaya, Kira V.; Kravchenko, Vladislav V. (February 2024, Mathematical Methods in the Applied Sciences)

   The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests. 

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5. Fluid-poroviscoelastic structure interaction problem with nonlinear coupling

   Jeffrey Kuan, Suncica Canic (December 2024, Jorunal de Mathematiques Pures and Appliques)

   We prove the existence of a weak solution to a fluid-structure interaction (FSI) problem between the flow of an incompressible, viscous fluid modeled by the Navier-Stokes equations, and a poroviscoelastic medium modeled by the Biot equations. The two are nonlinearly coupled over an interface with mass and elastic energy, modeled by a reticular plate equation, which is transparent to fluid flow. The existence proof is constructive, consisting of two steps. First, the existence of a weak solution to a regularized problem is shown. Next, a weak-classical consistency result is obtained, showing that the weak solution to the regularized problem converges, as the regularization parameter approaches zero, to a classical solution to the original problem, when such a classical solution exists. While the assumptions in the first step only require the Biot medium to be poroelastic, the second step requires additional regularity, namely, that the Biot medium is poroviscoelastic. This is the first weak solution existence result for an FSI problem with nonlinear coupling involving a Biot model for poro(visco)elastic media. 

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