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Although the isogeometric analysis has shown its great potential in achieving highly accurate numerical solutions of partial differential equations, its efficiency is the main factor making the method more competitive in practical simulations. In this paper, an integration of isogeometric analysis and a moving mesh method is proposed, providing a competitive approach to resolve the efficiency issue. Focusing on the Poisson equation, the implementation of the algorithm and related numerical analysis are presented in detail, including the numerical discretization of the governing equation utilizing isogeometric analysis, and a mesh redistribution technique developed via harmonic maps. It is found that the isogeometric analysis brings attractive features in the realization of moving mesh method, such as it provides an accurate expression for moving direction of mesh nodes, and allows for more choices for constructing monitor functions. Through a series of numerical experiments, the effectiveness of the proposed method is successfully validated and the potential of the method towards the practical application is also well presented with the simulation of a helium atom in Kohn--Sham density functional theory.

In this paper, a novel $h$-adaptive isogeometric solver utilizing high-order hierarchical splines is proposed to solve the all-electron Kohn--Sham equation. In virtue of the smooth nature of Kohn--Sham wavefunctions across the domain, except at the nuclear positions, high-order globally regular basis functions such as B-splines are well suited for achieving high accuracy. To further handle the singularities in the external potential at the nuclear positions, an $h$-adaptive framework based on the hierarchical splines is presented with a specially designed residual-type error indicator, allowing for different resolutions on the domain. The generalized eigenvalue problem raising from the discretized Kohn--Sham equation is effectively solved by the locally optimal block preconditioned conjugate gradient (LOBPCG) method with an elliptic preconditioner, and it is found that the eigensolver's convergence is independent of the spline basis order. A series of numerical experiments confirm the effectiveness of the $h$-adaptive framework, with a notable experiment that the numerical accuracy $10^{-3} \mathrm{~Hartree/particle}$ in the all-electron simulation of a methane molecule is achieved using only $6355$ degrees of freedom, demonstrating the competitiveness of our solver for the all-electron Kohn--Sham equation.

In this paper, we propose a novel eigenpair-splitting method, inspired by the divide-and-conquer strategy, for solving the generalized eigenvalue problem arising from the Kohn-Sham equation. Unlike the commonly used domain decomposition approach in divide-and-conquer, which solves the problem on a series of subdomains, our eigenpair-splitting method focuses on solving a series of subequations defined on the entire domain. This method is realized through the integration of two key techniques: a multi-mesh technique for generating approximate spaces for the subequations, and a soft-locking technique that allows for the independent solution of eigenpairs. Numerical experiments show that the proposed eigenpair-splitting method can dramatically enhance simulation efficiency, and its potential towards practical applications is also demonstrated well through an example of the HOMO-LUMO gap calculation. Furthermore, the optimal strategy for grouping eigenpairs is discussed, and the possible improvements to the proposed method are also outlined., Comment: arXiv admin note: text overlap with arXiv:2310.15651

In this paper, a gradient flow model is proposed for conducting ground state calculations in Wigner formalism of many-body system in the framework of density functional theory. More specifically, an energy functional for the ground state in Wigner formalism is proposed to provide a new perspective for ground state calculations of the Wigner function. Employing density functional theory, a gradient flow model is designed based on the energy functional to obtain the ground state Wigner function representing the whole many-body system. Subsequently, an efficient algorithm is developed using the operator splitting method and the Fourier spectral collocation method, whose numerical complexity of single iteration is $O(n_{\rm DoF}\log n_{\rm DoF})$. Numerical experiments demonstrate the anticipated accuracy, encompassing the one-dimensional system with up to $2^{21}$ particles and the three-dimensional system with defect, showcasing the potential of our approach to large-scale simulations and computations of systems with defect., Comment: 20 pages, 6 figures

Aerodynamic optimal design is crucial for enhancing performance of aircrafts, while calculating multi-target functionals through solving dual equations with arbitrary right-hand sides remains challenging. In this paper, a novel multi-target framework of DWR-based mesh refinement is proposed and analyzed. Theoretically, an extrapolation method is generalized to expand multi-variable functionals, which guarantees the dual equations of different objective functionals can be calculated separately. Numerically, an algorithm of calculating multi-target functionals is designed based on the multi-mesh approach, which can help to obtain different dual solutions simultaneously. One feature of our framework is the algorithm is easy to implement with the help of the hierarchical geometry tree structure and the calculation avoids the Galerkin orthogonality naturally. The framework takes a balance between different targets even when they are not the same orders of magnitude. While existing approach uses a linear combination of different components in multi-target functionals for adaptation, it introduces additional coefficients for adjusting. With each component calculated under a dual-consistent scheme, this multi-mesh framework addresses challenges such as the lift-drag ratio and other kinds of multi-target functionals, ensuring smooth convergence and precise calculations of dual solutions.

This paper is concerned with an inverse obstacle problem for the Laplace's equation. The aim is to recover the constant conductivity coefficient in the equation and the boundary of a Dirichlet polygonal obstacle from a single pair of Cauchy data. Uniqueness results are established under some a priori assumptions on the input boundary value data. A domain-defined sampling method, based on the factorization method originating from inverse acoustic scattering, has been proposed to recover both the constant conductivity coefficient and the polygonal obstacle. A hybrid strategy, which combines the sampling method and iterative scheme, is employed {\color{hgh}to reconstruct} the location and shape of the obstacle. Numerical examples indicate that our method is efficient.

This paper is concerned with inverse crack scattering problems for time-harmonic acoustic waves. We prove that a piecewise linear crack with the sound-soft boundary condition in two dimensions can be uniquely determined by the far-field data corresponding to a single incident plane wave or point source. We propose two non-iterative methods for imaging the location and shape of a crack. The first one is a contrast sampling method, while the second one is a variant of the classical factorization method but only with one incoming wave. Newton's iteration method is then employed for getting a more precise reconstruction result. Numerical examples are presented to show the effectiveness of the proposed hybrid method.

In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent field iterative algorithm which requires to solve the Kohn-Sham equation directly in each adaptive finite element space, our algorithm transforms the Kohn-Sham equation into some linear boundary value problems of the same scale in each adaptive finite element space, and then the wavefunctions derived from the linear boundary value problems are corrected by solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation directly, a significant improvement for the solving efficiency can be obtained. In addition, the adaptive moving mesh technique is used to generate the nonnested adaptive mesh for the nonnested augmented subspace method according to the singularity of the approximate wavefunctions. The modified Hessian matrix of the approximate wavefunctions is used as the metric matrix to redistribute the mesh. Through the moving mesh adaptive technique, the redistributed mesh is almost optimal. A number of numerical experiments are carried out to verify the efficiency and the accuracy of the proposed algorithm.

This paper is concerned with inverse scattering problems of determining the support of an isotropic and homogeneous penetrable body from knowledge of multi-static far-field patterns in acoustics and in linear elasticity. The normal derivative of the total fields admits no jump on the interface of the scatterer in the trace sense. If the contrast function of the refractive index function or the density function has a positive lower bound near the boundary, we propose a hyper-singular source method to prove uniqueness of inverse scattering with all incoming plane waves at a fixed energy. It is based on subtle analysis on the leading part of the scattered field when hyper-singular sources caused by the first derivative of the fundamental solution approach to a boundary point. As a by-product, we show that this hyper-singular method can be also used to determine the boundary value of a Holder continuous refractive index function in acoustics or a Holder continuous density function in linear elasticity.

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all observation directions in some neighborhood of a fixed direction $\hat{x}$ and for all wave numbers k belonging to a finite interval $(0,K)$. In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval $K>1$. In three dimensions we establish stability estimates of the $L^2$-norm and $H^{-1}$-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of H\"older type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.

This paper is concerned with inverse source problems for the acoustic wave equation in the full space R^3, where the source term is compactly supported in both time and spatial variables. The main goal is to investigate increasing stability for the wave equation in terms of the interval length of given parameters (e.g., bandwith of the temporal component of the source function). We establish increasing stability estimates of the L^2 -norm of the source function by using only the Dirichlet boundary data. Our method relies on the Huygens principle, the Fourier transform and explicit bounds for the continuation of analytic functions., Comment: 26pages,7figures. arXiv admin note: substantial text overlap with arXiv:2402.15973

This is a continuation of the authors' previous work (A. Kirsch, Math. Meth. Appl. Sci., 45 (2022): 5737-5773.) on well-posedness of time-harmonic scattering by locally perturbed periodic curves of Dirichlet kind. The scattering interface is supposed to be given by a non-self-intersecting Lipschitz curve. We study properties of the Green's function and prove new well-posedness results for scattering of plane waves at a propagative wave number. In such a case there exist guided waves to the unperturbed problem, which are also known as Bounded States in the Continuity (BICs) in physics. In this paper uniqueness of the forward scattering follows from an orthogonal constraint condition enforcing on the total field to the unperturbed scattering problem. This constraint condition, which is also valid under the Neumann boundary condition, is derived from the singular perturbation arguments and also from the approach of approximating a plane wave by point source waves. For the inverse problem of determining the defect, we prove several uniqueness results using a finite or infinite number of point source and plane waves, depending on whether a priori information on the size and height of the defect is available.

In this paper, a novel mechanism-driven reinforcement learning framework is proposed for airfoil shape optimization. To validate the framework, a reward function is designed and analyzed, from which the equivalence between the maximizing the cumulative reward and achieving the optimization objectives is guaranteed theoretically. To establish a quality exploration, and to obtain an accurate reward from the environment, an efficient solver for steady Euler equations is employed in the reinforcement learning method. The solver utilizes the B\'ezier curve to describe the shape of the airfoil, and a Newton-geometric multigrid method for the solution. In particular, a dual-weighted residual-based h-adaptive method is used for efficient calculation of target functional. To effectively streamline the airfoil shape during the deformation process, we introduce the Laplacian smoothing, and propose a B\'ezier fitting strategy, which not only remits mesh tangling but also guarantees a precise manipulation of the geometry. In addition, a neural network architecture is designed based on an attention mechanism to make the learning process more sensitive to the minor change of the airfoil geometry. Numerical experiments demonstrate that our framework can handle the optimization problem with hundreds of design variables. It is worth mentioning that, prior to this work, there are limited works combining such high-fidelity partial differential equatons framework with advanced reinforcement learning algorithms for design problems with such high dimensionality., Comment: 25 pages

This paper is concerned with an inverse wavenumber/frequency-dependent source problem for the Helmholtz equation. In two and three dimensions, the unknown source term is supposed to be compactly supported in spatial variables but independent on one spatial variable. The dependence of the source function on wavenumber/frequency is supposed to be unknown. Based on the Dirichlet-Laplacian and Fourier-Transform methods, we develop two effcient non-iterative numerical algorithms to recover the wavenumber-dependent source. Uniqueness proof and increasing stability analysis are carried out in terms of the boundary measurement data of Dirichlet kind. Numerical experiments are conducted to illustrate the effectiveness and efficiency of the proposed methods., Comment: 26 pages, 35 figures

The paper is concerned with well-posedness of TE and TM polarizations of time-harmonic electromagnetic scattering by perfectly conducting periodic surfaces and periodically arrayed obstacles with local perturbations. The classical Rayleigh Expansion radiation condition does not always lead to well-posedness of the Helmholtz equation even in unperturbed periodic structures. We propose two equivalent radiation conditions to characterize the radiating behavior of time-harmonic wave fields incited by a source term in an open waveguide under impenetrable boundary conditions. With these open waveguide radiation conditions, uniqueness and existence of time-harmonic scattering by incoming point source waves, plane waves and surface waves from locally perturbed periodic structures are established under either the Dirichlet or Neumann boundary condition. A DtN operator without using the Green's function is constructed for proving well-posedness of perturbed scattering problems.

In this paper, we investigate well-posedness of time-harmonic scattering of elastic waves by unbounded rigid rough surfaces in three dimensions. The elastic scattering is caused by an $L^2$ function with a compact support in the $x_3$-direction, and both deterministic and random surfaces are investigated via the variational approach. The rough surface in a deterministic setting is assumed to be Lipschitz and lie within a finite distance of a flat plane, and the scattering is caused by an inhomogeneous term in the elastic wave equation whose support lies within some finite distance of the boundary. For the deterministic case, a stability estimate of elastic scattering by rough surface is shown at an arbitrary frequency. It is noticed that all constants in {\it a priori} bounds are bounded by explicit functions of the frequency and geometry of rough surfaces. Furthermore, based on this explicit dependence on the frequency together with the measurability and $\mathbb{P}$-essentially separability of the randomness, we obtain a similar bound for the solution of the scattering by random surfaces.

Increasing the efficiency of charge transfer and separation efficiency of photogenerated carriers are still the main challenges in the field of semiconductor-based photocatalysts. Herein, we synthesized g-C3N4@V2C MXene photocatalyst by modifying g-C3N4 using V2C MXene. The prepared photocatalyst exhibited outstanding photocatalytic performance under visible light. The degradation efficiency of methyl orange by g-C3N4@V2C MXene photocatalyst was as high as 94.5%, which is 1.56 times higher than that by g-C3N4. This was attributed to the V2C MXene inhibiting the rapid recombination of photogenerated carriers and facilitating rapid transfer of photogenerated electrons (e) from g-C3N4 to MXene. Moreover, g-C3N4@V2C MXene photocatalyst showed good cycling stability. The photocatalytic performance was higher than 85% after three cycles. Experiments to capture free radicals revealed that superoxide radicals (02) are the main contributors to the photocatalytic activity. Thus, the proposed g-C3N4@V2C MXene photocatalyst is a promising visible-light catalyst., Comment: 20 pages, 9 figures

Chemical accuracy serves as an important metric for assessing the effectiveness of the numerical method in Kohn--Sham density functional theory. It is found that to achieve chemical accuracy, not only the Kohn--Sham wavefunctions but also the Hartree potential, should be approximated accurately. Under the adaptive finite element framework, this can be implemented by constructing the \emph{a posteriori} error indicator based on approximations of the aforementioned two quantities. However, this way results in a large amount of computational cost. To reduce the computational cost, we propose a novel multi-mesh adaptive method, in which the Kohn--Sham equation and the Poisson equation are solved in two different meshes on the same computational domain, respectively. With the proposed method, chemical accuracy can be achieved with less computational consumption compared with the adaptive method on a single mesh, as demonstrated in a number of numerical experiments., Comment: 19pages, 17 figures

In \cite{wang2023towards}, a dual-consistent dual-weighted residual-based $h$-adaptive method has been proposed based on a Newton-GMG framework, towards the accurate calculation of a given quantity of interest from Euler equations. The performance of such a numerical method is satisfactory, i.e., the stable convergence of the quantity of interest can be observed in all numerical experiments. In this paper, we will focus on the efficiency issue to further develop this method, since efficiency is vital for numerical methods in practical applications such as the optimal design of the vehicle shape. Three approaches are studied for addressing the efficiency issue, i.e., i). using convolutional neural networks as a solver for dual equations, ii). designing an automatic adjustment strategy for the tolerance in the $h$-adaptive process to conduct the local refinement and/or coarsening of mesh grids, and iii). introducing OpenMP, a shared memory parallelization technique, to accelerate the module such as the solution reconstruction in the method. The feasibility of each approach and numerical issues are discussed in depth, and significant acceleration from those approaches in simulations can be observed clearly from a number of numerical experiments. In convolutional neural networks, it is worth mentioning that the dual consistency plays an important role to guarantee the efficiency of the whole method and that unstructured meshes are employed in all simulations., Comment: In this papers, we use the CNNs architecture to solve the dual equations problem

In the classic implementation of the LOBPCG method, orthogonalization and the R-R (Rayleigh-Ritz) procedure cost nonignorable CPU time. Especially this consumption could be very expensive to deal with situations with large block sizes. In this paper, we propose an orthogonalization-free framework of implementing the LOBPCG method for SCF (self-consistent field) iterations in solving the Kohn-Sham equation. In this framework, orthogonalization is avoided in calculations, which can decrease the computational complexity. And the R-R procedure is implemented parallelly through OpenMP, which can further reduce computational time. During numerical experiments, an effective preconditioning strategy is designed, which can accelerate the LOBPCG method remarkably. Consequently, the efficiency of the LOBPCG method can be significantly improved. Based on this, the SCF iteration can solve the Kohn-Sham equation efficiently. A series of numerical experiments are inducted to demonstrate the effectiveness of our implementation, in which significant improvements in computational time can be observed., Comment: There are some mistakes in Fig 4.2 & 4.3, which can lead to some wrong conclusion dismatching the theory

This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a $\Theta$-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources.

In this paper, we introduce a frequency-domain approach to extract information on the trajectory of a moving point source. The method hinges on the analysis of multi-frequency near-field data recorded at one and sparse observation points in three dimensions. The radiating period of the moving point source is supposed to be supported on the real axis and a priori known. In contrast to inverse stationary source problems, one needs to classify observable and non-observable measurement positions. The analogue of these concepts in the far-field regime were firstly proposed in the authors' previous paper (SIAM J. Imag. Sci., 16 (2023): 1535-1571). In this paper we shall derive the observable and non-observable measurement positions for straight and circular motions in $\R^3$. In the near-field case, we verify that the smallest annular region centered at an observable position that contains the trajectory can be imaged for an admissible class of orbit functions. Using the data from sparse observable positions, it is possible to reconstruct the $\Theta$-convex domain of the trajectory. Intensive 3D numerical tests with synthetic data are performed to show effectiveness and feasibility of this new algorithm., Comment: arXiv admin note: substantial text overlap with arXiv:2212.14236

This paper is concerned with an inverse transmission problem for recovering the shape of a penetrable rectangular grating sitting on a perfectly conducting plate. We consider a general transmission problem with the coefficient \lambda\neq 1 which covers the TM polarization case. It is proved that a rectangular grating profile can be uniquely determined by the near-field observation data incited by a single plane wave and measured on a line segment above the grating. In comparision with the TE case (\lambda=1), the wave field cannot lie in H^2 around each corner point, bringing essential difficulties in proving uniqueness with one plane wave. Our approach relies on singularity analysis for Helmholtz transmission problems in a right-corner domain and also provides an alternative idea for treating the TE transmission conditions which were considered in the authors' previous work [Inverse Problem, 39 (2023): 055004.]

In this paper we study the variational method and integral equation methods for a conical diffraction problem for imperfectly conducting gratings modeled by the impedance boundary value problem of the Helmholtz equation in periodic structures. We justify the strong ellipticity of the sesquilinear form corresponding to the variational formulation and prove the uniqueness of solutions at any frequency. Convergence of the finite element method using the transparent boundary condition (Dirichlet-to-Neumann mapping) is verified. The boundary integral equation method is also discussed.

Suppose that a plane wave is incident onto an impenetrable grating profile of Dirichlet or Impedance type or a penetrable grating. The grating interface is assumed to be given by a Lipschitz function in two dimensions. We derive stability estimate of the grating diffraction problem via variational method with an explicit dependence of solutions on the incident wavenumber and incident angle.

In [Dai et al, Multi. Model. Simul., 2020], a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory, based on which a linearized method was developed in [Hu, et al, EAJAM, accepted] for further improving the numerical efficiency. In this paper, a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model. Temporally, the convergence, the asymptotic stability, as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works, while spatially, the convergence of the h-adaptive mesh method is demonstrated following [Chen et al, Multi. Model. Simul., 2014], with a key study on the boundedness of the Kohn-Sham potential for the all-electron Kohn-Sham model. Numerical examples confirm the theoretical results very well.

The dual consistency is an important issue in developing stable DWR error estimation towards the goal-oriented mesh adaptivity. In this paper, such an issue is studied in depth based on a Newton-GMG framework for the steady Euler equations. Theoretically, the numerical framework is redescribed using the Petrov-Galerkin scheme, based on which the dual consistency is depicted. A boundary modification technique is discussed for preserving the dual consistency within the Newton-GMG framework. Numerically, a geometrical multigrid is proposed for solving the dual problem, and a regularization term is designed to guarantee the convergence of the iteration. The following features of our method can be observed from numerical experiments, i). a stable numerical convergence of the quantity of interest can be obtained smoothly for problems with different configurations, and ii). towards accurate calculation of quantity of interest, mesh grids can be saved significantly using the proposed dual-consistent DWR method, compared with the dual-inconsistent one., Comment: In this work, we validated the dual consistency under the Newton-GMG framework. Based on the previous work, we further constructed the h-adaptivity method for the steady Euler equations in the AFVM4CFD package

We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.

We propose a multi-frequency algorithm for imaging the trajectory of a moving point source from one and sparse far-field observation directions in the frequency domain. The starting and terminal time points of the moving source are both supposed to be known. We introduce the concept of observable directions (angles) in the far-field region and derive all observable directions (angles) for straight and circular motions. At an observable direction, it is verified that the smallest trip containing the trajectory and perpendicular to the direction can be imaged, provided the orbit function possesses a certain monotonical property. Without the monotonicity one can only expect to recover a thinner strip. The far-field data measured at sparse observable directions can be used to recover the $\Theta$-convex domain of the trajectory. Both two- and three-dimensional numerical examples are implemented to show effectiveness and feasibility of the approach.

This paper is concerned with a direct sampling method for imaging the support of a frequency-dependent source term embedded in a homogeneous and isotropic medium. The source term is given by the Fourier transform of a time-dependent source whose radiating period in the time domain is known. The time-dependent source is supposed to be stationary in the sense that its compact support does not vary along the time variable. Via a multi-frequency direct sampling method, we show that the smallest strip containing the source support and perpendicular to the observation direction can be recovered from far-field patterns at a fixed observation angle. With multiple but sparse observation directions, the shape of the convex hull of the source support can be recovered. The frequency-domain analysis performed here can be used to handle inverse time-dependent source problems. Our algorithm has low computational overhead and is robust against noise. Numerical experiments in both two and three dimensions have proved our theoretical findings.

In a dual weighted residual method based on the finite element framework, the Galerkin orthogonality is an issue that prevents solving the dual equation in the same space as the one for the primal equation. In the literature, there have been two popular approaches to constructing a new space for the dual problem, i.e., refining mesh grids ($h$-approach) and raising the order of approximate polynomials ($p$-approach). In this paper, a novel approach is proposed for the purpose based on the multiple-precision technique, i.e., the construction of the new finite element space is based on the same configuration as the one for the primal equation, except for the precision in calculations. The feasibility of such a new approach is discussed in detail in the paper. In numerical experiments, the proposed approach can be realized conveniently with C++ \textit{template}. Moreover, the new approach shows remarkable improvements in both efficiency and storage compared with the $h$-approach and the $p$-approach. It is worth mentioning that the performance of our approach is comparable with the one through a higher order interpolation ($i$-approach) in the literature. The combination of these two approaches is believed to further enhance the efficiency of the dual weighted residual method.

This paper is concerned with the inverse diffraction problems by a periodic curve with Dirichlet boundary condition in two dimensions. It is proved that the periodic curve can be uniquely determined by the near-field measurement data corresponding to infinitely many incident plane waves with distinct directions at a fixed frequency. Our proof is based on Schiffer's idea which consists of two ingredients: i) the total fields for incident plane waves with distinct directions are linearly independent, and ii) there exist only finitely many linearly independent Dirichlet eigenfunctions in a bounded domain or in a closed waveguide under additional assumptions on the waveguide boundary. Based on the Rayleigh expansion, we prove that the phased near-field data can be uniquely determined by the phaseless near-field data in a bounded domain, with the exception of a finite set of incident angles. Such a phase retrieval result leads to a new uniqueness result for the inverse grating diffraction problem with phaseless near-field data at a fixed frequency. Since the incident direction determines the quasi-periodicity of the boundary value problem, our inverse issues are different from the existing results of [Htttlich & Kirsch, Inverse Problems 13 (1997): 351-361] where fixed-direction plane waves at multiple frequencies were considered.

We investigate inverse diffraction problems for penetrable gratings in a piecewise constant medium. In the TE polarization case, it is proved that a binary grating profile together with the refractive index beneath it can be uniquely determined by the near-field observation data incited by a single plane wave and measured on a line segment above the grating. Our approach relies on the expansion of solutions to the Helmholtz equation and the corner singularity analysis of solutions to the inhomogeneous Laplace equation with a piecewise continuous source term in a sector. This paper also contributes to corner scattering theory for the Helmholtz equation in a special non-convex domain., Comment: 25 pages,4 figures

This paper is concerned with the factorization method with a single far-field pattern to recover an arbitrary convex polygonal scatterer/source in linear elasticity. The approach also applies to the compressional (resp. shear) part of the far-field pattern excited by a single compressional (resp. shear) plane wave. The one-wave factorization is based on the scattering data for a priori given testing scatterers. It can be regarded as a domain-defined sampling method and does not require forward solvers. We derive the spectral system of the far-field operator for rigid disks and show that, using testing disks, the one-wave factorization method can be justified independently of the classical factorization method.

This paper studies the PML method for wave scattering in a half space of homogeneous medium bounded by a two-dimensional, perfectly conducting, and locally defected periodic surface, and develops a high-accuracy boundary-integral-equation (BIE) solver. Along the vertical direction, we place a PML to truncate the unbounded domain onto a strip and prove that the PML solution converges linearly to the true solution in the physical subregion of the strip with the PML thickness. Laterally, we divide the unbounded strip into three regions: a region containing the defect and two semi-waveguide regions, separated by two vertical line segments. In both semi-waveguides, we prove the well-posedness of an associated scattering problem so as to well define a Neumann-to-Dirichlet (NtD) operator on the associated vertical segment. The two NtD operators, serving as exact lateral boundary conditions, reformulate the unbounded strip problem as a boundary value problem onto the defected region. Due to the periodicity of the semi-waveguides, both NtD operators turn out to be closely related to a Neumann-marching operator, governed by a nonlinear Riccati equation. It is proved that the Neumann-marching operators are contracting, so that the PML solution decays exponentially fast along both lateral directions. The consequences culminate in two opposite aspects. Negatively, the PML solution cannot exponentially converge to the true solution in the whole physical region of the strip. Positively, from a numerical perspective, the Riccati equations can now be efficiently solved by a recursive doubling procedure and a high-accuracy PML-based BIE method so that the boundary value problem on the defected region can be solved efficiently and accurately. Numerical experiments demonstrate that the PML solution converges exponentially fast to the true solution in any compact subdomain of the strip., Comment: 31 pages, 9 figures

In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its solution and the solution of stationary Schr\"odinger equation. Then, a numerical method is designed for solving proposed eigenvalue problem in one dimensional case, which can be briefly described by i) a simplified model is derived based on a quantum hydrodynamic model [Z. Cai et al, J. Math. Chem., 2013] to reduce the dimension of the problem, ii) an imaginary time propagation method is designed for solving the model, and numerical techniques such as solution reconstruction are proposed for the feasibility of the method. Results of several numerical experiments verify our method, in which the potential application of the method for large scale system is demonstrated by examples with density functional theory.

Based on the numerical method proposed in [G. Hu, X. Xie, F. Xu, J. Comput. Phys., 355 (2018), 436-449.] for Kohn-Sham equation, further improvement on the efficiency is obtained in this paper by i). designing a numerical method with the strategy of separately handling the nonlinear Hartree potential and exchange-correlation potential, and ii).parallelizing the algorithm in an eigenpairwise approach. The feasibility of two approaches are analyzed in detail, and the new algorithm is described completely. Compared with previous results, a significant improvement of numerical efficiency can be observed from plenty of numerical experiments, which make the new method more suitable for the practical problems.

The factorization method by Kirsch (1998) provides a necessary and sufficient condition for characterizing the shape and position of an unknown scatterer by using far-field patterns of infinitely many time-harmonic plane waves at a fixed frequency. This paper is concerned with the factorization method with a single far-field pattern to recover a convex polygonal scatterer/source. Its one-wave version relies on the absence of analytical continuation of the scattered/radiated wave-fields in corner domains. It can be regarded as a domain-defined sampling method and does not require forward solvers. In this paper we provide a rigorous mathematical justification of the one-wave factorization method and present some preliminary numerical examples. In particular, the proposed scheme can be interpreted as a model-driven and data-driven method, because it essentially depends on the scattering model and a priori given \emph{sample data}.

It is proved that an inhomogeneous medium whose boundary contains a weakly singular point of arbitrary order scatters every incoming wave. Similarly, a compactly supported source term with weakly singular points on the boundary always radiates acoustic waves. These results imply the absence of non-scattering energies and non-radiating sources in a domain that is not $C^\infty$-smooth. Local uniqueness results with a single far-field pattern are obtained for inverse source and inverse medium scattering problems. Our arguments provide a sufficient condition of the surface under which solutions to the Helmholtz equation admits no analytical continuation.

All-electron calculations play an important role in density functional theory, in which improving computational efficiency is one of the most needed and challenging tasks. In the model formulations, both nonlinear eigenvalue problem and total energy minimization problem pursue orthogonal solutions. Most existing algorithms for solving these two models invoke orthogonalization process either explicitly or implicitly in each iteration. Their efficiency suffers from this process in view of its cubic complexity and low parallel scalability in terms of the number of electrons for large scale systems. To break through this bottleneck, we propose an orthogonalization-free algorithm framework based on the total energy minimization problem. It is shown that the desired orthogonality can be gradually achieved without invoking orthogonalization in each iteration. Moreover, this framework fully consists of Basic Linear Algebra Subprograms (BLAS) operations and thus can be naturally parallelized. The global convergence of the proposed algorithm is established. We also present a precondition technique which can dramatically accelerate the convergence of the algorithm. The numerical experiments on all-electron calculations show the efficiency and high scalability of the proposed algorithm., Comment: 20 pages, 7 figures, 4 tables

This paper concerns the Bayesian approach to inverse acoustic scattering problems of inferring the position and shape of a sound-soft obstacle from phaseless far-field data generated by point source waves. To improve the convergence rate, we use the Gibbs sampling and preconditioned Crank-Nicolson (pCN) algorithm with random proposal variance to implement the Markov chain Monte Carlo (MCMC) method. This usually leads to heavy computational cost, since the unknown obstacle is parameterized in high dimensions. To overcome this challenge, we examine a surrogate model constructed by the generalized polynomial chaos (gPC) method to reduce the computational cost. Numerical examples are provided to illustrate the effectiveness of the proposed method.

In this work, we investigate the shape identification and coefficient determination associated with two time-dependent partial differential equations in two dimensions. We consider the inverse problems of determining a convex polygonal obstacle and the coefficient appearing in the wave and Schr\"odinger equations from a single dynamical data along with the time. With the far field data, we first prove that the sound speed of the wave equation together with its contrast support of convex-polygon type can be uniquely determined, then establish a uniqueness result for recovering an electric potential as well as its support appearing in the Schr\"odinger equation. As a consequence of these results, we demonstrate a uniqueness result for recovering the refractive index of a medium from a single far field pattern at a fixed frequency in the time-harmonic regime.