arXiv subjects

Guanghui Hu

Publications and source records attributed to Guanghui Hu.

At least 37 records · Page 2Linked to original sources

arXiv2024

A multi-mesh approach for accurate computation of multi-target functionals in aerodynamics design

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.

math.NA↗

arXiv2024

An inverse obstacle problem with a single pair of Cauchy data: Laplace's equation case

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.

math.AP↗

arXiv2024

A mechanism-driven reinforcement learning framework for shape optimization of airfoils

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 Bézier 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 Bézier 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.

math.NA↗

arXiv2024

Detection of a piecewise linear crack with one incident wave

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.

math.NA↗

arXiv2024

A Nonnested Augmented Subspace Method for Kohn-Sham Equation

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.

math.NA↗

arXiv2024

Uniqueness to inverse acoustic and elastic medium scattering problems with hyper-singular source method

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.

math.AP↗

arXiv2024

Uniqueness, stability and algorithm for an inverse wave-number-dependent source problems

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.

math.NA↗

arXiv2024

Increasing stability for inverse acoustic source problems in the time domain

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.

math.AP↗

arXiv2024

Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

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 Hölder 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.

math.AP↗

arXiv2024

Direct and inverse time-harmonic scattering by Dirichlet periodic curves with local perturbations

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.

math.AP↗

arXiv2024

Time-harmonic scattering by locally perturbed periodic structures with Dirichlet and Neumann boundary conditions

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.

math.AP↗

arXiv2024

Time-harmonic elastic scattering by unbounded deterministic and random rough surfaces in three dimensions

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.

math.AP↗

arXiv2024

Inverse wave-number-dependent source problems for the Helmholtz equation with partial information on radiating period

This paper addresses a factorization method for imaging the support of a wave-number-dependent source function from multi-frequency data measured at a finite pair of symmetric receivers in opposite directions. The source function is given by the inverse Fourier transform of a compactly supported time-dependent source whose initial moment or terminal moment for radiating is unknown. Using the multi-frequency far-field data at two opposite observation directions, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the directions. A new parameter is incorporated into the design of test functions for indicating the unknown moment. The data from a finite pair of opposite directions can be used to recover the $Θ$-convex polygon of the support. Uniqueness in recovering the convex hull of the support is obtained as a by-product of our analysis using all observation directions. Similar results are also discussed with the multi-frequency near-field data from a finite pair of observation positions in three dimensions. We further comment on possible extensions to source functions with two disconnected supports. Extensive numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. The theoretical framework explored here should be seen as the frequency-domain analysis for inverse source problems in the time domain.

math.NA↗

arXiv2023

Inverse wave-number-dependent source problems for the Helmholtz equation

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 $Θ$-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.

math.NA↗

arXiv2023

Imaging a moving point source from multi-frequency data measured at one and sparse observation points (part II): near-field case in 3D

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 $Θ$-convex domain of the trajectory. Intensive 3D numerical tests with synthetic data are performed to show effectiveness and feasibility of this new algorithm.

math.NA↗

arXiv2023

V2C MXene-modified g-C3N4 for enhanced visible-light photocatalytic activity

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.

cond-mat.mtrl-sci↗

arXiv2023

Towards chemical accuracy using a multi-mesh adaptive finite element method in all-electron density functional theory

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.

physics.comp-ph↗

arXiv2023

Towards the efficient calculation of quantity of interest from steady Euler equations I: a dual-consistent DWR-based h-adaptive Newton-GMG solver

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.

math.NA↗