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Peijun Li

Publications and source records attributed to Peijun Li.

At least 55 records · Page 3Linked to original sources
arXiv2021

Inverse Elastic Scattering for a Random Potential

This paper is concerned with an inverse scattering problem for the time-harmonic elastic wave equation with a random potential. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic generalized Gaussian random field with the covariance operator being described by a classical pseudo-differential operator. The goal is to determine the principal symbol of the covariance operator from the scattered wave measured in a bounded domain which has a positive distance from the domain of the potential. For such a rough potential, the well-posedness of the direct scattering problem in the distribution sense is established by studying an equivalent Lippmann--Schwinger integral equation. For the inverse scattering problem, it is shown with probability one that the principal symbol of the covariance operator can be uniquely determined by the amplitude of the scattered waves averaged over the frequency band from a single realization of the random potential. The analysis employs the Born approximation in high frequency, asymptotics of the Green tensor for the elastic wave equation, and microlocal analysis for the Fourier integral operators.

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arXiv2021

Stability for an inverse source problem of the damped biharmonic plate equation

This paper is concerned with the stability of the inverse source problem for the damped biharmonic plate equation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the constant damping coefficient. The analysis employs Carleman estimates and time decay estimates for the damped plate wave equation to obtain an exact observability bound and depends on the study of the resonance-free region and an upper bound of the resolvent of the biharmonic operator with respect to the complex wavenumber.

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arXiv2021

Inverse random potential scattering for elastic waves

This paper is concerned with the inverse elastic scattering problem for a random potential in three dimensions. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic Gaussian random field whose covariance operator is a classical pseudo-differential operator. Given the potential, the direct scattering problem is shown to be well-posed in the distribution sense by studying the equivalent Lippmann--Schwinger integral equation. For the inverse scattering problem, we demonstrate that the microlocal strength of the random potential can be uniquely determined with probability one by a single realization of the high frequency limit of the averaged compressional or shear backscattered far-field pattern of the scattered wave. The analysis employs the integral operator theory, the Born approximation in the high frequency regime, the microlocal analysis for the Fourier integral operators, and the ergodicity of the wave field.

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arXiv2021

Stability for an inverse source problem of the biharmonic operator

In this paper, we study for the first time the stability of the inverse source problem for the biharmonic operator with a compactly supported potential in $\mathbb R^3$. Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr$\ddot{\rm o}$dinger operator $Δ^2 + V(x)$ on a ball which contains the support of the potential $V$. We prove a Weyl-type law for the upper bounds of spherical normal derivatives of both the eigenfunctions $ϕ$ and their Laplacian $Δϕ$ corresponding to the bi-Schr$\ddot{\rm o}$dinger operator. This type of upper bounds was proved by Hassell and Tao for the Schr$\ddot{\rm o}$dinger operator. Secondly, we investigate the meromorphic continuation of the resolvent of the bi-Schr$\ddot{\rm o}$dinger operator and prove the existence of a resonance-free region and an estimate of $L^2_{\rm comp} - L^2_{\rm loc}$ type for the resolvent. As an application, we prove a bound of the analytic continuation of the data from the given data to the higher frequency data. Finally, we derive the stability estimate which consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases.

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arXiv2021

An adaptive finite element DtN method for the elastic wave scattering problem in three dimensions

Consider the elastic scattering of an incident wave by a rigid obstacle in three dimensions, which is formulated as an exterior problem for the Navier equation. By constructing a Dirichlet-to-Neumann (DtN) operator and introducing a transparent boundary condition, the scattering problem is reduced equivalently to a boundary value problem in a bounded domain. The discrete problem with the truncated DtN operator is solved by using the a posteriori error estimate based adaptive finite element method. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to converge exponentially with respect to the truncation parameter. Moreover, the generalized Woodbury matrix identity is utilized to solve the resulting linear system efficiently. Numerical experiments are presented to demonstrate the superior performance of the proposed method.

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arXiv2021

An inverse source problem for the stochastic wave equation

This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.

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arXiv2020

Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLift

This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by examining the well-posedness and regularity of the solution for the equivalent stochastic two-point boundary value problem in the frequency domain. For the inverse problem, the Fourier modulus of the diffusion coefficient of the random source is proved to be uniquely determined by the variance of the Fourier transform of the boundary data. As a phase retrieval for the inverse problem, the PhaseLift method with random masks is applied to recover the diffusion coefficient from its Fourier modulus. Numerical experiments are reported to demonstrate the effectiveness of the proposed method.

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arXiv2020

Lipschitz stability for an inverse source scattering problem at a fixed frequency

This paper is concerned with an inverse source problem for the three-dimensional Helmholtz equation by a single boundary measurement at a fixed frequency. We show the Lipschitz stability under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains..

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arXiv2020

An inverse random source problem for the one-dimensional Helmholtz equation with attenuation

This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation with attenuation. The source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The random sources under consideration are equivalent to the generalized fractional Gaussian random fields which include rough fields and can be even rougher than the white noise, and hence should be interpreted as distributions. The well-posedness of the direct source scattering problem is established in the distribution sense. The micro-correlation strength of the random source, which appears to be the strength in the principal symbol of the covariance operator, is proved to be uniquely determined by the wave field in an open measurement set. Numerical experiments are presented for the white noise model to demonstrate the validity and effectiveness of the proposed method.

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arXiv2020

A fast algorithm for the electromagnetic scattering from a large rectangular cavity in three dimensions

The paper is concerned with the three-dimensional electromagnetic scattering from a large open rectangular cavity that is embedded in a perfectly electrically conducting infinite ground plane. By introducing a transparent boundary condition, the scattering problem is formulated into a boundary value problem in the bounded cavity. Based on the Fourier expansions of the electric field, the Maxwell equation is reduced to one-dimensional ordinary differential equations for the Fourier coefficients. A fast algorithm, employing the fast Fourier transform and the Gaussian elimination, is developed to solve the resulting linear system for the cavity which is filled with either a homogeneous or a layered medium. In addition, a novel scheme is designed to evaluate rapidly and accurately the Fourier transform of singular integrals. Numerical experiments are presented for large cavities to demonstrate the superior performance of the proposed method.

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arXiv2020

An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an open domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.

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arXiv2020

A highly accurate boundary integral method for the elastic obstacle scattering problem

Consider the scattering of a time-harmonic plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in two dimensions. In this paper, a novel boundary integral formulation is proposed and its highly accurate numerical method is developed for the elastic obstacle scattering problem. More specifically, based on the Helmholtz decomposition, the model problem is reduced to a coupled boundary integral equation with singular kernels. A regularized system is constructed in order to handle the degenerated integral operators. The semi-discrete and full-discrete schemes are studied for the boundary integral system by using the trigonometric collocation method. Convergence is established for the numerical schemes in some appropriate Sobolev spaces. Numerical experiments are presented for both smooth and nonsmooth obstacles to demonstrate the superior performance of the proposed method.

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arXiv2020

An adaptive finite element PML method for the open cavity scattering problems

Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse electric and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element method with the perfectly matched layer (PML) technique is developed to solve the reduced problem. The estimate takes account both of the finite element approximation error and the PML truncation error, where the latter is shown to decay exponentially with respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBC method for both polarizations to illustrate the competitive behavior of the proposed method.

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arXiv2020

An adaptive finite element DtN method for the open cavity scattering problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper concerns the numerical solutions of the open cavity scattering problems in both transverse magnetic and transverse electric polarizations. Based on the Dirichlet-to-Neumann (DtN) map for each polarization, a transparent boundary condition is imposed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed. The estimate consists of the finite element approximation error and the truncation error of the DtN operator, which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented for both polarizations to illustrate the competitive behavior of the adaptive method.

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arXiv2020

An inverse random source problem for Maxwell's equations

This paper is concerned with an inverse random source problem for the three-dimensional time-harmonic Maxwell equations. The source is assumed to be a centered complex-valued Gaussian vector field with correlated components, and its covariance operator is a pseudo-differential operator. The well-posedness of the direct source scattering problem is established and the regularity of the electromagnetic field is given. For the inverse source scattering problem, the micro-correlation strength matrix of the covariance operator is shown to be uniquely determined by the high frequency limit of the expectation of the electric field measured in an open bounded domain disjoint with the support of the source. In particular, we show that the diagonal entries of the strength matrix can be uniquely determined by only using the amplitude of the electric field. Moreover, this result is extended to the almost surely sense by deducing an ergodic relation for the electric field over the frequencies.

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arXiv2020

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

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arXiv2019

Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc.) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.

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