# American Institute of Mathematical Sciences

eISSN:
2688-1594

All Issues

## Electronic Research Archive

### Special issue on Analysis, Sci. Comp. and Appl. of PDEs

Jingzhi Li, Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China li.jz@sustech.edu.cn

Hongyu Liu, Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China hongyliu@cityu.edu.hk

Xiaoming Wang, Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China wangxm@sustech.edu.cn

2020, 28(2): 837-851 doi: 10.3934/era.2020043 +[Abstract](9)+[HTML](10) +[PDF](1291.07KB)
Abstract:

In this paper, we introduce a dimension splitting method for simulating the air flow state of the aeroengine turbine fan. Based on the geometric model of the fan blade, the dimension splitting method establishes a semi-geodesic coordinate system. Under such coordinate system, the Navier-Stokes equations are reformulated into the combination of membrane operator equations on two-dimensional manifolds and bending operator equations along the hub circle. Using Euler central difference scheme to approximate the third variable, the new form of Navier-Stokes equations is splitting into a set of two-dimensional sub-problems. Solving these sub-problems by alternate iteration, it follows an approximate solution to Navier-Stokes equations. Furthermore, we conduct a numerical experiment to show that the dimension splitting method has a good performance by comparing with the traditional methods. Finally, we give the simulation results of the pressure and flow state of the fan blade.

2020, 28(2): 961-976 doi: 10.3934/era.2020051 +[Abstract](18)+[HTML](10) +[PDF](19092.29KB)
Abstract:

In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. A-posteriori error estimators based on the recovery type and residual type are proposed, respectively. Based on our a-posteriori error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed a-posteriori error estimations for the adaptive method.

2020, 28(2): 977-1000 doi: 10.3934/era.2020052 +[Abstract](26)+[HTML](12) +[PDF](443.55KB)
Abstract:

This paper develops efficient numerical algorithms for the optimal control problem constrained by Poisson equations with uncertain diffusion coefficients. We consider both unconstrained condition and box-constrained condition for the control. The algorithms are based upon a multi-mode expansion (MME) for the random state, the finite element approximation for the physical space and the alternating direction method of multipliers (ADMM) or two-block ADMM for the discrete optimization systems. The compelling aspect of our method is that, target random constrained control problem can be approximated to one deterministic constrained control problem under a state variable substitution equality. Thus, the computing resource, especially the memory consumption, can be reduced sharply. The convergence rates of the proposed algorithms are discussed in the paper. We also present some numerical examples to show the performance of our algorithms.

2020, 28(2): 1001-1022 doi: 10.3934/era.2020053 +[Abstract](21)+[HTML](7) +[PDF](883.29KB)
Abstract:

Efficient numerical methods for solving Poisson equation constraint optimal control problems with random coefficient are discussed in this paper. By applying the finite element method and the Monte Carlo approximation, the original optimal control problem is discretized and transformed into an optimization problem. Taking advantage of the separable structures, Algorithm 1 is proposed for solving the problem, where an alternating direction method of multiplier is used. Both computational and storage costs of this algorithm are very high. In order to reduce the computational cost, Algorithm 2 is proposed, where the multi-modes expansion is introduced and applied. Further, for reducing the storage cost, we propose Algorithm 3 based on Algorithm 2. The main idea is that the random term is shifted to the objective functional, which could be computed in advance. Therefore, we only need to solve a deterministic optimization problem, which could reduce all the costs significantly. Moreover, the convergence analyses of the proposed algorithms are established, and numerical simulations are carried out to test the performances of them.

2020, 28(2): 1107-1121 doi: 10.3934/era.2020061 +[Abstract](23)+[HTML](20) +[PDF](1258.3KB)
Abstract:

In this paper, we are concerned with the three-dimensional (3D) geometric body shape generation with several well-selected characteristic values. Since 3D human shapes can be viewed as the support of the electromagnetic sources, we formulate a scheme to regenerate 3D human shapes by inverse scattering theory. With the help of vector spherical harmonics expansion of the magnetic far field pattern, we build on a smart one-to-one correspondence between the geometric body space and the multi-dimensional vector space that consists of all coefficients of the spherical vector wave function expansion of the magnetic far field pattern. Therefore, these coefficients can serve as the shape generator. For a collection of geometric body shapes, we obtain the inputs (characteristic values of the body shapes) and the outputs (the coefficients of the spherical vector wave function expansion of the corresponding magnetic far field patterns). Then, for any unknown body shape with the given characteristic set, we use the multivariate Lagrange interpolation to get the shape generator of this new shape. Finally, we get the reconstruction of this unknown shape by using the multiple-frequency Fourier method. Numerical examples of both whole body shapes and human head shapes verify the effectiveness of the proposed method.

2020, 28(2): 1123-1142 doi: 10.3934/era.2020062 +[Abstract](28)+[HTML](19) +[PDF](841.28KB)
Abstract:

This paper proposes a near-field shape neural network (NSNN) to determine the shape of a sound-soft cavity based on a single source and several measurements placed on a curve inside the cavity. The NSNN employs the near-field measurements as input, and the output is the shape parameters of the cavity. The self-attention mechanism is employed to obtain the feature information of the near-field data, as well as the correlations among them. The weights and biases of the NSNN are updated through the gradient descent algorithm, which minimizes the error of the reconstructed shape of the cavity. We prove that the loss function sequence related to the weights is a monotonically bounded non-negative sequence, which indicates the convergence of the NSNN. Numerical experiments show that the shape of the cavity can be effectively reconstructed with the NSNN.

2020, 28(3): 1239-1255 doi: 10.3934/era.2020068 +[Abstract](9)+[HTML](8) +[PDF](367.52KB)
Abstract:

We consider a particular type of inverse problems where an unknown source embedded in an inhomogeneous medium, and one intends to recover the source and/or the medium by knowledge of the wave field (generated by the unknown source) outside the medium. This type of inverse problems arises in many applications of practical importance, including photoacoustic and thermoacoustic tomography, brain imaging and geomagnetic anomaly detections. We survey the recent mathematical developments on this type of inverse problems. We discuss the mathematical tools developed for effectively tackling this type of inverse problems. We also discuss a related inverse problem of recovering an embedded obstacle and its surrounding medium by active measurements.

2020, 28(3): 1257-1272 doi: 10.3934/era.2020069 +[Abstract](10)+[HTML](7) +[PDF](361.94KB)
Abstract:

We consider the anomalous localized resonance induced by negative elastic metamaterials and its application in invisibility cloaking. We survey the recent mathematical developments in the literature and discuss two mathematical strategies that have been developed for tackling this peculiar resonance phenomenon. The first one is the spectral method, which explores the anomalous localized resonance through investigating the spectral system of the associated Neumann-Poincaré operator. The other one is the variational method, which considers the anomalous localized resonance via calculating the nontrivial kernels of a non-elliptic partial differential operator. The advantages and the relationship between the two methods are discussed. Finally, we propose some open problems for the future study.

2021, 29(1): 1753-1782 doi: 10.3934/era.2020090 +[Abstract](796)+[HTML](419) +[PDF](492.02KB)
Abstract:

Unique identifiability by finitely many far-field measurements in the inverse scattering theory is a highly challenging fundamental mathematical topic. In this paper, we survey some recent progress on the inverse obstacle scattering problems and the inverse medium scattering problems associated with time-harmonic waves within a certain polyhedral geometry, where one can establish the unique identifiability results by finitely many measurements. Some unique identifiability issues on the inverse diffraction grating problems are also considered. Furthermore, the geometrical structures of Laplacian and transmission eigenfunctions are reviewed, which have important applications in the unique determination for inverse obstacle and medium scattering problems with finitely many measurements. We discuss the mathematical techniques and methods developed in the literature. Finally, we raise some intriguing open problems for the future investigation.

2021, 29(2): 2007-2028 doi: 10.3934/era.2020102 +[Abstract](611)+[HTML](313) +[PDF](3144.54KB)
Abstract:

We study decoupled numerical methods for multi-domain, multi-physics applications. By investigating various stages of numerical approximation and decoupling and tracking how the information is transmitted across the interface for a typical multi-modeling model problem, we derive an approximate intrinsic or inertial type Robin condition for its semi-discrete model. This new interface condition is justified both mathematically and physically in contrast to the classical Robin interface condition conventionally introduced for decoupling multi-modeling problems. Based on the intrinsic or inertial Robin condition, an equivalent semi-discrete model is introduced, which provides a general framework for devising effective decoupled numerical methods. Numerical experiments also confirm the effectiveness of this new decoupling approach.

2021, 29(2): 2029-2045 doi: 10.3934/era.2020103 +[Abstract](533)+[HTML](297) +[PDF](435.38KB)
Abstract:

The main aim of this paper is to study the \begin{document}$P_1$\end{document} nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate \begin{document}$O(h)$\end{document} can be obtained by a refined analysis when the exact solution belongs to \begin{document}$H^{2}(\Omega)$\end{document} without any assumption. Furthermore, we also study the optimal convergence for the case \begin{document}$u\in H^{1+\nu}(\Omega)$\end{document} with \begin{document}$\frac{1}{2}<\nu<1$\end{document}.

2021, 29(2): 2047-2061 doi: 10.3934/era.2020104 +[Abstract](764)+[HTML](347) +[PDF](378.87KB)
Abstract:

In this paper, a unified theoretical method is presented to implement the finite/fixed-time synchronization control for complex networks with uncertain inner coupling. The quantized controller and the quantized adaptive controller are designed to reduce the control cost and save the channel resources, respectively. By means of the linear matrix inequalities technique, two sufficient conditions are proposed to guarantee that the synchronization error system of the complex networks is finite/fixed-time stable in virtue of the Lyapunov stability theory. Moreover, two types of setting time, which are dependent and independent on the initial values, are given respectively. Finally, the effectiveness of the control strategy is verified by a simulation example.

2021, 29(2): 2149-2165 doi: 10.3934/era.2020110 +[Abstract](622)+[HTML](269) +[PDF](335.94KB)
Abstract:

This article is an overview on some recent advances in the inverse scattering problems with phaseless data. Based upon our previous studies on the uniqueness issues in phaseless inverse acoustic scattering theory, this survey aims to briefly summarize the relevant rudiments comprising prototypical model problems, major results therein, as well as the rationale behind the basic techniques. We hope to sort out the essential ideas and shed further lights on this intriguing field.

2021, 29(3): 2391-2415 doi: 10.3934/era.2020121 +[Abstract](531)+[HTML](265) +[PDF](535.93KB)
Abstract:

We consider the recovery of some statistical quantities by using the near-field or far-field data in quantum scattering generated under a single realization of the randomness. We survey the recent main progress in the literature and point out the similarity among the existing results. The methodologies in the reformulation of the forward problems are also investigated. We consider two separate cases of using the near-field and far-field data, and discuss the key ideas of obtaining some crucial asymptotic estimates. We pay special attention on the use of the theory of pseudodifferential operators and microlocal analysis needed in the proofs.

2020 Impact Factor: 1.833
5 Year Impact Factor: 1.833