Abstract : The recent advances in wave equations and its fast numerical methods have provided useful tools for many applications ranging from nano-optics to medical imaging and geosciences. This mini-symposium will discuss the challenges in the formulations of forward and inverse problems, cutting edge fast algorithms and their efficient implementation and applications in various fields. At the same time, it will provide opportunities to promote interdisciplinary research collaboration between computational scientists and other fields.
[00347] Accurate evaluation of Helmholtz layer potentials using Quadrature by two expansions
Format : Talk at Waseda University
Author(s) :
Min Hyung Cho (University of Massachusetts Lowell)
Jared Weed (University of Massachusetts Lowell)
Lingyun Ding (University of California Los Angeles)
Jingfang Huang (University of North Carolina at Chapel Hill)
Abstract : The Helmholtz layer potentials are evaluated using the Quadrature by two expansions (QB2X). The QB2X method uses local complex Taylor expansion and planewave-type expansions to achieve a representation that is numerically accurate at all target points inside the leaf box in the hierarchical tree structure. Compared with the original quadrature by the expansion that uses only the local expansion, the QB2X includes explicit nonlinearity from the boundary geometry in the planewave type expansions and it follows standard fast multipole method error analysis. Therefore, QB2X overcomes many challenges quadrature by expansion has and is suitable for complex geometry. The main ideas of the derivation of QB2X using Fourier extension and contour integrals, and numerical results showing the efficiency of QB2X compared with the quadrature by expansion will be presented for both flat and curved boundaries.
[00411] On the Robustness of Inverse Scattering for Penetrable, Homogeneous Objects
Format : Talk at Waseda University
Author(s) :
Borges Carlos (University of Central Florida)
Manas Rachh (Flatiron Institute)
Leslie Greengard (New York University and Flatiron Institute)
Abstract : In the inverse obstacle scattering problem, one determines the shape of a domain from measurements of the scattered field due to a set of incident fields. For a penetrable obstacle with known sound speed, this can be accomplished by treating the boundary alone as an unknown curve. Alternatively, one can treat the entire object as an unknown and use a volumetric representation, without making use of the known sound speed. Both lead to strongly nonlinear and nonconvex optimization problems for which recursive linearization provides a useful framework. After extending our shape optimization approach developed earlier for impenetrable bodies to penetrable obstacles, we carry out a systematic study of both methods and compare their performance on a variety of examples. Our findings indicate that the volumetric approach is more robust, even though the number of degrees of freedom is significantly larger.
[00490] A Neural Network Warm-Start Approach for the Inverse Acoustic Obstacle Scattering Problem
Format : Talk at Waseda University
Author(s) :
Manas Rachh (Flatiron Institute)
Mo Zhou (Duke University)
Jiequn Han (Flatiron Institute)
Borges Carlos (University of Central Florida)
Abstract : We consider the inverse acoustic obstacle problem for sound-soft star-shaped obstacles in two dimensions. The inverse problem is computationally challenging since the local set of convexity shrinks with increasing frequency and results in an increasing number of local minima in the vicinity of the true solution. In this talk, we present a neural network warm-start approach for solving the inverse scattering problem, where an initial guess is obtained using a trained neural network.
[04073] Optimal Transportation for Electrical Impedance Tomography
Format : Talk at Waseda University
Author(s) :
Gang BAO (Zhejiang University)
Yixuan Zhang (Zhejiang University)
Abstract : A new framework is introduced for solving inverse boundary problems with the geodesic based quadratic Wasserstein distance. A general form of the Fréchet gradient is systematically derived by optimal transportation (OT) theory. A fast algorithm based on the new formulation of OT is developed to solve the corresponding optimal transport problem with much improved computational complexity. This framework provides a new computational approach for solving the challenging electrical impedance tomography problem.
[03446] A high-accuracy boundary integral equation method for wave scattering by 3D analytic surfaces
Format : Talk at Waseda University
Author(s) :
Wangtao Lu (Zhejiang University)
Jun Lai (Zhejiang University)
Abstract : In this talk, we present a high accuracy boundary integral equation method for wave scattering by a closed analytic surface. The surface is assumed to parameterize in earth-like latitude and longitude coordinates. Any analytic function on it can be firstly approximated by its Fourier series in the latitude variable, and then piecewise Legendre polynomials in the longitude variable. By doing so, we can accurately approximate the standard single-, double-, and adjoint double-layer integral operators in two stages. First, the integrals in the latitude direction are approximated by a nearly optimal algorithm for computing the Fourier transforms of the related kernel functions. Second, the resulting single-variable integrals in the latitude direction have weakly singular kernels, and can be spectrally discretized by pre-designed high-accuracy quadrature rules. We study exterior and interior boundary value problems to validate efficiency and accuracy of the proposed method.
[03520] Fast algorithms for multiple elastic obstacles scattering and inverse scattering
Format : Talk at Waseda University
Author(s) :
Jun Lai (Zhejiang University)
Abstract : Elastic wave scattering and inverse scattering have been appeared in a lot of important applications, including non-destructive testing, seismic inversion, and medical imaging, etc. Integral equation method provides an effective tool for solving elastic wave scattering and inverse scattering problems. In this talk, fast and high order numerical methods based on integral equations will be presented for elastic wave equations in the presence of multiple obstacles. In particular, I will talk about the numerical algorithms using high order discretization of singular integrals and the fast multipole method for evaluating the multiple elastic scattering problem, as well as their applications in the inverse elastic wave scattering based on the time reversal method.
[05069] Exploring inverse obstacle scattering with an impedance model
Format : Talk at Waseda University
Author(s) :
Travis Askham (New Jersey Institute of Technology)
Manas Rachh (Flatiron Institute)
Carlos Borges (University of Central Florida)
Jeremy Hoskins (University of Chicago)
Abstract : It is well known that in certain limits the impedance boundary condition can mimic the sound hard, sound soft, and transmission boundary conditions. Here we explore the performance of this approximation in inverse obstacle scattering problems. We find that for certain problems a relaxation of the usual measurement of scattering error improves the quality of obstacle recovery.
[03858] Hybrid methods for the application of singular integral operators
Format : Talk at Waseda University
Author(s) :
Leslie Greengard (New York University and Flatiron Institute)
Shidong Jiang (Flatiron Institute)
Jun Wang (Tsinghua University)
Fredrik Fryklund (Courant Institute, NYU)
Samuel F Potter (Courant Institute, NYU)
Abstract : We present hybrid asymptotic/numerical methods for the accurate computation of elliptic and parabolic volume and layer potentials in two and three dimensions.
[04029] Obstacles and interfaces composite scattering in a multilayered medium
Format : Talk at Waseda University
Author(s) :
Lei Zhang (Zhejiang University of Technology)
Abstract : This talk will focus on the mathematical analysis and numerical methods for the composite scattering problem from obstacles and interfaces in a multilayered medium. We will highlight some recent progress in this area. Specifically, we will address how to handle obstacles and rough surfaces, and how to model scattering from unbounded surfaces. We will also discuss the well-posedness and numerical methods for solving these problems based on the characteristic of the scattering problems.
[04887] Lippmann Schwinger integral equation for fiber optics analysis
Format : Talk at Waseda University
Author(s) :
Felipe Vico (UPV)
Miguel Ferrando-Bataller (UPV)
Eva Antonino-Daviu (UPV)
Marta Cabedo-Fabrés (UPV)
Abstract : In this talk, we will present a Limpann-Schwinger integral formulation for accurately calculating the propagating modes in fiber optics. Our formulation is based on a second-kind integral equation, which ensures stability in the presence of noise and uncertainties. Furthermore, our discretization scheme exhibits superalgebraic convergence for smooth refractive index fibers, making it suitable for graded-index fibers.
[05178] Single-excitation quantum optics: analysis and algorithms
Format : Talk at Waseda University
Author(s) :
Jeremy Graeme Hoskins (University of Chicago)
Jeremy Hoskins (University of Chicago)
Manas Rachh (Flatiron Institute)
John Schotland (Yale University)
Jason Kaye (Flatiron Institute, Simons Foundation)
Abstract : Recent progress in experimental quantum optics has facilitated the physical construction of systems of increasing complexity. Of particular importance are experiments involving the scattering of one or two photons from a collection of atoms. In this context, a central question is to understand the time evolution of the entanglement between atoms, mediated by the field. In this talk we will discuss analytical results on the properties of these systems, and how those properties depend on disorder or distribution of the locations of the atoms.
[05639] Poisson Solver for Complicated Geometries in R3 Using Function Extension
Author(s) :
Fredrik Fryklund (New York University)
Charles Epstein (Flatiron Institute, Simons Foundation)
Shidong Jiang (Flatiron Institute, Simons Foundation)
Leslie Greengard (Flatiron Institute, Simons Foundation)
Abstract : We describe a new, adaptive solver for the three-dimensional Poisson equation in complicated geometries. The solution is represented as the sum of a volume potential and a double layer potential. The source data is extended with high order accuracy along the normals to the surface, to a geometrically simpler region. This allows us to accelerate the evaluation of the volume potential using an FMM.
