A brief introduction to recent developments in both the analysis of and numerical methods for time-dependent wave scattering, and the connections therebetween.
On the numerical side, we propose a frequency/time hybrid integral-equation method for transient wave scattering. The method uses Fourier-time transformation, resulting in required solution of a fixed set (with size independent of the desired solution time) of frequency-domain integral equations to evaluate transient solutions for arbitrarily long times. Two main concepts are introduced, namely 1) A smoothly-windowed time-partitioning methodology that enables accurate band-limited representations for arbitrary long time signals, and 2) A novel Fourier transform approach which delivers dispersionless spectrally-accurate solutions. The proposed algorithm is computationally parallelizable and exhibits high-order convergence for scattering from complex geometries while, crucially, enabling time-parallel solution with an O(1)-cost of sampling at large times T.
On the analysis side, some recent results in scattering theory are outlined. It becomes useful to study temporal decay of wave solutions (including in “trapping” scenarios), a classical question treated by the well-known Lax-Phillips scattering theory. We develop (computationally-amenable) “domain-of-dependence” bounds on solutions to wave scattering problems and establish rapid decay estimates using only (existing) Helmholtz resolvent estimates on the real frequency axis, for geometries that have previously posed as barriers to proving rapid decay. This includes the first rapid decay rate for wave scattering for connected “trapping” obstacles and, additionally, for scattering in contexts where periodic trapped orbits span the full volume of a physical cube.
The virtual element method (VEM) is a generalization of the classic finite element method (FEM). In the VEM framework the shape functions used to approximate the solution to PDE systems can be proven to exist but no explicit formula can be attained, thus they are said to be virtual.
The "name of the game" in this method is to define a series of projectors onto polynomial spaces where an explicit basis is known and can be used to come up with approximations to mass and stiffness matrices.
In this presentation we will discuss the design of a VEM for the kinematics of magnetohydrodynamics (MHD) which is a coupling between electromagnetics and fluid flow describing the behaviour of magnetized fluids. Implementations of VEM for MHD present two major advantages, the first is the possibility of its implementation in a very general class of meshes making VEM well-suited for problems posed in oddly shaped domains or with irregularly shaped material interfaces. The second involves the divergence of the magnetic field, it should remain close to zero at the discrete level in order to prevent the appearance of fictitious forces that render simulations unfaithful to the true physics involved, VEM captures this feature exactly as we will prove theoretically and verify numerically.
Collective behaviors (clustering, flocking, milling, swarming, etc.) are among the most interesting and
challenging phenomena to comprehend from the mathematical point of view. We offer a non-parametric
learning approach to discover the governing structure, i.e. the interaction functions between agents,
of collective dynamics from observation of the trajectory data. Our learning approach can aid in
validating and improving the modeling of collective dynamics.
Having established the convergent properties of our learning approach in [1], in [2] we explore in
three different directions to expand our learning theory: steady states behaviors of the estimated
dynamics, learning of more complicated dynamics with two-variable-dependence interaction functions,
and discovery of hidden parametric structure from dynamics driven by parametric family of interaction
functions. Then, we apply our extended learning approach to study the planetary motion of our
solar system using the NASA JPL's development Ephemeris. We are able to reproduce trajectory data with a
precession rate of 540'' per Earth-century for Mercury's orbit. Compared to Newton's theoretical 532'' rate,
we are able to learn portion of the general relativity effect directly from the data.
Convergence properties of the extended learning approaches are also studied.
Examples of learning collective dynamics on non-Euclidean manifolds are being investigated.
[1]: Lu, Zhong, Tang, Maggioni, PNAS, 2019.
[2]: Zhong, Miller, Maggioni, Physica D, 2020.