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This talk introduces fast algorithms of the matvec $g=Kf$ for $K\in \mathbb{C}^{N\times N}$, which is the discretization of the oscillatory integral transform $g(x) = \int K(x,\xi) f(\xi)d\xi$ with a kernel function $K(x,\xi)=\alpha(x,\xi)e^{2\pi i\Phi(x,\xi)}$, where $\alpha(x,\xi)$ is a smooth amplitude function , and $\Phi(x,\xi)$ is a piecewise smooth phase function with $O(1)$ discontinuous points in $x$ and $\xi$. A unified framework is proposed to compute $Kf$ with $O(N\log N)$ time and memory complexity via the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an $O(N)$ fast algorithm to determine whether NUFFT or BF is more suitable. This framework works for two cases: 1) explicite formulas for the amplitude and phase functions are known; 2) only indirect access of the amplitude and phase functions are available. Especially in the case of indirect access, our main contributions are: 1) an $O(N\log N)$ algorithm for recovering the amplitude and phase functions is proposed based on a new low-rank matrix recovery algorithm; 2) a new stable and nearly optimal BF with amplitude and phase functions in form of a low-rank factorization (IBF-MAT) is proposed to evaluate the matvec $Kf$. Numerical results are provided to demonstrate the effectiveness of the proposed framework. | This talk introduces fast algorithms of the matvec $g=Kf$ for $K\in \mathbb{C}^{N\times N}$, which is the discretization of the oscillatory integral transform $g(x) = \int K(x,\xi) f(\xi)d\xi$ with a kernel function $K(x,\xi)=\alpha(x,\xi)e^{2\pi i\Phi(x,\xi)}$, where $\alpha(x,\xi)$ is a smooth amplitude function , and $\Phi(x,\xi)$ is a piecewise smooth phase function with $O(1)$ discontinuous points in $x$ and $\xi$. A unified framework is proposed to compute $Kf$ with $O(N\log N)$ time and memory complexity via the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an $O(N)$ fast algorithm to determine whether NUFFT or BF is more suitable. This framework works for two cases: 1) explicite formulas for the amplitude and phase functions are known; 2) only indirect access of the amplitude and phase functions are available. Especially in the case of indirect access, our main contributions are: 1) an $O(N\log N)$ algorithm for recovering the amplitude and phase functions is proposed based on a new low-rank matrix recovery algorithm; 2) a new stable and nearly optimal BF with amplitude and phase functions in form of a low-rank factorization (IBF-MAT) is proposed to evaluate the matvec $Kf$. Numerical results are provided to demonstrate the effectiveness of the proposed framework. | ||

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=== Eric Keaveny (Imperial College London) === | === Eric Keaveny (Imperial College London) === |

## Revision as of 11:36, 1 April 2018

## Contents

- 1 ACMS Abstracts: Spring 2018
- 1.1 Thomas Fai (Harvard)
- 1.2 Michael Herty (RWTH-Aachen)
- 1.3 Lee Panetta (Texas A&M)
- 1.4 Francois Monard (UC Santa Cruz)
- 1.5 Haizhao Yang (National University of Singapore)
- 1.6 Eric Keaveny (Imperial College London)
- 1.7 Anne Gelb (Dartmouth)
- 1.8 Molei Tao (Georgia Tech)
- 1.9 William Irvine (U Chicago)
- 1.10 Boualem Khouider (UVic)
- 1.11 Anru Zhang (UW-Madison, statistics)

# ACMS Abstracts: Spring 2018

### Thomas Fai (Harvard)

*The Lubricated Immersed Boundary Method*

Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.

### Michael Herty (RWTH-Aachen)

*Opinion Formation Models and Mean field Games Techniques*

Mean-Field Games are games with a continuum of players that incorporate the time dimension through a control-theoretic approach. Recently, simpler approaches relying on reply strategies have been proposed. Based on an example in opinion formation modeling we explore the link between differentiability notions and mean-field game approaches. For numerical purposes a model predictive control framework is introduced consistent with the mean-field game setting that allows for efficient simulation. Numerical examples are also presented as well as stability results on the derived control.

### Lee Panetta (Texas A&M)

*Traveling waves and pulsed energy emissions seen in numerical simulations of electromagnetic wave scattering by ice crystals*

The numerical simulation of single particle scattering of electromagnetic energy plays a fundamental role in remote sensing studies of the atmosphere and oceans, and in efforts to model aerosol "radiative forcing" processes in a wide variety of models of atmospheric and climate dynamics, I will briefly explain the main challenges in the numerical simulation of single particle scattering and describe how work with 3-d simulations of scattering of an incident Gaussian pulse, using a Pseudo-Spectral Time Domain method to numerically solve Maxwell’s Equations, led to an investigation of episodic bursts of energy that were observed at various points in the near field during the decay phase of the simulations. The main focus of the talk will be on simulations in dimensions 1 and 2, simple geometries, and a single refractive index (ice at 550 nanometers). The periodic emission of pulses is easy to understand and predict on the basis of Snell’s laws in the 1-d case considered. In much more interesting 2-d cases, simulations show traveling waves within the crystal that give rise to pulsed emissions of energy when they interact with each other or when they enter regions of high surface curvature. The time-dependent simulations give a more dynamical view of "photonic nanojets" reported earlier in steady-state simulations in other contexts, and of energy release in "morphology-dependent resonances."

### Francois Monard (UC Santa Cruz)

*Inverse problems in integral geometry and Boltzmann transport*

The Boltzmann transport (or radiative transfer) equation describes the transport of photons interacting with a medium via attenuation and scattering effects. Such an equation serves as the model for many imaging modalities (e.g., SPECT, Optical Tomography) where one aims at reconstructing the optical parameters (absorption/scattering) or a source term, out of measurements of intensities radiated outside the domain of interest.

In this talk, we will review recent progress on the inversion of some of the inverse problems mentioned above. In particular, we will discuss an interesting connection between the inverse source problem (where the optical parameters are assumed to be known) and a problem from integral geometry, namely the tensor tomography problem (or how to reconstruct a tensor field from knowledge of its integrals along geodesic curves).

### Haizhao Yang (National University of Singapore)

*A Unified Framework for Oscillatory Integral Transform: When to use NUFFT or Butterfly Factorization?*

This talk introduces fast algorithms of the matvec $g=Kf$ for $K\in \mathbb{C}^{N\times N}$, which is the discretization of the oscillatory integral transform $g(x) = \int K(x,\xi) f(\xi)d\xi$ with a kernel function $K(x,\xi)=\alpha(x,\xi)e^{2\pi i\Phi(x,\xi)}$, where $\alpha(x,\xi)$ is a smooth amplitude function , and $\Phi(x,\xi)$ is a piecewise smooth phase function with $O(1)$ discontinuous points in $x$ and $\xi$. A unified framework is proposed to compute $Kf$ with $O(N\log N)$ time and memory complexity via the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an $O(N)$ fast algorithm to determine whether NUFFT or BF is more suitable. This framework works for two cases: 1) explicite formulas for the amplitude and phase functions are known; 2) only indirect access of the amplitude and phase functions are available. Especially in the case of indirect access, our main contributions are: 1) an $O(N\log N)$ algorithm for recovering the amplitude and phase functions is proposed based on a new low-rank matrix recovery algorithm; 2) a new stable and nearly optimal BF with amplitude and phase functions in form of a low-rank factorization (IBF-MAT) is proposed to evaluate the matvec $Kf$. Numerical results are provided to demonstrate the effectiveness of the proposed framework.

### Eric Keaveny (Imperial College London)

*Linking the micro- and macro-scales in populations of swimming cells*

Swimming cells and microorganisms are as diverse in their collective dynamics as they are in their individual shapes and swimming mechanisms. They are able to propel themselves through simple viscous fluids, as well as through more complex environments where they must interact with other microscopic structures. In this talk, I will describe recent simulations that explore the connection between dynamics at the scale of the cell with that of the population in the case where the cells are sperm. In particular, I will discuss how the motion of the sperm’s flagella can greatly impact the overall dynamics of their suspensions. Additionally, I will discuss how in complex environments, the density and stiffness of structures with which the cells interact impact the effective diffusion of the population.

### Anne Gelb (Dartmouth)

*Reducing the effects of bad data measurements using variance based weighted joint sparsity *

We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.

### Molei Tao (Georgia Tech)

*Explicit high-order symplectic integration of nonseparable Hamiltonians: algorithms and long time performance*

Symplectic integrators preserve the phase-space volume and have favorable performances in long time simulations. Methods for an explicit symplectic integration have been extensively studied for separable Hamiltonians (i.e., H(q,p)=K(p)+V(q)), and they lead to both accurate and efficient simulations. However, nonseparable Hamiltonians also model important problems, such as non-Newtonian mechanics and nearly integrable systems in action-angle coordinates. Unfortunately, implicit methods had been the only available symplectic approach for general nonseparable systems.

This talk will describe a recent result that constructs explicit and arbitrary high-order symplectic integrators for arbitrary Hamiltonians. Based on a mechanical restraint that binds two copies of phase space together, these integrators have good long time performance. More precisely, based on backward error analysis, KAM theory, and some additional multiscale analysis, a pleasant error bound is established for integrable systems. This bound is then demonstrated on a conceptual example and the Schwarzschild geodesics problem. For nonintegrable systems, some numerical experiments with the nonlinear Schrodinger equation will be discussed.

### William Irvine (U Chicago)

*Spinning top-ology: Order, disorder and topology in mechanical gyro-materials and fluids*

Geometry, topology and broken symmetry often play a powerful role in determining the organization and properties of materials. A recent example is the discovery that the excitation spectra of materials -- be they electronic, optical, or mechanical -- may be topologically non-trivial. I will explore the use of `spinning tops' to explore this physics. In particular I will discuss an experimental and theoretical study of a simple kind of active meta-material – coupled gyroscopes – that naturally encodes non-trivial topology in its vibrational spectrum. These materials have topologically protected edge modes which we observe in experiment. Crucially, the geometry of the underlying lattice controls the presence of time reversal symmetry that is essential to the non-trivial topology of the spectrum. We exploit this to control the chirality of the edge modes by simply deforming the lattice. Moving beyond ordered lattices we show that amorphous gyroscopic networks are naturally topological. If time permits I will conclude with a brief foray into gyrofluids: the liquid counterpart of these topological solids.

### Boualem Khouider (UVic)

*Using a stochastic convective parametrization to improve the simulation of tropical modes of variability in a GCM*

Convection in the tropics is organized into a hierarchy of scales ranging from the individual cloud of 1 to 10 km to cloud clusters and super-clusters of 100’s km and 1000’s km, respectively, and their planetary scale envelopes. These cloud systems are strongly coupled to large scale dynamics in the from of wave disturbances going by the names of meso-scale systems, convectively coupled equatorial waves (CCEW), and intraseasonal oscillations, including the eastward propagating Madden Julian Oscillation (MJO) and poleward moving monsoon intraseasonal oscillation (MISO). Coarse resolution climate models (GCMs) have serious difficulties in representing these tropical modes of variability, which are known to impact weather and climate variability in both the tropics and elsewhere on the globe. Atmospheric rivers, for example, such the pineapple express that brings heavy rainfall to the Pacific North West, are believed to be directly connected to the MJO.

The deficiency in the GCMs is believed to be rooted from the inadequateness of the underlying cumulus parameterizations to represent the variability at the multiple spatial and temporal scales of organized convection and the associated two-way interactions between the wave flows and convection; these parameterizations are based on the quasi-equilibrium closure where convection is basically slaved to the large scale dynamics. To overcome this problem we employ a stochastic multi-cloud model (SMCM) convective parametrization, which mimics the interactions at sub-grid scales of multiple cloud types, as seen in observations. The new scheme is incorporated into the National Centers for Environmental Prediction (NCEP) Climate Forecast System version 2 (CFSv2) model (CFSsmcm) in lieu of the pre-existing simplified Arakawa-Schubert (SAS) cumulus scheme.

Significant improvements are seen in the simulation of MJO, CCEWs as well as the Indian MISO. These improvements appear in the form of improved variability, morphology and physical features of these wave flows. This particularly confirms the multicloud paradigm of organized tropical convection, on which the SMCM design was based, namely, congestus, deep and stratiform cloud decks that interact with each other to form the building block for multiscale convective systems. An adequate account for the dynamical interactions of this cloud hierarchy thus constitutes an important requirement for cumulus parameterizations to succeed in representing atmospheric tropical variability. SAS fails to fulfill this requirement evident in the unrealistic physical structures of the major intra-seasonal modes simulated by the default CFSv2.

### Anru Zhang (UW-Madison, statistics)

*Singular value decomposition for high-dimensional high-order data*

High-dimensional high-order data arise in many modern scientific applications including genomics, brain imaging, and social science. In this talk, we consider the methods, theories, and computations for tensor singular value decomposition (tensor SVD), which aims to extract the hidden low-rank structure from high-dimensional high-order data. First, comprehensive results are developed on both the statistical and computational limits for tensor SVD under the general scenario. This problem exhibits three different phases according to signal-noise-ratio (SNR), and the minimax-optimal statistical and/or computational results are developed in each of the regimes. In addition, we further consider the sparse tensor singular value decomposition which allows more robust estimation under sparsity structural assumptions. A novel sparse tensor alternating thresholding algorithm is proposed. Both the optimal theoretical results and numerical analyses are provided to guarantee the performance of the proposed procedure.