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− | + | Abstract: The large-scale numerical solution of PDEs is an essential part of scientific research. Decades of work have been put into developing fast numerical schemes for specific equations, but computational research in many fields is still largely software-limited. Here I will discuss how algorithmic flexibility and composability can enable new science, as illustrated by the Dedalus Project. Dedalus is an open-source Python framework that automates the solution of general PDEs using spectral methods. High-level abstractions allow users to symbolically specify equations, parallelize and scale their solvers to thousands of cores, and perform arbitrary analysis with the computed solutions. I will provide an overview the code’s design and the underlying sparse spectral algorithms, and show how they are enabling novel simulations of diverse hydrodynamical systems. I will include astrophysical and geophysical applications using new bases for tensor-valued equations in spherical domains, immersed boundary methods for multiphase flows, and multi-domain simulations interfacing Dedalus with other PDE and integral equation solvers. |

## Revision as of 17:25, 1 April 2020

## Contents

# ACMS Abstracts: Spring 2020

### Hung Tran

Title: Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel

Abstract: We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of mass-conserving solutions to the Coagulation-Fragmentation equation. Joint work with Truong-Son Van (CMU).

### Svetlana Avramov-Zamurovic

Title: Experiments with Structured Light

Abstract: Complete understanding of laser light propagation through random complex media, including theoretical models and experimental verifications, is relevant for numerous contemporary communication and sensors applications. Light radiation is the most suitable for transmitting high data rates due to its wide bandwidth, but it is significantly impacted by the state of the propagation media. To mitigate the deterioration of laser light along a propagation path, various independent characteristics of light could be manipulated, most notably: spatial coherence, intensity, wavelength, polarization, as well as the orbital angular momentum of light. Much of the research has focused on laser propagation through turbulent atmospheric conditions, but with the development of distributed sensor networks and autonomous underwater vehicles, achieving high performance data transmission in the ocean is becoming exceptionally valuable. The propagation of laser light in water is influenced by high attenuation rates caused by scattering from organic and inorganic particulates as well as change in refractive index due to temperature and salinity fluctuations. Structured light offers a tool to combat some of the mentioned deteriorations.

The talk will focus on experiments with structured light propagating in maritime environment. First, the underwater communication system that uses the superposition of coherent beams carrying orbital angular momentum, will be presented. The design objective is the creation of a family of dissimilar images suitable for fast and accurate classification using only the intensity patterns imaged by a camera. Next, the measurements from the field experiments with spatially partially coherent light as well as polarization diversity, propagating at the Academy grounds, will be given. The talk emphasis will be on the physical aspects of the experiments with structured laser light, and the relationship to the data obtained.

### Vadim Gorin

Title: Integrability of KPZ equation.

Abstract: Kardar-Parisi-Zhang stochastic partial differential equation is a prototypical model for the random growth of one-dimensional interfaces. I will review how it appeared and present various exact formulas, which allow the large time asymptotic analysis of the solutions to the equation and hint on its connections to other stochastic objects.

### John Harlim

Title: Modeling Dynamical Systems with Machine Learning

Abstract: The recent success of machine learning has drawn tremendous interest in applied mathematics and scientific computations. In the first part of the talk, I will discuss recent efforts in using an unsupervised learning algorithm (a branch of machine learning) to estimate time-dependent densities of Ito diffusion from time series of the stochastic processes. The second part of the talk is on the topic of model error arises in modeling of dynamical systems. Particularly, I will discuss a general framework to compensate for the model error. The proposed framework reformulates the model error problem into a supervised learning task to approximate a very high-dimensional target function involving the Mori-Zwanzig representation of projected dynamical systems. Connection to traditional parametric approaches will be clarified as specifying the appropriate hypothesis space for the target function. Theoretical convergence and numerical demonstration on modeling problems arising from PDE's will be discussed.

### Qiu Yang

Title: Upscale Impact of Mesoscale Convective Systems on the MJO and Its Parameterization in Coarse-Resolution GCMs

Abstract: The multi-scale organization of tropical convection has emerged as an exciting interdisciplinary research area where many applied math methods can be used to address real-world problems. For example, Madden-Julian oscillation (MJO), the holy grail of tropical atmospheric dynamics, is organized in a hierarchical structure that the eastward-moving planetary-scale envelope usually contains multiple synoptic-scale superclusters with numerous embedded mesoscale convective systems (MCSs). Present-day global climate models (GCMs) fail to explicitly resolve small-scale MCSs due to their coarse resolutions. We hypothesized that such inadequate treatment of MCSs and their upscale impact leads to the poorly simulated MJOs in the GCMs. Here we tackled this challenging problem from three different perspectives based on models in a hierarchy of complexity. We first simulated the multi-scale organization of tropical convection in a 2D global cloud-resolving model and demonstrated the crucial upscale impact of MCSs on the MJO through energy budget analysis. Then we used a multi-scale theoretical model to describe the observed scenario where a convectively coupled Kelvin wave on the synoptic-scale typically contains numerous embedded MCSs. The eddy transfer of momentum and temperature that stands out from the model is interpreted as the upscale impact of MCSs on large-scale circulation. Finally, we developed a basic parameterization for the upscale impact of MCSs based on the multi-scale theoretical model. We tested the effect of this parameterization in both an idealized testbed and a coarse-resolution GCM. The results show that the parameterization promotes persistent eastward propagation of the MJO and recover its realistic features of spatiotemporal variability.

### Curt A. Bronkhorst

Title: Computational Prediction of Shear Banding and Deformation Twinning in Metals

Abstract: The high deformation rate mechanical loading of polycrystalline metallic materials, which have ready access to plastic deformation mechanisms, generally involve an intense process of several deformation mechanisms within the material: dislocation slip (thermally activated and phonon drag dominated), recovery (annihilation and recrystallization), mechanical twinning, porosity, and shear banding depending upon the material. For this class of ductile materials, depending upon the boundary conditions imposed, there are varying degrees of porosity or adiabatic shear banding taking place at the later stages of the deformation history. Each of these two processes are as yet a significant challenge to predict accurately. This is true for both material models to represent the physical response of the material or the computational framework to represent accurately the creation of new surfaces or interfaces in a topologically independent way. Within this talk, I will present an enriched element technique to represent the adiabatic shear banding and deformation twinning process within a traditional Lagrangian finite element framework. A rate-dependent onset criterion for the initiation of a band is defined based upon a rate and temperature dependent material model. Once the bifurcation condition is met, the location and orientation of an embedded field zone is computed and inserted within a computational element. Once embedded the boundary conditions between the localized and unlocalized regions of the element are enforced and the composite sub-grid element follows a weighted average representation of both regions. Continuity in shear band growth is ensured by employing a non-local level-set technique connected to the displacement field within the finite-element solver. The material inside the band is able to be represented independent from the outside material and the thickness of the band can be assigned by any appropriate method. Dynamic recrystallization (DRX) is often observed in conjunction with adiabatic shear banding (ASB) in polycrystalline materials and is believed to be a critical softening mechanism contributing to the material instability. The recrystallized nanograins in the shear band have few dislocations compared to the material outside of the shear band. We reformulate a recently developed continuum theory of polycrystalline plasticity and include the creation of grain boundaries. While the shear-banding instability emerges because thermal heating is faster than heat dissipation, recrystallization is interpreted as an entropic effect arising from the competition between dislocation creation and grain boundary formation and is a significant softening mechanism. We show that our theory closely matches recent results in sheared 316L stainless steel. The theory thus provides a thermodynamically consistent way to systematically describe the formation of shear bands and recrystallized grains therein. The numerical tool has recently been applied to the modeling of deformation twinning in high-purity Ti which will be briefly discussed.

### Keaton Burns (MIT)

Title: Flexible spectral methods and high-level programming for PDEs

Abstract: The large-scale numerical solution of PDEs is an essential part of scientific research. Decades of work have been put into developing fast numerical schemes for specific equations, but computational research in many fields is still largely software-limited. Here I will discuss how algorithmic flexibility and composability can enable new science, as illustrated by the Dedalus Project. Dedalus is an open-source Python framework that automates the solution of general PDEs using spectral methods. High-level abstractions allow users to symbolically specify equations, parallelize and scale their solvers to thousands of cores, and perform arbitrary analysis with the computed solutions. I will provide an overview the code’s design and the underlying sparse spectral algorithms, and show how they are enabling novel simulations of diverse hydrodynamical systems. I will include astrophysical and geophysical applications using new bases for tensor-valued equations in spherical domains, immersed boundary methods for multiphase flows, and multi-domain simulations interfacing Dedalus with other PDE and integral equation solvers.