9:30 Randy Price (George Mason University):
NINNs: Nudging Induced Neural Networks
Training accurate neural networks is a challenging task requiring time consuming trial-and-error. To combat this we have developed NINNs which improve the accuracy of pre-existing neural networks. NINNs are based on the idea that neural networks can be nudged during forward propagation towards given data. In this talk we propose two method types that are based on the nudging algorithm for ODEs and PDEs. The two method types are then applied to two numerical studies. First, data assimilation results for the Lorenz 63 and 96 ODEs are presented. Second, NINNs are applied to a chemically reacting flow example.
  9:50 Abhishek Balakrishna (University of Maryland, Baltimore County):
Data Assimilation and Determining Functionals for Three-Dimensional Fluids
Data assimilation is a technique that combines observational data with a given model to improve the model's accuracy. We begin by discussing the motivation and the idea behind data assimilation; we then proceed to apply a particular data assimilation technique (AOT algorithms) to the 3-D Boussinesq equation (3D BE). The key point to note being that the data assimilation is performed using only “finitely-observed data”. We then observe that such a data assimilated system has a “strong” solution (or a regular solution) when the observed data satisfies certain conditions. Also, the data assimilated solution asymptotically approaches the actual solution at an exponential rate. Lastly, leveraging on the proximity of the 3-D Navier-Stokes equation (3D NSE) to the 3D BE, we conclude by stating a regularity criterion for the 3D NSE that is based on finitely observed data.
10:10 Lu Zhang (Columbia University):
Data-Driven Joint Inversions for PDE Models
The task of simultaneously reconstructing multiple physical coefficients in partial differential equations from observed data is ubiquitous in applications. In this work, we propose an integrated machine learning and model-based iterative reconstruction framework for such joint inversion problems where additional data on the unknown coefficients are supplemented for better reconstructions. Our method couples the supplementary data with the PDE model to make the data-driven modeling process consistent with the model-based reconstruction procedure. We characterize the impact of learning uncertainty on the joint inversion results for two typical model inverse problems. Numerical evidences are provided to demonstrate the feasibility of using data-driven models to improve joint inversion of physical models.
11:00 Lucas Bouck (University of Maryland, College Park):
Finite Element Approximation of a Membrane Model for Liquid Crystal Polymeric Networks
Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. The derivation is similar to derivations in Ozenda, Sonnet, and Virga (2020) and Cirak et. al. (2014). We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy. For minimizing the discrete problem, employ a nonlinear gradient flow scheme, which is energy stable. Finally, we present computations, which include configurations that arise from liquid crystal defects as well as nonisometric origami.
11:20 Yukun Yue (Carnegie Mellon University):
Convergence Analysis For A Semi-discrete Energy Stable Scheme For Hydrodynamic Q-tensor Model
We present convergence analysis of an unconditional energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model based on Invariant Quadratization Method(IEQ). This model couples a Navier-Stokes system for the flows and a parabolic type Q-tensor system governing the nematic crystal director fields. We prove the stability properties of the scheme and show convergence to weak solutions of this coupled liquid crystal system.
11:40 Farjana Siddiqua (University of Pittsburgh):
A second-order symplectic method for an advection-diffusion-reaction problem in Bioseparation
An advection-diffusion-reaction problem with non-homogeneous boundary conditions is considered that modes the chromatography process, a vital stage in bioseparation. We prove stability and error estimates for both constant and affine adsorption, using the midpoint method for time discretization and finite elements for spatial discretization. In addition, we did the stability analysis for nonlinear, explicit adsorption in the continuous case. The numerical tests are performed that validate our theoretical results.
12:00 Madhu Gupta (George Mason University):
Sparsity-based nonlinear reconstruction of optical parameters in two-photon photoacoustic computed tomography
We discuss a new nonlinear optimization approach for the sparse reconstruction of single-photon absorption and two-photon absorption coefficients in photoacoustic computed tomography (PACT). This framework comprises minimizing an objective functional involving a least squares fit of the interior pressure field data corresponding to two boundary source functions, where the absorption coefficients and the photon density are related through a semi-linear elliptic partial differential equation (PDE) arising in photoacoustic tomography. Further, the objective functional consists of an \(L^1\) regularization term that promotes sparsity patterns in absorption coefficients. We also discuss theoretical results and several numerical experiments to demonstrate the effectiveness of the proposed framework.
  3:00 Daniel Hayes (University of Delaware):
Saddle Point Least Squares for Convection-Reaction-Diffusion
We use a saddle point least squares discretization of Convection-Reaction-Diffusion equations. Ideas of optimal test norms, and specific choices for the discrete test and trial spaces are considered in order to have a stable and efficient solving strategy of the problem. The discretization is robust for certain right hand side data and can be combined with other known methods for solving more general (practical) problems.
  3:20 Zhaopeng Hao (Purdue University):
Pseudo-spectral methods for the integral fractional Laplacian
We study the numerical solution for fractional elliptic equations with the integral fractional Laplacian. The bottleneck for the pseudo-differential operator is the numerical evaluation of the hyper-singular integral. To address this issue, we derive a pseudo-spectral formula for the global integral fractional Laplacian operator, which is based on the fractional order-dependent, generalized multi-quadratic radial basis functions. The proposed formula provides a new starting point to design a simple, easy-to-implement, nearly integration-free numerical method for solving the challenging multi-dimensional boundary value problems with the integral fractional Laplacian. Numerical results are presented to demonstrate the accuracy of the novel method. In particular, we observe that our approach enjoys spectral accuracy when the solution is smooth.
  3:40 Fruzsina Agocs (Center for Computational Mathematics, Flatiron Institute):
A fast and accurate solver for highly oscillatory ODEs
Oscillatory systems are ubiquitous in physics: they arise in celestial and quantum mechanics, electrical circuits, molecular dynamics, and beyond. Yet even in the simplest case, when the frequency of oscillations changes slowly but is large, the vast majority of numerical methods struggle to solve such equations. Methods based on approximating the solution with polynomials are forced to take \(\mathcal{O}(k)\) timesteps, where \(k\) is the characteristic frequency of oscillations. This scaling can generate unacceptable computational costs when the ODE in question needs to be solved billions of times, e.g. as the forward modelling step of Bayesian parameter estimation. In this talk I will introduce an efficient method for solving 2nd order, linear ODEs with highly oscillatory solutions. The solver employs two methods: in regions where the solution varies slowly, it uses a spectral method based on Chebyshev nodes and with an adaptive stepsize, but in the highly oscillatory phase it automatically switches over to an asymptotic method. The asymptotic method constructs a nonoscillatory phase function solution of the Riccati equation associated with the ODE. In the talk I will present how the method fits in the landscape of oscillatory solvers, the theoretical underpinnings of the asymptotic solver, a summary of the switching and stepsize-update algorithms, some examples, and a brief error analysis.
  4:00 Joseph Nakao (University of Delaware):
A structure preserving, conservative, low-rank tensor scheme for solving the 1D2V Vlasov-Fokker-Planck equation
We propose a hybrid low-rank tensor scheme for solving the 1D2V Vlasov-Fokker-Planck equation in Cartesian physical space and cylindrical velocity space. The solution is full rank in physical space and low rank in velocity space. By incorporating several robust methods into our proposed algorithm, we attain a scheme that is conservative, equilibrium preserving, relative entropy dissipative, and low-rank with low storage complexity. A kinetic ion-fluid electron model is assumed; the Leonard-Bernstein-Fokker-Planck operator is discretized using a structure preserving Chang-Cooper method; and the updated solution is truncated using a local macroscopic conservative low rank tensor method. Preliminary numerical results are presented to demonstrate these properties.
  4:50 Amir Sagiv (Columbia University):
Density Estimation in Uncertainty Propagation - Approximating Pushforward Measures
In many scientific areas, uncertainty propagation is employed to account for the effect of uncertain parameters in an otherwise deterministic model. Traditionally, the analysis of such problems is done through the lens of moment approximation. However, in many applications, the "full statistics" are required, i.e., we wish to approximate the probability density function (PDF). Underlying this computational problem is a fundamental question - if two "similar" functions pushforward the same measure, would the new resulting measures be close, and if so, in what sense? We will show how the PDF of the quantity of interest can be approximated, first using a spline-based method and then using spectral methods, both with theoretical guarantees. We will then present an alternative viewpoint: through optimal transport theory, a Wasserstein-distance formulation of our problem yields a much simpler and widely applicable theory.
  5:10 Jiaxing Liang (University of Maryland, College Park):
Surrogate approximation of the Grad-Shafranov free boundary problem via stochastic collocation on sparse grids
In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations in the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. The use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30.
  5:30 Mingkai Yu (University of Maryland, Baltimore County):
State and parameter estimation from partial state observations in stochastic reaction networks
We consider chemical reaction networks modeled by a discrete state and continuous in time Markov process for the vector copy number of the species. We describe new Monte Carlo methods for the accurate and efficient estimation of unobserved states and parameters of a stochastic reaction network based on exact partial state observations in continuous time. We present evolution equations for the conditional distribution, provide particle filter algorithms and demonstrate our approach with numerical examples. Time permitting, we will also cover scenarios in which observations are made in snapshots of time.
  5:50 END