Sayas Numerics Seminar Archives for Fall 2021 to Spring 2022
From integrating to learning dynamics: new studies on Linear Multistep Methods
When: Tue, September 8, 2020 - 3:30pmWhere: ONLINE https://sayasseminar.math.umd.edu/
Speaker: Qiang Du (Columbia University) -
Abstract: Numerical integration of given dynamic systems can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. Solving both forward and inverse problems forms the loop of informative and intelligent scientific computing.
This lecture is concerned with the application of Linear multistep methods (LMMs) in the inverse problem setting that has been gaining importance in data-driven modeling of complex dynamic processes via deep/machine learning. While a comprehensive mathematical theory of LMMs as popular numerical integrators of prescribed dynamics has been developed over the last century and has become textbook materials in numerical analysis, there seems to be a new story when LMMs are used in a black box machine learning formulation for learning dynamics from observed states.
A natural question is concerned with whether a convergent LMM for integrating known dynamics is also suitable for discovering unknown dynamics. We show that the conventional theory of consistency, stability and convergence of LMM for time integration must be reexamined for dynamics discovery, which leads to new results on LMM that have not been studied before. We present refined concepts and algebraic criteria to assure stable and convergent discovery of dynamics in some idealized settings. We also apply the theory to some popular LMMs and make some interesting observations on their second characteristic polynomials.
(This is part of a joint work with Rachael Keller of Columbia).
How the SVD Saves the Universe
When: Tue, September 15, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Cleve Moler (MathWorks) -
Abstract: What is the matrix SVD? How did Spock use the SVD to save the Universe in the first Star Trek movie? How can you use the SVD to study the human gait, to find a date, to win elections, and to face Covid-19?
Cleve Moler is a former math professor, the author of the first MATLAB, a cofounder of MathWorks, and a SIAM visiting lecturer.
Data Driven Governing Equations Recovery with Deep Neural Networks
When: Tue, September 22, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Dongbin Xiu (Ohio State University) -
Abstract: We present effective numerical algorithms for recovering unknown governing equations from measurement data. Upon recasting the problem into a function approximation problem, we discuss the importance of using a large number of short bursts of trajectory data, rather than using data from a small number of long trajectories. Several recovery strategies using deep neural networks (DNNs) are presented. We demonstrate that residual network (ResNet) is particularly suitable for equation discovery, as it can produce exact time integrator for numerical prediction. We then present a set of applications of the DNN learning of unknown dynamical systems, which may contain random parameters or missing variables, as well as learning of unknown partial differential equations. The numerical examples demonstrate that DNNs can be a highly effective tool for data driven physics recovery.
Deep Learning Interpretation: Flip Points and Homotopy Methods
When: Tue, September 29, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Roozbeh Yousefzadeh (Yale University) -
Abstract: This talk concerns methods for studying deep learning models and interpreting their outputs and their functional behavior. A trained model is a function that maps inputs to outputs. Deep learning has shown great success in performing different machine learning tasks; however, these models are complicated mathematical functions, and their interpretation remains a challenging research question. We formulate and solve optimization problems to answer questions about the models and their outputs. A deep classifier partitions its domain and assigns a class to each of those partitions. Partitions are defined by the decision boundaries, but such boundaries are geometrically complex. Specifically, we study the decision boundaries of models using flip points, points on those boundaries. The flip point closest to a given input is of particular importance, and this point is the solution to a well-posed optimization problem. To compute the closest flip point, we develop a homotopy algorithm that transforms the deep learning function in order to overcome the issues of vanishing and exploding gradients. We show that computing closest flip points allows us to systematically investigate the model, identify decision boundaries, interpret and audit the models with respect to individual inputs and entire datasets, and find vulnerability against adversarial attacks. We demonstrate that flip points can help identify mistakes made by a model, improve their accuracy, and reveal the most influential features for classifications.
Fractional Deep Neural Network via Constrained Optimization
When: Tue, September 29, 2020 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Ratna Khatri (U.S. Naval Research Laboratory) -
Abstract: In this talk, we will introduce a novel algorithmic framework for a deep neural network (DNN) which allows us to incorporate history (or memory) into the network. This DNN, called Fractional-DNN, can be viewed as a time-discretization of a fractional in time nonlinear ordinary differential equation (ODE). The learning problem then is a minimization problem subject to that fractional ODE as constraints. We test our network on datasets for classification problems. The key advantage of the fractional-DNN is a significant improvement to the vanishing gradient issue, due to the memory effect.
Fractional Optimal Control Problems with States Constraints: Algorithm and Analysis
When: Tue, September 29, 2020 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Deepanshu Verma (George Mason University) -
Abstract: Motivated by several applications in geophysics and machine learning, in this talk, we introduce a novel class of optimal control problems with fractional PDEs. The main novelty is due to the obstacle type constraints on the state. The analysis of this problem has required us to create several new, widely applicable, mathematical tools such as characterization of dual of fractional Sobolev spaces, regularity of PDEs with measure-valued datum. We have created a Moreau-Yosida based algorithm to solve this class of problems. We establish convergence rates with respect to the regularization parameter. Finite element discretization is carried out and a rigorous convergence of the numerical scheme is established. Numerical examples confirm our theoretical findings.
Practical Leverage-Based Sampling for Low-Rank Tensor Decomposition
When: Tue, October 6, 2020 - 3:30pmWhere: ONLINE https://sayasseminar.math.umd.edu/
Speaker: Tamara G. Kolda (Sandia National Laboratories) -
Abstract: Conventional algorithms for finding low-rank canonical polyadic (CP) tensor decompositions are unwieldy for large sparse tensors. The CP decomposition can be computed by solving a sequence of overdetermined least problems with special Khatri-Rao structure. In this work, we present an application of randomized algorithms to fitting the CP decomposition of sparse tensors, solving a significantly smaller sampled least squares problem at each iteration with probabilistic guarantees on the approximation errors. Prior work has shown that sketching is effective in the dense case, but the prior approach cannot be applied to the sparse case because a fast Johnson-Lindenstrauss transform (e.g., using a fast Fourier transform) must be applied in each mode, causing the sparse tensor to become dense. Instead, we perform sketching through leverage score sampling, crucially relying on the fact that the structure of the Khatri-Rao product allows sampling from overestimates of the leverage scores without forming the full product or the corresponding probabilities. Naive application of leverage score sampling is ineffective because we often have cases where a few scores are quite large, so we propose a novel hybrid of deterministic and random leverage-score sampling which consistently yields improved fits. Numerical results on real-world large-scale tensors show the method is significantly faster than competing methods without sacrificing accuracy.
This is joint work with Brett Larsen, Stanford University.
Machine Learning meets Optimal Transport: Old solutions for new problems and vice versa
When: Tue, October 13, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Lars Ruthotto (Emory University) -
Abstract: This talk presents new connections between optimal transport (OT), which has been a critical problem in applied mathematics for centuries, and machine learning (ML), which has been receiving enormous attention in the past decades. In recent years, OT and ML have become increasingly intertwined. This talk contributes to this booming intersection by providing efficient and scalable computational methods for OT and ML.
The first part of the talk shows how neural networks can be used to efficiently approximate the optimal transport map between two densities in high dimensions. To avoid the curse-of-dimensionality, we combine Lagrangian and Eulerian viewpoints and employ neural networks to solve the underlying Hamilton-Jacobi-Bellman equation. Our approach avoids any space discretization and can be implemented in existing machine learning frameworks. We present numerical results for OT in up to 100 dimensions and validate our solver in a two-dimensional setting.
The second part of the talk shows how optimal transport theory can improve the efficiency of training generative models and density estimators, which are critical in machine learning. We consider continuous normalizing flows (CNF) that have emerged as one of the most promising approaches for variational inference in the ML community. Our numerical implementation is a discretize-optimize method whose forward problem relies on manually derived gradients and Laplacian of the neural network and uses automatic differentiation in the optimization. In common benchmark challenges, our method outperforms state-of-the-art CNF approaches by reducing the network size by 8x, accelerate the training by 10x- 40x and allow 30x-50x faster inference.
Application of adaptive ANOVA and reduced basis methods to the stochastic Stokes-Brinkman problem
When: Tue, October 20, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Kevin Williamson (University of Maryland, Baltimore County) -
Abstract: The Stokes-Brinkman equations model fluid flow in highly heterogeneous porous media. In this presentation, we consider the numerical solution of the Stokes-Brinkman equations with stochastic permeabilities, where the permeabilities in subdomains are assumed to be independent and uniformly distributed within a known interval. We employ a truncated anchored ANOVA decomposition alongside stochastic collocation to estimate the moments of the velocity and pressure solutions. Through an adaptive procedure selecting only the most important ANOVA directions, we reduce the number of collocation points needed for accurate estimation of the statistical moments. However, for even modest stochastic dimensions, the number of collocation points remains too large to perform high-fidelity solves at each point. We use reduced basis methods to alleviate the computational burden by approximating the expensive high-fidelity solves with inexpensive approximate solutions on a low-dimensional space. We furthermore develop and analyze rigorous a posteriori error estimates for the reduced basis approximation. We apply these methods to 2D problems considering both isotropic and anisotropic permeabilities.
Parallel Implicit-Explicit General Linear Methods
When: Tue, October 20, 2020 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Steven Roberts (Virginia Tech) -
Abstract: High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This presentation describes two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multistage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.
On optimal coarse grid correction for the Optimized Schwarz Method
When: Tue, October 20, 2020 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Faycal Chaouqui (Temple University) -
Abstract: We present a new optimal coarse space correction for the optimized Restricted Additive Schwarz method. We use coarse spaces defined by harmonic extensions of interface and surface functions to the subdomains' interior. In particular, we show that these coarse spaces yield convergence in a single iteration when fully used. We then explain how to choose and implement approximations of these coarse spaces utilizing the operator's spectral information. Numerical examples are provided to illustrate the performance of the ideas presented.
Multivariate positive quadrature rules and computational Tchakaloff theorems
When: Tue, October 27, 2020 - 3:30pmWhere: ONLINE https://sayasseminar.math.umd.edu/
Speaker: Akil Narayan (University of Utah) -
Abstract: The design of interpolatory quadrature rules with positive weights is of great interest in approximation theory and scientific computing: Such quadrature rules achieve near-optimal approximation of integrals and are associated with well-behaved Lebesgue constants. Such quadrature rules are used heavily in applications like uncertainty quantification and computational finance.
Computing such quadrature rules in more than one dimension is an arduous task, frequently attempted with non-convex optimization schemes. The success or failure of such approaches typically depends on the type of domain, dimension, or approximation space. We present a new procedure whose implementation is a probabilistic algorithm based on a novel constructive proof of Tchakaloff's theorem. In particular, we can successfully compute size-N quadrature rules in high dimensions for general approximation spaces. The main feature of our algorithm is a complexity that depends algebraically on N, and in some cases is strictly linear in the dimension. We illustrate that our procedure is effective even for quadrature on complex and unbounded multivariate domains.
ALESQP: An Augmented Lagrangian Equality-constrained SQP Method for Function-space Optimization with General Constraints
When: Tue, November 3, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Denis Ridzal (Sandia National Laboratories) -
Abstract: We present a new algorithm for infinite-dimensional optimization with general constraints, called ALESQP. In a nutshell, ALESQP is an augmented Lagrangian method that penalizes inequality constraints and solves equality-constrained nonlinear optimization subproblems at every iteration. The subproblems are solved using a matrix-free trust-region sequential quadratic programming (SQP) method that takes advantage of iterative, i.e., inexact linear solvers and is suitable for PDE-constrained optimization and other large-scale applications.
We analyze convergence of ALESQP under different assumptions. We show that strong accumulation points are stationary, i.e., in finite dimensions ALESQP converges to a stationary point. In infinite dimensions we establish that weak accumulation points are feasible in many practical situations. Under additional assumptions we show that weak accumulation points are stationary.
In the context of optimal control problems, e.g., in PDE-constrained optimization, ALESQP provides a unified framework to efficiently handle general constraints on both the state variables and the control variables. A key algorithmic feature is a constraint decomposition strategy that allows ALESQP to exploit problem-specific variable scalings and inner products. We present several examples with state and control inequality constraints where ALESQP shows remarkable mesh-independent performance, requiring only a handful of outer (AL) iterations to meet constraint tolerances at the level of machine precision. At the same time, ALESQP uses the inner (SQP) loop economically, requiring only a few dozen SQP iterations in total.
Efficient solvers for nonlinear Bayesian statistical inverse problems
When: Tue, November 10, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Akwum Onwunta (George Mason University) -
Abstract: Bayesian statistical inverse problems are often solved with Markov chain Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale discrete nonlinear partial differential equations (PDEs), they are computationally challenging because one would then need to solve the forward problem at every
sample point. In this talk, the use of reduced-order models (ROMs), as well as deep neural network techniques, is considered for the forward solves within an MCMC routine. In particular, a preconditioning strategy for the ROMs is also proposed to accelerate the forward solves. Numerical experiments are provided to demonstrate the efficiency of the approach for solving forward problems and the associated statistical inverse problems.
Preconditioned accelerated gradient descent methods for locally Lipschitz smooth objectives with applications to the solution of nonlinear partial differential equations
When: Tue, November 10, 2020 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Jeahyun Park (University of Tennessee) -
Abstract: We talk about preconditioned Nesterov’s accelerated gradient descent methods (PAGD) for approximating the minimizer of locally Lipschitz smooth, strongly convex objective functionals. We introduce a second order ordinary differential equation (ODE) as the limiting case of PAGD as the step size tends to zero. Using a simple energy argument, we will show an exponential convergence of the ODE solution to its steady state. The PAGD method may be viewed as an explicit-type time-discretization scheme of the ODE system, which requires a natural time step restriction for energy stability. Assuming this restriction, an exponential rate of convergence of the PAGD sequence is demonstrated by mimicking the convergence of the solution to the ODE via energy methods. Applications of the PAGD method are made in the context of solving certain nonlinear elliptic PDE using Fourier collocation methods, and several numerical experiments are conducted. The results confirm the global geometric and h-independent convergence of the PAGD method, with an accelerated rate that is improved over the preconditioned gradient descent (PGD) method.
L infinity error estimates of HDG methods for Poisson equations
When: Tue, November 10, 2020 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Yangwen Zhang (University of Delaware) -
Abstract: We prove quasi-optimal L infinity norm error estimates (up to logarithmic factors) for the solution of Poisson's problem in two dimensional space by the standard Hybridizable Discontinuous Galerkin (HDG) method. Although such estimates are available for conforming and mixed finite element methods, this is the first proof for HDG. The method of proof is motivated by known L infinity norm estimates for mixed finite elements. We show two applications: the first is to prove optimal convergence rates for boundary flux estimates, and the second is to prove that numerically observed convergence rates for the solution of a Dirichlet boundary control problem are to be expected theoretically. Numerical examples show that the predicted rates are seen in practice.
Scalable High-Order Finite Elements for Compressible Hydrodynamics
When: Tue, November 17, 2020 - 3:30pmWhere: ONLINE https://sayasseminar.math.umd.edu/
Speaker: Tzanio Kolev (Lawrence Livermore National Laboratory) -
Abstract: The discretization of the Euler equations of gas dynamics ("compressible hydrodynamics") in a moving material frame is at the heart of many multi-physics simulation codes. The Arbitrary Lagrangian-Eulerian (ALE) framework is frequently applied in these settings in the form of a Lagrange phase, where the hydrodynamics equations are solved on a moving mesh, followed by a three-part "advection phase" involving mesh optimization, field remap and multi-material zone treatment.
This talk presents a general Lagrangian framework [1] for discretization of compressible shock hydrodynamics using high-order finite elements. The use of high-order polynomial spaces to define both the mapping and the reference basis functions in the Lagrange phase leads to improved robustness and symmetry preservation properties, better representation of the mesh curvature that naturally develops with the material motion and significant reduction in mesh imprinting. We will discuss the application of the curvilinear technology to the "advection phase" of ALE, including a DG-advection approach for conservative and monotonic high-order finite element interpolation (remap), as well as to coupled physics, such as electromagnetic diffusion. We will also review progress in robust and efficient algorithms for high-order mesh optimization, matrix-free preconditioning, high-order time integration and matrix-free monotonicity, which are critical components for the successful use of high-order methods in the compressible ALE settings
In addition to their mathematical benefits, high-order finite element discretizations are a natural fit for modern HPC hardware, because their order can be used to tune the performance, by increasing the FLOPs/bytes ratio, or to adjust the algorithm for different hardware. In this direction, we will present some of our work on scalable high-order finite element software that combines the modular finite element library MFEM [2], the high-order shock hydrodynamics code BLAST [3] and its miniapp Laghos [4], where we will demonstrate the benefits of our approach with respect to strong scaling and GPU acceleration. Finally, we will give a brief update on related efforts in the co-design Center for Efficient Exascale Discretizations (CEED) in the Exascale Computing Project (ECP) of the DOE [5].
[1] High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics, V. Dobrev and Tz. Kolev and R. Rieben, SIAM Journal on Scientific Computing, (34) 2012, pp.B606-B641.
[2] MFEM: Modular finite element library
[3] BLAST: High-order shock hydrodynamics
[4] Laghos: Lagrangian high-order solver
[5] Center for Efficient Exascale Discretizations
Numerical methods and analysis of the stochastic Navier-Stokes equations
When: Tue, December 1, 2020 - 3:30pmWhere: ONLINE https://sayasseminar.math.umd.edu/
Speaker: Andreas Prohl (Universitaet Tuebingen, Germany) -
Jet Marching Methods for Solving the Eikonal Equation
When: Tue, December 8, 2020 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Samuel Potter (University of Maryland, College Park) -
Abstract: Solvers for the eikonal equation fall into one of two camps: 1) direct solvers, which are typically based on Dijkstra's algorithm, the classical example being the fast marching method, and 2) iterative solvers, best represented by the fast sweeping method. Direct solvers run in optimal, output-sensitive time, but are generally only first-order accurate. Iterative methods can be made higher-order, but require wide stencils; additionally, their performance degrades when applied to complicated domains. We develop high-order, semi-Lagrangian direct solvers for the eikonal equation, which use compact stencils. These solvers march the partial derivatives of the eikonal up to some order in addition to the eikonal itself. Semi-Lagrangian updates approximate the eikonal and updating characteristics using local Hermite interpolants. We present convergence guarantees, thorough numerical results, and numerical examples related to room acoustic modeling, which is our motivating application.
LDG approximation of deformations of bilayer plates
When: Tue, December 8, 2020 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Shuo Yang (University of Maryland, College Park) -
Abstract: Bilayer plates are slender structures made of two thin layers of different materials glued together. These layers react differently to non-mechanical stimuli and develop large bending deformations. We design a LDG approach for the constrained minimization problem of bilayer plates. With this new discretization, we prove the Gamma-convergence and design a fully practical gradient flow scheme. We also prove the energy stability and the control of constraint defect for this scheme.
An unfitted method for harmonic map flows into S^2
When: Tue, December 8, 2020 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Vladimir Yushutin (University of Maryland, College Park) -
Abstract: We present a novel second-order scheme for gradient flows of the Dirichlet energy of maps from a three-dimensional domain to the unit sphere. The unit-length constraint is treated with help of a Lagrange multiplier, and Dirichlet conditions are imposed weakly on piecewise quadratic level sets defined on an unfitted regular mesh. The method is applied to study trapping pits for colloid particles immersed in a liquid crystal.
Using Flip Points to Understand and Debug Deep Learning Models
When: Tue, January 26, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Dianne O'Leary (University of Maryland, College Park) -
Abstract: Deep learning models for classification have been criticized for their lack of easy interpretation, which undermines confidence in their use for important applications. We demonstrate the power of flip points in interpreting and debugging these models.
A flip point is a point on the boundary between two output classes. Finding the flip point that is closest to a given input is a tractable optimization problem.
We demonstrate the use of closest flip points to identify flaws in a neural network model, to generate synthetic training data to correct the flaws, to assess uncertainty in the model's output, and to provide individual and group-level interpretations of the model.
This is joint work with Roozbeh Yousefzadeh.
Detailed Simulation of Viral Propagation and Mitigation in the Built Environment
When: Tue, February 2, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Rainald Lohner (George Mason University) -
Abstract: A summary is given of the mechanical characteristics of virus contaminants and the transmission via droplets and aerosols. The ordinary and partial differential equations describing the physics of these processes with high fidelity are presented, as well as appropriate numerical schemes to solve them.
Several examples taken from recent evaluations of the built environment are shown, as well as mitigation options such as UV radiation. Thereafter, the optimal placement of sensors for post Cov-19 opening scenarios is discussed.
Hybrid frequency-time analysis and numerical methods for time-dependent wave propagation
When: Tue, February 9, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Thomas Anderson (University of Michigan) -
Abstract: 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.
A Virtual Element Method for Magnetohydrodynamics
When: Tue, February 9, 2021 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Sebastian Naranjo (Oregon State University) -
Abstract: 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.
Learning Collective Dynamics from Trajectory Data
When: Tue, February 9, 2021 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Ming Zhong (Johns Hopkins University) -
Abstract: 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.
Parallel preconditioning for time-dependent PDEs and PDE control
When: Tue, February 16, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Andy Wathen (University of Oxford, UK) -
Abstract: We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation. This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated.
The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple computational results will be shown for the heat equation and the wave equation which indicate the potential as a parallel-in-time method.
Direct Sampling Algorithms for Inverse Scattering
When: Tue, February 23, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Isaac Harris (Purdue University) -
Abstract: In this talk, we will discuss a recent qualitative imaging method referred to as the Direct Sampling Method for inverse scattering. This method allows one to recover a scattering object by evaluating an imaging functional that is the inner-product of the far-field data and a known function. It can be shown that the imaging functional is strictly positive in the scatterer and decays as the sampling point moves away from the scatterer. The analysis uses the factorization of the far-field operator and the Funke-Hecke formula. This method can also be shown to be stable with respect to perturbations in the scattering data. We will discuss the inverse scattering problem for acoustic waves.
This is joint work with A. Kleefeld.
Time Fractional Gradient Flows: Theory and Numerics
When: Tue, February 23, 2021 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Wenbo Li (University of Tennessee, Knoxville) -
Abstract: This talk is concerned with a generalization of the classical gradient flow problem to the case where the time derivative is replaced by the fractional Caputo derivative. We first define the energy solution for this problem and prove its existence and uniqueness, under suitable assumptions, via a generalized minimizing movements scheme. We then introduce and discuss a semi-discrete numerical scheme where we only discretize in time. We then present a posteriori error estimates, which allow to adaptively choose time steps. We also give convergence rates. Several numerical experiments are also presented in the end to help the understanding of the problem and the numerical scheme.
Quadrature by Zeta Correction
When: Tue, February 23, 2021 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Bowei Wu (University of Texas at Austin) -
Abstract: We consider the approximation of singular integrals on closed smooth contours and surfaces. Such integrals frequently arise in the solution of Boundary Integral Equations. We introduce a new quadrature method for these integrals that attains high-order accuracy, numerical stability, ease of implementation, and compatibility with the "fast" algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Our quadrature method exploits important connections between the punctured trapezoidal rule and the Riemann zeta function, leading to remarkably simple construction procedures.
On learning kernels for numerical approximation and learning
When: Tue, March 2, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Houman Owhadi (California Institute of Technology) -
Abstract: There is a growing interest in solving numerical approximation problems as learning problems. Popular approaches can be divided into (1) Kernel methods, and (2) methods based on variants of Artificial Neural Networks.
We illustrate the importance of using adapted kernels in kernel methods and discuss strategies for learning kernels from data. We show how ANN methods can be formulated and analyzed as (1) kernel methods with warping kernels learned from data (2) discretized solvers for a generalization of image registration algorithms in which images are replaced by high dimensional shapes.
Finite Element and Neural Network
When: Tue, March 9, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Jinchao Xu (Pennsylvania State University) -
Abstract: Piecewise polynomials with certain global smoothness can be given by traditional finite element methods and also by neural networks with some power of ReLU as activation function.
In this talk, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.
Efficient quantum algorithm for dissipative nonlinear differential equations
When: Tue, March 16, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Andrew Childs (Department of Computer Science and Institute for Advanced Computer Studies University of Maryland, College Park) -
Abstract: While there has been extensive previous work on efficient quantum algorithms for linear differential equations, analogous progress for nonlinear differential equations has been severely limited due to the linearity of quantum mechanics.
Despite this obstacle, we develop a quantum algorithm for initial value problems described by dissipative quadratic n-dimensional ordinary differential equations. Assuming R
Column Partition based Distributed Algorithms for Coupled Convex Sparse Optimization: Dual and Exact Regularization Approaches
When: Tue, March 23, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: E. K. Hathibelagal Kammara (University of Maryland, Baltimore County) -
Abstract: In this talk, we discuss column partition based distributed schemes for a class of large-scale convex sparse optimization problems, e.g., basis pursuit (BP), LASSO, basis pursuit denosing (BPDN), and their extensions, e.g., fused LASSO. We are particularly interested in the cases where the number of (scalar) decision variables is much larger than the number of (scalar) measurements, and each agent has limited memory or computing capacity such that it only knows a small number of columns of a measurement matrix. These problems in consideration are densely coupled and cannot be formulated as separable convex programs using column partition.
To overcome this difficulty, we consider their dual problems which are separable or locally coupled. Once a dual solution is attained, it is shown that a primal solution can be found from the dual of corresponding regularized BP-like problems under suitable exact regularization conditions. A wide range of existing distributed schemes can be exploited to solve the obtained dual problems.
This yields two-stage column partition based distributed schemes for LASSO-like and BPDN-like problems; the overall convergence of these schemes is established using sensitivity analysis techniques. Numerical results illustrate the effectiveness of the proposed schemes.
Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance
When: Tue, March 23, 2021 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Jin-Peng Liu (University of Maryland, College Park) -
Abstract: Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.
https://arxiv.org/abs/2012.06283
RCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems
When: Tue, March 23, 2021 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Chao Chen (University of Texas at Austin) -
Abstract: We introduce a randomized algorithm, namely RCHOL, to construct an approximate Cholesky factorization for a given sparse Laplacian matrix (a.k.a., graph Laplacian). The (exact) Cholesky factorization for the matrix introduces a clique in the associated graph after eliminating every row/column. By randomization, RCHOL samples a subset of the edges in the clique. We prove RCHOL is breakdown free and apply it to solving linear systems with symmetric diagonally-dominant matrices. In addition, we parallelize RCHOL based on the nested dissection ordering for shared-memory machines. Numerical experiments demonstrated the robustness and the scalability of RCHOL. For example, our parallel code scaled up to 64 threads on a single node for solving the 3D Poisson equation, discretized with the 7-point stencil on a 1024 by 1024 by 1024 grid, or one billion unknowns.
https://arxiv.org/abs/2011.07769
Iterative methods in quantum chemistry and first-principle materials science
When: Tue, March 30, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Eric Cances (Ecole des Ponts ParisTech and INRIA Paris, France) -
Abstract: Electronic structure calculation is one of the major application fields of scientific computing. It is used daily in any chemistry or materials science department, and it accounts for a high percentage of machine occupancy in supercomputing centers. Current challenges include the study of complex molecular systems and processes (e.g. photosynthesis, high-temperature superconductivity...), and the building of large, reliable databases for the design of materials and drugs.
The most commonly used models are the Kohn-Sham Density Functional Theory (DFT), and the (post) Hartree-Fock models. The Hartree-Fock and Kohn-Sham models have similar mathematical structures. They consist in minimizing an energy functional on the Sobolev space H^1(R^3)^N under L^2-orthonormality constraints. The associated Euler-Lagrange equations are systems on nonlinear elliptic PDEs. After discretization in a Galerkin basis, one obtains smooth optimization problems on matrix manifolds, or on convex hulls of matrix manifolds.
Solving these problems is easy for small simple molecular systems, but very difficult for large or complex systems. Two classes of numerical methods compete in the field: constrained direct minimization of the energy functional, and self-consistent field (SCF) iterations to solve the Euler-Lagrange equations. In this talk, I will present a comparative study of these two approaches, as well as new efficient algorithms for systems with spin symmetries.
Operator Inference: Bridging model reduction and scientific machine learning
When: Tue, April 6, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Karen Willcox (University of Texas at Austin) -
Abstract: Model reduction methods have grown from the computational science community, with a focus on reducing high-dimensional models that arise from physics-based modeling, whereas machine learning has grown from the computer science community, with a focus on creating expressive models from black-box data streams. Yet recent years have seen an increased blending of the two perspectives and a recognition of the associated opportunities.
This talk presents our work in operator inference, where we learn effective reduced-order operators directly from data. The physical governing equations define the form of the model we should seek to learn. Thus, rather than learn a generic approximation with weak enforcement of the physics, we learn low-dimensional operators whose structure is defined by the physics. This perspective provides new opportunities to learn from data through the lens of physics-based models and contributes to the foundations of Scientific Machine Learning, yielding a new class of flexible data-driven methods that support high-consequence decision-making under uncertainty for physical systems.
Inverse Problems Without Adjoints
When: Tue, April 13, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Andrew Stuart (California Institute of Technology) -
Abstract: I will describe the use of ensemble based particle methods to solve Bayesian inverse problems, including ensemble Kalman methods, methods based on multiscale SDEs, and conjunctions of the two approaches.
Space-Time Multi-patch Discontinuous Galerkin Isogeometric Analysis for Parabolic Evolution Problems
When: Tue, April 20, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Stephen Moore (University of Cape Coast, Ghana) -
Abstract: We present and analyze a stable space-time multi-patch discontinuous Galerkin Isogeometric Analysis (dG-IGA) scheme for the numerical solution of parabolic evolution equations in moving space-time computational domains. Following Langer et. al, 2016, we use a time-upwind test function and apply multi-patch discontinuous Galerkin IGA methodology for discretizing the evolution problem both in space and in time. This yields a discrete bilinear form which is elliptic on the IGA space with respect to a space-time dG norm. This property together with a corresponding boundedness property, consistency and approximation results for the IGA spaces yields an a priori discretization error estimate with respect to the space-time dG norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IGA spaces.
Flux-stability for conservation laws with discontinuous flux and convergence rates of the front tracking method
When: Tue, April 20, 2021 - 3:50pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Adrian Ruf (ETH Zurich, Switzerland) -
Abstract: In this talk we consider conservation laws with discontinuous flux. Such equations have a wide range of applications including vehicle traffic flow in the presence of abruptly varying road conditions and two-phase flow through heterogeneous porous media.
We consider the class of adapted entropy solutions where the spatial flux dependency is piecewise constant. This setting allows the flux to change abruptly across finitely many points in space. We prove that adapted entropy solutions are L1 stable with respect to changes in the initial datum, the flux function, and the spatial dependency parameter. This result allows us to derive a convergence rate estimate for the front tracking method - a numerical method which is widely used in the field of conservation laws with discontinuous flux. To the best of our knowledge, both of these results are the first of their kind in the literature on conservation laws with discontinuous flux.
We briefly present numerical experiments motivated by uncertainty quantification for two-phase reservoir simulations for reservoirs with varying geological properties.
Best low-rank approximations and Kolmogorov n-widths
When: Tue, April 20, 2021 - 4:10pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Espen Sande (University of Rome Tor Vergata, Italy) -
Abstract: It is well known that if the singular values of a matrix A are distinct, then the best rank-n approximation to A is uniquely determined in the Frobenius norm and given by the truncated singular value decomposition. On the other hand, this uniqueness is in general not true for best rank-n approximations in the spectral norm.
In this talk we relate the problem of finding best rank-n approximations in the spectral norm to Kolmogorov n-widths and corresponding optimal spaces. By providing new criteria for optimality of spaces with respect to the n-width, we describe a large family of best rank-n approximations to A. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties. Special attention is paid to the case of rank-1 approximation.
This talk is based on joint work with Michael S. Floater, Carla Manni and Hendrik Speleers.
Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction
When: Tue, April 27, 2021 - 3:30pmWhere: ONLINE http://sayasseminar.math.umd.edu
Speaker: Christoph Schwab (ETH Zurich, Switzerland) -
Abstract: Centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded domains in Euclidean space or smooth, compact and orientable manifolds are determined by their covariance operators. We analyze the numerical approximation of centered GRFs given sample-wise as variational solutions to coloring operator equations driven by spatial white noise, with coloring operator being elliptic, self-adjoint and positive from the Hormander class. This includes the Matern class of GRFs as a special case.
Precision and covariance operators can be represented as bi-infinite matrices. Finite sections of these may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. and compressed ("tapered" in the language of graphical statistical models). We prove that tapering by thresholding as e.g. in [1] applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations.
"Numerical sparsity" signifies that, asymptotically, a number of nonzero matrix entries that grows linearly with the number of GRF parameters. The tapering strategy is non-adaptive and the locations of these nonzero matrix entries are known a priori.
Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. This extends [2] to estimation of (finite sections of) pseudodifferential covariances for GRFs by this fast MLMC method.
This is joint work with
H. Harbrecht (University of Basel, Switzerland)
L. Herrmann (RICAM, Linz, Austria)
K. Kirchner (TU Delft, The Netherlands)
Preprint (SAM Report 2021-09)
https://math.ethz.ch/sam/research/reports.html?id=951
References
[1] P.J. Bickel and E. Levina: Covariance regularization by thresholding, Ann. Statist., 36 (2008), 2577-2604
[2] P.J. Bickel and E. Levina: Regularized Estimation of Large Covariance Matrices, Ann. Stat., 36 (2008), pp. 199-227