Southern Ontario Numerical Analysis Day 2019
Friday, May 3
ONTechU UA 1220
Program
Abstracts
10:10 – 10:55 : Dr. Felicia Maria G. Magpantay , Queen's University
Title: Modeling imperfect vaccines: theory and statistical inference methods
Abstract: The dynamics of vaccine-preventable diseases depend on the underlying disease process and the nature of the vaccine. I will present a general model of an imperfect vaccine and the epidemiological consequences of different modes of vaccine failure. I will also discuss likelihood-based statistical inference methods (e.g. trajectory matching, sequential Monte Carlo methods) that can be used to estimate the parameters of these models. The methods used can be extended to study and parametrize mechanistic, stochastic models of complex systems beyond those in disease ecology.
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11:00 – 11:10 : Abdullah Ali Sivas, University of Waterloo
Title: A preconditioner for a statically condensed HDG discretization of the Navier--Stokes equations
Abstract: The Navier-Stokes equations play an essential role in many practical and engineering design applications. Hence the efficient and high fidelity solution of the problem is one of the foci of research in scientific computing. In this work, we generalize the pressure convection-diffusion (PCD) preconditioner [1] to the hybridized discontinuous Galerkin (HDG) discretization of Rhebergen and Wells [2]. We will discuss the discretization and why we prefer it over other methods. We then introduce the idea behind the preconditioner and the challenges specific to the HDG discretization. For example, while this discretization lends itself to static condensation through which we can obtain a significantly smaller linear system of equations, it comes at the cost of difficulty in designing preconditioners. On the other hand, if we do not use static condensation it is straightforward to employ the PCD preconditioner, however, we have to solve a larger linear system of equations. Finally, we compare the results (static condensation vs. full) in terms of efficiency.
- H. C. Elman, D. J. Silvester, and A. J. Wathen. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, USA, 2014.
- S. Rhebergen and G.N. Wells, A hybridizable discontinuous Galerkin method for the Navier-Stokes equations with pointwise divergence-free velocity field, J. Sci. Comput., 76/3 (2018), pp 1484-1501.
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11:15 – 11:25 :
Title: Inverse Modeling of Solid-State Diffusion by Multi-objective Optimization
Abstract: We use inverse modelling via multi-objective optimization to determine, from different phase transformation models (Fickian with constant diffusion, Cahn-Hilliard, and Fickian with concentration dependence), which one can better describe the concentration and cell voltage experimental data for a graphite electrode, obtained by MRI and NMR measurements. For this we formulate the inverse problem as an optimization where we minimize a multi-objective functional that represents a convex combination of a potential-dependent functional with a concentration-dependent one. The solution of the multi-objective optimization problem is a family of reconstructed parameter values (parametrizing this family by means of the convex combination coefficient ρ). Among other parameters reconstructed in the optimization, we obtain the solid-state diffusion coefficient/function for our different models.
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11:30 – 11:40 : Reza Zolfaghari, McMaster University
Title: Numerical Integration of Stiff High-Index DAEs
Abstract: The Daets solver by Nedialkov and Pryce integrates numerically high- index DAE systems. This solver is based on explicit Taylor series and is efficient on non-stiff to mildly stiff problems, but can have severe stepsize restrictions on (very) stiff problems. Hermite-Obreschkoff (HO) methods can be viewed as a generalization of Taylor series methods. The former have smaller truncation error than the latter and can be A- or L- stable. We develop an HO method for numerical solution of stiff high-index DAEs. As in Daets, our method employs Pryce's structural analysis to determine the constraints of the problem and to organize the computations of higher-order derivatives and their gradients. We discuss this method and its ingredients: nding a consistent initial point, computing an initial guess for Newton's method, automatic differentiation for constructing the needed Jacobians, and error estimation and control. We report numerical results on stiff DAE and ODE systems illustrating the performance of our method, and in particular, its ability to take large steps on stiff problems
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11:45 – 11:55 : Chufeng Hu, University of Waterloo
Title:: Algorithms for local graph clustering
Abstract: Local spectral graph clustering methods are used to find small and medium-scale clusters without touching the whole graph. The optimal solution of [ℓ1]-regularized Page-Rank problem has many non-zero coordinates, which can be used to find those local clusters. In this talk, we present a randomized proximal coordinate descent algorithm and an accelerated version of it for solving the [ℓ1] -regularized Page-Rank problem. The proposed methods find the optimal non-zero coordinates of the problem without touching coordinates that are zero in the solution. We show that our methods have running time proportional to the number of non-zero coordinates in the optimal solution.
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1:10 – 1:55 : Dr. Adam Stinchcombe, University of Toronto
Title: Simulating the Mammalian Circadian Clock
Abstract: In order to address questions related to shift-work, jet lag, and a variety of diseases, a detailed simulation of the circadian (24-hour) timekeeping mechanism is essential. The site of the master clock is the suprachiasmatic nucleus, which consists of 20,000 electrically and chemically coupled neurons. The molecular clock is a transcription-translation feedback loop within each neuron that is modelled by a system of 180 ordinary differential equations. On a faster timescale, the electrical activity of these neurons is driven by voltage-gated ion channels, internal calcium dynamics, and synaptic currents, which are described by a ten-variable ordinary differential equation. In this talk, I will discuss some of the details of our model, but will focus on the numerical challenges associated with solving this large and multi-scale differential equation. I will describe how we use splitting, custom time-stepping schemes, pre-computation, and parallization with GPUs to create a simulation of the mammalian circadian clock that can be used to gain insight into biological questions.
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2:00 – 2:10 : Andrew Nagel, Ontario Tech University
Title: Solving Partial Differential Equations in Nanobiophysics with Deep Neural Networks
Abstract: Partial differential equations (PDEs) in nanobiophysics (NBP) often arise in complicated geometries. Typically, such problems would be solved with mesh-based numerical solvers. However, the stochastic many-body systems common to NBP are described by high-dimensional PDEs. Mesh-based solvers fail for such problems, so instead these PDEs are solved indirectly using particle simulations. Still, these particles are often subject to force fields, which are themselves described by similar PDEs. Furthermore, to establish how observables, like molecular mobility, depend on problem parameters, like molecular size, simulations must be repeated many times. A new method for solving PDEs is to approximate solutions with deep neural networks (DNNs). DNNs can even learn solutions directly from the PDE problem statement, without using any external data. In this talk, I will illustrate some benefits of this method for solving PDEs in NBP. DNNs are memory-efficient, enabling complicated electric fields to be used in GPU-accelerated particle simulations. Surprisingly, DNNs can actually solve high-dimensional PDEs directly, as an alternative to particle simulations. Finally, this method can naturally be extended to express target observables as differentiable functions of problem parameters.
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2:15 – 2:25 : Yuwei Chen, University of Toronto
Title: A penalty-like method for pricing bilateral XVA by a PDE model
Abstract: Counterparty risk is the risk to each party of a contract that a counterparty may not live up to its contractual obligations. Counterparty risk must be evaluated properly, and the risk neutral value of a derivative must be adjusted accordingly. For the pricing of the credit valuation adjustment (CVA), or, more generally, the total valuation adjustment, known as XVA, we adopt a Black-Scholes PDE model with additional non-linear terms. For the discretization, we use standard second order differences in space and Crank-Nicolson timestepping. For the treatment of the nonlinearity, we formulate a penalty-like iteration. We present numerical experiments indicating that the penalty method converges in about one iteration per timestep, irrespectively of the discretization size, and that second order convergence is exhibited
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2:30 – 2:40 : John Ernsthausen, McMaster University
Title:: Stepsize Selection in the Rigorous Defect Control of Taylor Series Methods
Abstract: Modern numerical methods for initial-value problems (IVPs) in ordinary differential equations (ODE) produce a (piecewise) differentiable numerical solution. The defect or residual evaluated at it induces a perturbed ODE that is satisfied exactly by this solution. Defect control methods try to ensure that a norm of the defect is bounded by a user-specified tolerance. This talk outlines and discusses a simple and effective stepsize selection strategy and an overall method for both validated and non-validated defect control of an explicit Taylor series method for IVP ODEs. In validated mode, this method guarantees that the defect is bounded by a user-specified tolerance.
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3:15 – 3:25 : Brady Metherall, Ontario Tech University
Title: A Nonlinear Functional Model for a Tuneable Laser
Abstract: A new nonlinear model is proposed for tuneable lasers. Using the generalized non-linear Schrödinger equation as a starting point, expressions for the transformations undergone by the pulse are derived for each component in the cavity. These transformations are then composed to give the overall effect of one trip around the cavity. The nonlinear model is solved numerically. A consequence of this model being nonlinear is that it is able to exhibit wave breaking which prior models could not. We highlight the rich structure of the boundary of stability for a particular plane of the parameter space.
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3:30 – 3:40 : Tony Joseph, Ontario Tech University
Title: Joint Spatial and Layer Attention for Convolutional Networks
Abstract: In this paper, we propose a novel approach that learns to sequentially attend to different Convolutional Neural Networks (CNN) layers (i.e., “what” feature abstraction to attend to) and different spatial locations of the selected feature map (i.e., “where”) to perform the task at hand. Specifically, at each Recurrent Neural Network (RNN) step, both a CNN layer and localized spatial region within it are selected for further processing. We demonstrate the effectiveness of this approach on two computer vision tasks: (i) image-based six degree of freedom camera pose regression and (ii) indoor scene classification. Empirically, we show that combining the “what” and “where” aspects of attention improves network performance on both tasks. We evaluate our method on standard benchmarks for camera localization (Cambridge, 7-Scenes, and TUM-LSI) and for scene classification (MIT-67 Indoor Scenes). For camera localization our approach reduces the median error by 18.8% for position and 8.2% for orientation (averaged over all scenes), and for scene classification, it improves the mean accuracy by 3.4% over previous methods.
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3:45 – 3:55 : Kyle Bower and Gonzalo García A. E, University of Toronto
Title:: Geometric Integration of the Outer Solar System with Symplectic Integrators
Abstract: We introduce the theory of symplectic integrators by simulating the Kepler problem and the N-body problem for the solar system. We build up the theory of Hamiltonian systems to demonstrate the superiority of symplectic methods in solving such systems. We numerically test several symplectic methods including the symplectic Euler, the Störmer-Verlet, and the Blanes-Moan method. We find that symplectic methods have three main advantages over non-symplectic methods: they conserve both energy and angular momentum, and preserve the qualitative behaviour of the solutions.
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4:00 – 4:10 : Caroline (Ruo yi) Lin, University of Toronto
Title:: Stochastically Fitting the Spring Constants on a System of Coupled VdP Oscillators Simulated Annealing
Abstract: Stochastically Fitting the Spring Constants on a System of Coupled VdP Oscillators Simulated Annealing is a stochastic (Monte Carlo) method which can be used for global optimization. They can be used to obtain results in large search spaces, where non-stochastic methods are prohibitively time-consuming. In this talk, we present the results of a simulated annealing algorithm from the scipy toolkit used to fit the spring constants of a system of linearly coupled Van der Pol oscillators. The constants are fit to a relatively high degree of precision for a short (minute-scale) runtime. We demonstrate the robustness of the method in the presence of noisy input data and show how other methods fail to scale well for this problem.
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