"Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations"
Abstract: We study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise kth degree polynomials, the error between the DG solution and the exact solution is (2k+1)th order superconvergent in suitable negative-order norm, (k + 2)th order superconvergent at the downwind-biased Radau points, (2k+1)th order superconvergence at the downwind points. The extension in two space dimensions will also be mentioned.
Abstract: The adjoint method, among other sensitivity analysis methods, can fail in chaotic dynamical systems. The result from these methods can be too large, often by orders of magnitude, when the result is the derivative of a long time averaged quantity. This failure is known to be caused by ill-conditioned initial value problems. This paper overcomes this failure by replacing the initial value problem with the well-conditioned "least squares shadowing (LSS) problem". The LSS problem is then linearized in our sensitivity analysis algorithm, which computes a derivative that converges to the derivative of the infinitely long time average. We demonstrate our algorithm in several dynamical systems exhibiting both periodic and chaotic oscillations.
Joint Work of: Blonigan, Patrick; Gomez, St, even; Hu, Rui; Gauger, Nico; Diskin, Boris; Nielson, Eric; Farrell, Patrick
Main reference: Wang, Qiqi, Rui Hu, and Patrick Blonigan. "Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations." Journal of Computational Physics 267 (2014): 210-224.
"Near-field radiative energy; energy; and momentum transfer in fluctuational electrodynamics"
Abstract: The ability to control the radiative properties of objects is of great interest in diverse areas like solar and thermophotovoltaic energy conversion, selective thermal emission and absorption, and camouflage in military applications. Thermal radiation at the nanometer scale is significantly different from classical or macroscopic radiative energy transport since near-field effects such as interference, diffraction, and tunneling of surface waves play a significant role.
My talk will focus mainly on small-scale energy, entropy and momentum transfer via electromagnetic waves, especially due to thermal and quantum fluctuations. A dyadic Green’s function formalism has been developed to determine near-field radiative energy and momentum transfer between objects of arbitrary shapes and sizes. Momentum transfer due to electromagnetic fluctuations is responsible for van der Waals and Casimir forces, which are important in many different fields such as adhesion and stiction of materials, bioengineering, and phase change heat transfer. I will talk about how we model van der Waals forces, and show how my work provides a new interpretation, and a better understanding, of this historical problem. In addition, entropy associated with near-field radiative transfer has been studied for the first time. It can be used to determine the maximum work that can be extracted and a thermodynamic limit of energy conversion efficiency that can be obtained in near-field thermal radiation. Experimental investigation will focus on the thermal and optical properties of 2D or 3D nanostructured materials, and it leads to new types of thermophotovoltaic solar cells and selective thermal emitters using metamaterials and nanoparticles. Small-scale thermal transport has shown great potential and applications for use in manipulating macroscale energy systems and energy harvesting.
This presentation provides an overview of the different data analytics applications in the financial sector, with a focus on the Property and Casulty Insurance industry. Methodologies will be discussed for marketing and sales applications. The second part of the presentation will cover a case study, based on the most recent Liberty Mutual Kaggle Competition. The winning team is going to share their insights to address this difficult problem. Keys to their success include capping, downsampling, feature selection and a bit of feature engineering to extract value and remove noise from blocks of features. Xavier and Owen also explored diverse combinations of feature transformations and ML algorithms. An ensemble of RandomForest, ExtraTrees and Linear models generated by DataRobot proved to be the lucky winner!
Qiuyan Xu has a PhD in statistics from University of California, Davis. She has been applying data science in different areas, including genetic study, risk management and Insurance. Currently she works as a director of Advanced Anlytics in Liberty Mutual Inuruance, with a focus on inproving sales and service results in personal line.
Xavier Conort is the Chief Data Scientist at DataRobot. Xavier has applied statistical learning techniques to diverse business problems ranging from churn prediction to claims modelling, to flight arrival prediction, to essay scoring, to sales forecasting, and biological response prediction. Actuary by training, Xavier sharpens his skills in Machine Learning by competing on Kaggle where he has won 9 prizes. Before becoming a data science enthusiast, Xavier held different roles (actuary, CFO, risk manager) in the insurance industry in France, Brazil, China and Singapore. Xavier obtained in 1996 two Masters Degrees in Actuarial Science and Statistics from ENSAE ParisTech and Paris Denis Diderot University and is a Fellow of the Institut des Actuaires and a Chartered Enterprise Risk Analyst (CERA). Xavier received in 2013 honours and accolades from Singapore\'s Agency for Science, Technology and Research.
Owen Zhang is the Chief Product Officer at DataRobot. Owen has spent most of his career in US P&C insurance industry. After spending several years in IT building transnational system for commercial insurance product, Owen discovered his passion in machine learning and moved into advanced analytics to building underwriting, pricing, and claim models. Before joining DataRobot in August 2014, Owen was Vice President of Modeling in the newly formed AIG science team. Owen has a Masters degree in electrical engineering from University of Toronto and a Bachelor degree from University of Science and Technology of China. Owen currently holds the #1 rank on Kaggle.
"Reduced order modeling and domain decomposition methods for uncertainty quantification"
Abstract: Traditionally, terms in PDEs such as permeabilities, viscosities or boundary conditions have been treated as known deterministic quantities. However, these quantities are not always known with certainty, and there is much interest today in treating them as random fields. In this talk, I will present a reduced basis collocation method for efficiently solving PDEs with random coefficients, which is joint work with Howard Elman of University of Maryland. I will also present a domain-decomposed uncertainty quantification approach for complex systems, which is joint work with Karen Willcox of Massachusetts Institute of Technology.
"High order meshless methods for arbitrary Lagrangian-Eulerian schemes"
Abstract: By discretizing the Navier-Stokes equations in a Lagrangian framework, flows with complex deforming boundaries can be studied while avoiding the non-linearity associated with the advection term in the momentum equations. For mesh-based methods this approach is only applicable to flows undergoing mild mesh deformation before expensive remeshing takes place. Meshless methods have been used successfully in the past to investigate more complex deformations, but generally rely on a low order discretization and conservation arguments to get meaningful results. In the present work, the moving least squares discretization is introduced and used to discretize a new stiffly-stable high order projection method, which allows high order discretization in both space and time efficiently through a collocation scheme.
"Uncertainty Quantification in Computational Models"
Abstract: Models of physical systems typically involve inputs/parameters that are determined from empirical measurements, and therefore exhibit a certain degree of uncertainty. Estimating the propagation of this uncertainty into computational model output predictions is crucial for purposes of model validation, design optimization, and decision support.
Recent years have seen significant developments in probabilistic methods for efficient uncertainty quantification (UQ) in computational models. These methods are grounded in the use of functional representations for random variables. In particular, Polynomial Chaos (PC) expansions have seen significant use in this context. The utility of PC methods has been demonstrated in a range of physical models, including structural mechanics, porous media, fluid dynamics, aeronautics, heat transfer, and chemically reacting flow. While high-dimensionality remains a challenge, great strides have been made in dealing with moderate dimensionality along with non-linearity and oscillatory dynamics.
In this talk, I will give an overview of UQ in computational models. I will cover the two key classes of UQ activities, namely: estimation of uncertain input parameters from empirical data, and forward propagation of parametric uncertainty to model outputs. I will cover the basics of PC UQ methods with examples of their use in both forward and inverse UQ problems.
"GPU Spectral Method and Stable Parareal Method for Large-scale Scientific Computing"
Abstract: Large-scale problems in scientific simulation and data analysis lead to high computational costs in both spatial and temporal dimensions. There is an increasing demand of new mathematical algorithms scalable on modern supercomputing architectures for this growing complexity. However, the global nature in space and the sequential nature in time pose a great challenge for the parallelization of related algorithms. In this presentation, we first discuss recent efforts to develop GPU-suited spectral methods. We introduce a decoupling strategy that maps spectral methods to the Nvidia CUDA programming model. We then introduce the parallel-in-time approach to break the sequential bottleneck of the time direction. Motivated by the instability of the original parareal method for wave problems, we propose an adjoint parareal method and a reduced basis parareal method. We outline the theoretical ideas behind these new parareal methods and discuss their robustness and efficiency.
Throughout the presentation we shall illustrate the performance with computational examples in order to highlight the major advantages of the proposed approach. In the end of the talk, we shall mention future work and challenges associated with large-scale scientific computing. Part of the work is done in collaboration with Jie Shen, Jan Hesthaven, Xueyu Zhu, and Yvon Maday.
"A Path Forward to Exascale Computing: High-order Numerical Methods for Large-scale Scientific Computing and Plasma Simulations"
Abstract: Modeling diverse physical processes using mathematical algorithms has become a successful tool in modern science and engineering. The underlying mathematical models are carefully designed to perform large-scale computer simulations that involve disparate scales of space and time. Such complexities often arise when incorporating various multi-physics components that can be represented by classes of partial differential equations (PDEs).
In the first part, I will discuss key components of mathematical algorithms to solve PDEs in order to construct numerical solutions for computational fluid dynamics, gas dynamics and plasma physics. I describe mathematical algorithms with special attention to two numerical approaches: first, the traditional formulations based on high-order polynomials; second, a new innovative exponentially converging formulation based on Gaussian Process Modeling. Moreover, I will show the importance of fast convergent, high-order accurate numerical methods and how they are crucial for future high performance exascale computing architectures. In the second part, I will present laboratory astrophysics scientific simulations using the numerical algorithms introduced in the first part.
The key ideas and challenges of computational mathematics in this talk have been developed within the framework of the University of Chicago's FLASH code. FLASH is a highly capable, massively parallel, publicly available open source scientific code with a wide user base in the fields of astrophysics, cosmology, and high-energy-density physics.
"Numerical simulation of boundary layer separation for incompressible fluid over an irregular domain"
Abstract: The development of boundary layer separation for incompressible flow, subject to no-slip boundary conditions, is a complicated process; many small scale fluid structures areinvolved.In particular, a numerical study of such a separation over an irregular domain becomes even more challenging, due to the boundary complexity. In this talk, we present a numerical simulation of a driven cavity flow over a triangular domain, using a simple finite element numerical scheme based on the vorticity-stream function formulation. Such a numerical scheme decouples the Stokes solver into two Poisson-like solvers at each time stage in the Runge-Kutta temporal discretization. As a result, the LBB condition is avoided and the numerical efficiency is greatly improved. Some numerical results are also provided.
"Challenges and Applications of Modern Quantum Chemistry"
Abstract: Nowadays, theory and computation have become indispensable in various fields of chemical research and development due to its ability to provide important insights into the structures, properties, and reactivities of molecular and biological systems. In this talk, I will discuss recent theoretical and computational efforts which address some of the major challenges in quantum chemistry. The availability of high-performance computers and development of novel methods are critical to realize these challenges. In particular, I will describe developing affordable highly accurate methods for open shell and excited states as well as for large systems with thousands of electrons. Two computational research applications will be presented: (1) the photodissociation dynamics of methane and (2) mechanical properties of carbon nanotubes.
"Discrete two-dimensional waves: from integrals to sources"
Abstract: I will start the talk with some explanations of a simple scattering problem: an incident acoustic wave hits a sound-soft obstacle which modifies its path. With a simulation, I will show how monochromatic signals produce monochromatic responses, thus justifying the frequency-domain analysis of linear waves. The next ingredient to be introduced is the single layer acoustic potential. I will present it in the two-dimensional case for a smooth parametrizable scatterer at a given frequency, and I will give some hints at how to discretize it properly. Some mathematical arguments will next be derived to justify some particular choices of parameters in the otherwise very intuitive way of dealing with boundary integral equations for scattering problems. In the final part of the talk we will move back to the time-domain and will observe some interesting wave behavior for multiple obstacles or penetrable scatterers. Time permitting, I will also mention the treatment of elastic waves.
"A robust high order finite difference method for strong multispecies detonation"
Abstract: In this talk, we propose a high order finite difference WENO method with Harten's ENO subcell resolution idea for the chemical reactive flows. In the reaction problems, when the reaction time scale is very small, the problems will become very stiff. Wrong propagation of discontinuity occurs due to the underresolved numerical solutions in both the space and time. The proposed method is a modified fractional step method which solves the convection step and reaction step separately. A fifth-order WENO is used in convection step. In the reaction step, a modified ODE solver is applied but with the flow variables in the discontinuity region modified by the subcell resolution idea.
"Single Domain Hybrid Fourier Continuation Method and Weighted Essentially Non-oscillatory Finite Difference Scheme for Conservation Law"
Abstract: In this talk, we introduce a hybrid Fourier continuation (FC) method and weighted essentially non-oscillatory (WENO) finite difference scheme, together with the high order multi-resolution algorithm by Harten to determine the smoothness of a solution of hyperbolic conservation laws at a given stencil, in a single domain framework (Hybrid), as opposed to in a given subdomain in a multi-domain framework. The Hybrid scheme conjugates an efficient shock-capturing WENO-Z nonlinear scheme in discontinuous stencils with an essentially non-dispersive and non-dissipative linear FC method in smooth stencils, yielding a high fidelity scheme for applications containing both discontinuous and complex smooth structures. Several critical numerical issues in the implementation (such as FFT and Symmetry preserving), accuracy, and efficiency of the Hybrid scheme will be illustrated and resolved. Examples, including the one dimensional classical Riemann IV problems, shock-entropy wave interaction and two dimensional March 3 double Mach reflection problems, regarding the efficiency and accuracy of the Hybrid scheme will be shown. This is a joint work with Prof. Gao Zhen, Prof. Xie and Li Peng during the First Summer Workshop in Advanced Research in Applied Mathematics and Scientific Computing 2013 In the School of Mathematical Sciences at Ocean University of China.
This seminar is sponsored jointly by the Engineering and Applied Sciences program at UMassD.
Abstract: MATLAB is generally considered to be the leading software package for scientific computing. In this talk we consider a number of computational examples where MATLAB gives or appears to give wrong answers. These examples are useful to help better understand the inner workings, evolution, limits and tradeoffs of a software package such as MATLAB. The talk should be accessible to both graduate and undergraduate students that have some background in either linear algebra or numerical analysis. The examples should help students gain greater insight into finite precision arithmetic and related concepts such as round off error, machine precision, numerical stability, and conditioning.