Center seminars are organized by the Mathematics Department and the College of Engineering. The Center gratefully acknowledges support from the UMassD Office of the Provost.

"Beyond Automated Multilevel Substructuring: Domain Decomposition with Rational Filtering"

Abstract: This talk discusses a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem associated with each subdomain into two disjoint subproblems. The first subproblem is associated with the interface variables and accounts for the interaction among neighboring subdomains. To compute the solution of the original eigenvalue problem at the interface variables we leverage ideas from contour integral eigenvalue solvers. The second subproblem is associated with the interior variables in each subdomain and can be solved in parallel among the different subdomains using real arithmetic only. Compared to rational filtering projection methods applied to the original matrix pencil, the proposed technique integrates only a part of the matrix resolvent while it applies any orthogonalization necessary to vectors whose length is equal to the number of interface variables. In addition, no estimation of the number of eigenvalues located inside the interval of interest is needed. Numerical experiments performed in distributed memory architectures illustrate the competitiveness of the proposed technique against rational filtering Krylov approaches.

"Modeling and analysis of chromatin and RNA structures"

Abstract: Higher-order structures of chromatin and RNA play key roles in many biological processes and are important in health and diseases. Advancements in high-throughput sequencing have provided unprecedented opportunities to characterize chromatin and RNA structures at the genome scale. The complexity of the relevant data types brings challenges on their modeling and analysis. In this talk, I will introduce statistical and computational methods for genome-wide reconstruction of chromatin and RNA structures. Bio: Dr. Zhengqing Ouyang is an assistant professor of computational biology and systems genomics at The Jackson Laboratory for Genomic Medicine in Farmington, Connecticut. He received his B.S. and M.S. degrees from Peking University. He then obtained his Ph.D. from Stanford University under the advisement of Prof. Wing Hung Wong in 2010, and did postdoctoral training with Profs. Howard Chang and Michael Snyder at Stanford from 2010 to 2012. He is a recipient of the Genome Technology Young Investigators of the Year Award from GenomeWeb in 2013; the Research Starter Grant in Informatics Awards from PhRMA Foundation in 2016; and the NIH/NIGMS Maximizing Investigators' Research Award in 2017.

"L1-Reduced basis method and its applications to waves, nonlocal problems, and an ultra-efficient linear system solver"

Abstract: Abstract: Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Reduced basis method (RBM) can improve efficiency by several orders of magnitudes leveraging an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces. It solves a reduced problem posed in a surrogate space. A critical ingredient to guarantee the accuracy of the surrogate solution and guide the construction of the surrogate space is an error estimation procedure that typically involves the residual norm. Unfortunately, its efficient evaluation is delicate or outright infeasible.

We propose a new error indicator based on the Lebesgue function in interpolation theory. This error indicator circumvents the need for a posteriori analysis of numerical methods. In particular, it does not require computation of residual norms, and instead only requires the ability to compute the L1 norm of the RBM solution coefficients in the surrogate space. After introducing this new L1-based RBM, this talk will present some of our recent applications including to wave problems in solar cells, fractional partial differential equations, and a new iterative linear solver that converges faster than multigrid-preconditioned conjugate gradient method for symmetric and positive definite systems.

These are joint works with Jiahua Jiang, Akil Narayan, Peter Monk, Manuel Solano, Harbir Antil, and Cuong Nguyen.

"Deep Learning for Large-Scale Spatiotemporal Data"

Abstract: In many real-world applications, such as climate science, transportation and physics, machine learning is applied to large-scale spatiotemporal data. Such data is often nonlinear, high-dimensional, and demonstrates complex spatial and temporal correlations. Deep learning provides a powerful framework for feature extraction, but existing deep learning models are still insufficient to handle the challenges posed by spatiotemporal data. In this talk, I will show how to design deep learning models to learn from large-scale spatiotemporal data. In particular, I will present our recent results on 1) High-Order Tensor RNNs for modelling nonlinear dynamics, and 2) Diffusion Convolutional RNNs for modelling spatiotemporal patterns, applied to real-world climate and traffic data. I will also discuss the opportunities and challenges of applying deep learning to large-scale spatiotemporal data.

September 26, 2018

Wednesday

Juan Cheng

Institute Of Applied Physics And Computational Mathematics

1:00 PM

Textile 105A

"Positivity-preserving and symmetry-preserving Lagrangian schemes for compressible multi-material fluid flows"

Abstract: In applications such as astrophysics and inertial confinement fusion, there are many three-dimensional cylindrical-symmetric multi-material problems which are usually simulated by Lagrangian schemes in the two-dimensional cylindrical coordinates. For this type of simulation, the critical issues for the schemes include keeping positivity of physically positive variables such as density and internal energy and keeping spherical symmetry in the cylindrical coordinate system if the original physical problem has this symmetry. In this talk, we will introduce our recent work on high order positivity-preserving and symmetry-preserving Lagrangian schemes solving compressible Euler equations. The properties of positivity-preserving and symmetry-preserving are proven rigorously. One- and two-dimensional numerical results are provided to verify the designed characteristics of these schemes.

Abstract: The Northeast Cyberteam Initiative is an NSF-sponsored program whose goal is to make high performance computing more readily accessible to researchers at small and medium sized institutions in the upper northeastern states of Maine, New Hampshire, Vermont and Massachusetts. There are two main goals of the program: 1) Launch a series of short (3-6 month) projects where a researcher requesting assistance is assigned a student-facilitator, paired with a mentor with subject matter expertise, who assist with applying high performance computing to the researcher’s project; and 2) Develop a set of tools and resources to provide researchers and facilitators self-service help when utilizing high performance computing resources.

Julie Ma from MGHPCC is the project leader for the Northeast Cyberteam Initiative. She will briefly introduce MGHPCC and then give an overview of the Northeast Cyberteam Initiative; discuss a few example projects and tools that have been developed including Ask.CI, the recently launched Q and A platform for people who do research computing; and finally show audience members how to submit a project proposal for consideration.

"Optimal convergence order of a spectral method for fractional advection-diffusion-reaction equations"

Abstract: We consider the regularity of solutions to a fractional advection-diffusion-reaction equation with fractional Laplacian in one dimension. We show that the regularity of the solution in weighted Sobolev spaces can be significantly improved compared to that in standard Sobolev spaces. We present a spectral Galerkin method and its optimal error estimate. Numerical results are presented to verify our regularity estimate. This work is joint work with Zhaopeng Hao.

"Reduced Basis Methods and Hybridizable Discontinuous Galerkin method for efficient forward solvers"

Abstract: Many problems arising in computational science and engineering are described by mathematical models of high complexity – involving multiple disciplines, characterized by a large number of parameters, and impacted by multiple sources of uncertainty. The central theme of accurate, reliable and efficient forward solvers investigated in this thesis concerns with Reduced Basis Method (RBM) for model order reduction, and Hybridizable Discontinuous Galerkin (HDG) for Finite Element Method (FEM) scheme. The essential ingredients are described.

March 5, 2018

Monday

Zachary Grant

Umassd

11:00 AM

Textile 105A

"On implicit and explicit high order SSP methods with a variety stages, steps, And derivatives for linear and nonlinear problems"

Abstract: High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge--Kutta methods exist only up to fourth order, and implicit ones only up to sixth order, while multistep SSP methods have small allowable time step. To break these order barriers and SSP time-step bounds, we turn to general (multistep, multi-stage) linear methods of order two and above. Order conditions and monotonicity conditions for such methods were worked out in terms of the method coefficients, and we formulated a numerical optimization to find optimal methods of up to five steps, eight stages, and tenth order. Another approach to breaking the order barrier is to restrict ourselves to solving only linear autonomous problems. In this case the order conditions simplify and this order barrier is lifted: explicit SSP Runge--Kutta methods of any linear order exist, but these methods reduce to second order when applied to nonlinear problems. We found explicit SSP Runge--Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. Finally, we turn to multiderivative time-stepping methods which have recently been implemented with hyperbolic PDEs. We describe sufficient conditions for a two-derivative multistage method to be SSP, and formulate an optimization problem that were used to find optimal SSP multistage two-derivative methods. These methods were tested on a variety of scalar hyperbolic partial differential equations, which demonstrate the need for the SSP condition and the sharpness of the SSP time-step.

"Networks and Dynamics Based Approaches in Data Analysis"

Abstract: In the first half of the talk, I will present a novel method to analyze large spatiotemporal rainfall data sets. This method employs the nonlinear correlation measure known as event synchronization to transform the high dimensional data into a network. Furthermore, by studying different topological properties of the inferred network, I will provide insights into the dynamics of South Asian summer Monsoon—a climatic phenomenon of substantial socio-economic significance to the 1.5 billion inhabitants of the region. In the second half of the talk, I will present a new, hybrid approach to network classification, combining a manual selection of features of potential interest with automated classification methods. I will demonstrate the broad applicability of this approach by classifying days of the week from call detail records and diagnosing types of cancer tumors based on their transcription factor-gene regulatory networks.

"Modeling 2D Multilayer Materials: Non-Commutative Geometry to the Rescue of Numerical Computation"

Abstract: The recent discovery of a whole family of two-dimensional crystalline materials such as graphene, hexagonal boron nitride (h-BN) and many others leads to study the properties of their combinations, particularly by stacking a number of layers vertically. Such structures are generally non-periodic, with interesting geometric properties such as moiré effects.
We first recall the usual description for electronic structure and conduction phenomena in periodic as well as disordered systems, using quantum models of tight-binding type. We show how a unified framework, formulated by Bellissard et al. in the context of noncommutative geometry to model disordered systems, extends to non-periodic systems with multiple layers and allows to write an explicit formula for their macroscopic electrical conductivity. This abstract framework surprisingly leads to a new type of numerical scheme going beyond traditional methods.

January 29, 2018

Monday

Olaniyi Iyiola

Minnesota State University-Moorhead

11:00 AM

Textile 105A

"Exponential Time Differencing for Nonlinear (Fractional) Diffusion-Reaction Systems"

Abstract: Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this talk, I will present a novel Exponential Time Differencing (ETD) scheme for nonlinear reaction-diffusion fractional models. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. Due to these real distinct poles, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second order convergent; and proven to be robust for problems involving non-smooth or mismatched initial and boundary conditions and steep solution gradients. This scheme combined with fractional central differencing is used for simulating some nonlinear space fractional models. The superiority of the proposed ETD-RDP scheme over competing second-order ETD schemes and BDF2 scheme is demonstrated. The numerical tests show that the proposed scheme is computationally more efficient (in terms of cpu time).

Abstract: Every minute, the humankind produces about 2000 Terabytes of data and learning from this data has the potential to improve many aspects of our lives. Doing so requires exploiting the geometric structure hidden within the data. Our overview of different models in data and computational sciences starts with the ubiquitous linear subspace model, where we will present an alternative formulation of the Principal Component Analysis (a statistical tool for detecting linear structure in data) and describe its remarkable implications. We will also present state-of-the-art algorithms for streaming and distributed PCA, accompanied with an example in structural health monitoring.

Next up is learning from data with nonlinear structure. In this part, we will discuss the powerful continuum-of-subspaces model and (known) manifold models, reinforced with applications in super-resolution and nonlinear time-series analysis. Sometimes, however, the nonlinear structure in data has to be learned itself from the data. In the last part of the talk, we will discuss manifold learning from incomplete data and training neural networks, both leading to highly nonconvex optimization programs. Finally, we will list a number of pressing challenges in data and computational sciences, and lay out a path towards addressing them.

"A high-order meshfree framework for solving PDEs on irregular domains and surfaces"

Abstract: We present meshfree methods based on Radial Basis Function (RBF) interpolation for solving partial differential equations (PDEs) on irregular domains and surfaces; such domains are of great importance in mathematical models of biological processes. First, we present a generalized high-order RBF-Finite Difference (RBF-FD) method that exploits certain approximation properties of RBF interpolants to achieve improved computational complexity, both in serial and in parallel. Like all RBF-FD methods, our method requires stabilization when applied to solving PDEs. We present a robust and automatic hyperviscosity-based stabilization technique to rectify the spectra of RBF-FD differentiation matrices. The amount of hyperviscosity is determined quasi-analytically in two stages: first, we develop a novel mathematical model of spurious eigenvalues, and second, we use simple 1D Von Neumann analysis to analytically cancel out these spurious growth terms. When combined with a ghost node technique, we obtain a high-order meshfree framework for solving PDEs on irregular domains. Finally, we present a powerful new RBF-FD technique that allows for the solution of PDEs on manifolds using scattered nodes and Cartesian coordinate systems. In all cases, our methods achieve O(N) complexity for N nodes

"Data-Driven Optimization for Bike-Share Networks"

Abstract: As bike-share systems expand in urban areas, the wealth of publicly available data has drawn researchers to the novel operational challenges these systems face. Bike-share systems are inherently complex because of the many stations and riders. Further, users can directly change the state of the system, possibly filling up or emptying out bike stations. This leads to many interesting optimization problems for how to better meet user demand. First, we will study the problem of redistributing bikes over the system to anticipate demand the next day. Second, we will look at scheduling dock repairs. We will present computational results to show that the proposed methods, which combine integer programming and heuristics, perform well in practice.

"Multiscale Convergence Properties for Spectral Approximations of a Kinetic Model"

Abstract: In this work, we prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form $N^{-q}$, where $N$ is the number of modes and $q>0$ depends on the regularity of the solution. However, in the multiscale setting, the error estimate can be expressed in terms of the scaling parameter $\epsilon$, which measures the ratio of the mean-free-path to the characteristic domain length. We show that, for isotropic initial conditions, the error in the spectral approximation is $\mathcal{O}(\epsilon^{N+1})$. More surprisingly, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the $\ell^{th}$ coefficient of the expansion scales like $\mathcal{O}(\epsilon^{2N})$ when $\ell =0$ and $\mathcal{O}(\epsilon^{2N+2-\ell})$ for all $1\leq \ell \leq N$. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. Numerical tests will also be presented to support the theoretical results.