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.

Computational Science Seminars (highlighted in gray) are technical talks on a particular research topic.

Lunchtime Computing Talks (highlighted in light blue) will introduce a new computational tool or technique to the broader UMassD community. These informal introductions assume no prior experience and will often feature a hands-on tutorial, so make sure to bring your laptops!

May 8, 2019

Wednesday

Leah Isherwood

Umassd

1:00 PM

Textile 105A

"Strong Stability Preserving Integrating Factor One-step and Two-step Runge--Kutta Methods"

Abstract: Strong stability preserving (SSP) Runge--Kutta methods are often desirable when evolving in time problems with components that have very different time scales. In situations where the SSP property is needed, it has been shown that implicit and implicit-explicit Runge--Kutta methods have very restrictive time steps and are therefore not efficient. For this reason, SSP integrating factor methods may offer an attractive alternative to traditional time stepping methods for problems with a stiff linear component and a non-stiff nonlinear component. This work defines strong stability properties of integrating factor Runge--Kutta methods. It shows that it is possible to define explicit integrating factor Runge--Kutta methods that preserve the desired strong stability properties satisfied by each of the components when coupled with forward Euler time stepping, or even given weaker conditions. Sufficient conditions are defined for explicit integrating factor one-step and two-step Runge--Kutta methods to be SSP, namely that they are based on explicit SSP Runge--Kutta methods with non-decreasing abscissas or with operators replaced by the downwinded operator when the abscissas decrease. We find such one-step methods of up to fourth order and two-step methods up to eighth order, analyze their SSP coefficients, prove their optimality in a few cases, and investigate downwinding approaches to preserve strong stability. These methods are tested to demonstrate their convergence, to show that the SSP time step predicted by the theory is generally sharp, and that the non-decreasing abscissa condition or downwinded modification is needed in our test cases. Finally, this research shows that on typical total variation diminishing linear and nonlinear test cases our new SSP integrating factor Runge--Kutta methods out-perform the corresponding explicit SSP Runge--Kutta methods, implicit-explicit SSP Runge--Kutta methods, and some well-known exponential time differencing methods.

May 1, 2019

Wednesday

Donghui Yan

Umassd

12:00 PM

Textile 105A

"Lunchtime Computing: Introduction to R"

Abstract: Come and meet R! Over lunch, I will introduce to you one of the most popular programming languages for data science. It has also been used widely by many other fields that involve data analysis, such as engineering, social sciences, lab sciences, business etc. In this talk, I will briefly introduce the concept of R programming, and discuss its application in data visualization, statistical computing, as well as packages for machine learning (such as Random Forests and deep neural networks etc).

This lunchtime computing talk is intended for anyone with little or some understanding of statistical data analysis. After this talk, you will know how to do basic R programming, data visualization, statistical inference, statistical modeling and machine learning with R. Bring your laptops if possible!

"A seamless homogenization method for multiscale diffusion and advection operators"

Abstract: Analytical and numerical homogenizations typically require scale separation between the larger scales and the small scales to be homogenized. We propose a methodology that overcomes this hurdle for a class of differential equations. In particular, we investigate a specific decomposition structure of diffusion tensors that allows a small scale homogenization independent of larger scale components. An advection operator is handled as a subclass of such tensors. Using the particular structure, we propose a seamless numerical method for diagonal multiscale diffusion tensors that find applications in the modeling of heterogeneous media. The method decomposes the multiscale diffusion tensor into different scale components and homogenizes each component iteratively. A transformation technique minimizes the interaction between different scale components to enables the computational complexity to only increase linearly in the number of different scale components.To demonstrate the efficiency and robustness of the proposed method, we present theory and provide several numerical tests including a non-separable scale diffusion tensor. This is joint work with Professor Engquist at the University of Texas at Austin.

Abstract: It is widely believed that long after gravitational collapse, the external geometry of the resulting black hole relaxes to the Kerr-Newman solution. In contrast, the interior geometry depends on initial conditions. Nevertheless, there can exist universal features, such as singularities, whose structure is independent of initial conditions. I will discuss gravitational collapse of a massless scalar field in asymptotically flat spacetime. I will argue that due to decaying fluxes through the horizon, such black holes must contain a central spacelike singularity at r = 0 and a null singularity on the Cauchy Horizon.

"Projected Nonlinear Least Squares for H2 Model Reduction"

Abstract: Optimal H2 model reduction seeks to build a reduced order model of a linear, time invariant (LTI) system that minimizes the mismatch of the transfer function along the imaginary axis in the L2-norm. Here we propose a new approach to the H2 model reduction problem that only requires access to samples of the transfer function. Our approach projects the infinite dimensional optimization problem onto a finite dimensional subspace via the sampling operator that results in a nonlinear weighted least squares program. Rational approximation in a conventional pole-residue parameterization often possesses many spurious local minima; therefore, we propose Gauss-Newton initialized with the Adaptive Antoulas-Anderson (AAA) Algorithm that frequently avoids these spurious minima. Finally, we provide an iteration that improves this subspace until the suboptimal solutions converge to the optimal H2 solution. Currently, the main technique for constructing an optimal reduced order model in the H2-norm is the Iterative Rational Krylov Algorithm (IRKA), requiring access to the full order system in state-space form which is not always available in experimental settings. We compare our algorithm to two data-driven approaches: Transfer Function IRKA (TF-IRKA) and quadrature-based vector fitting (QuadVF). Since this projection-based approach is based on numerical optimization techniques, our approach has the potential to be extended to new classes of transfer function realizations.

April 10, 2019

Wednesday

Sidafa Conde

Sandia National Lab

1:00 PM

Textile 105A

"Embedded Error Estimation and Adaptive Step-size Control for Optimal Explicit Strong Stability Preserving Runge–Kutta Methods"

Abstract: We construct a family of embedded pairs for optimal strong stability preserving explicit Runge–Kutta methods of order 2≤p≤4 to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective methods, large region of absolute stability, and optimal error measurement. The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size based on the local error estimation while maintaining their inherent nonlinear stability properties. Through several numerical experiments, we assess the overall effectiveness in terms of precision versus work while also taking into consideration accuracy and stability.

April 3, 2019

Wednesday

Sergei Artamoshin

Umassd

11:30 AM

Textile 105A

"At Most Two Radii Theorem For A Real Eigenvalue Of The Hyperbolic Laplacian"

Abstract: We study a $(k+1)$-dimensional hyperbolic space of a negative constant sectional curvature $\kappa=-1/\rho^2$. Let $\lambda$ be a real eigenvalue and $f_{\lambda} (x)$ be an eigenfunction of the hyperbolic Laplacian assuming a non-zero value at $x_0$. Then the average value of $f_{\lambda}(x)$ over any sphere centered at $x_0$ allows to identify the corresponding eigenvalue $\lambda$ uniquely as long as that average value is large enough. Otherwise, to identify the corresponding eigenvalue uniquely, we need to make sure that the computed average value is not zero and then we need to compute an additional average value of $f_{\lambda}(x)$ over a small enough sphere centered at the same point $x_0$.

"Introduction to Topological Data Analysis with Applications"

Abstract: Topological Data Analysis (TDA) is a topology-based approach to identifying and analyzing features in complex data sets. In this talk we will present an introduction to basic concepts in TDA, including simplicial complexes, filtrations, persistence diagrams and persistence bar codes. In addition we will briefly consider recent TDA projects by undergraduate and graduate students on the analysis of antibody uptake in mouse tumors, tortuosity in retinal vasculature, diurnal cycles in tropical cyclones, and genetic recombination rates.