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.

"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.

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 Monsoonâ€”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.

"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.