Pavel Solin’s research while affiliated with University of Nevada, Reno and other places

We present a novel self-paced, instructor-assisted approach to teaching Python programming in a college classroom environment. Instead of listening to a lecture and doing homework on their own later, students work actively 100% of the time, using an advanced interactive learning platform combined with real-time individual assistance of their instructor. Instead of lecturing, the instructor walks through the classroom and assists students individually. This makes a huge positive difference for the students, and it also allows the instructor to understand much better how each student performs and where they need help. After numerous years of using this approach, we are convinced that nothing can replace a one-to-one interaction between the student and his/her instructor. Unfortunately, traditional classroom lectures are not consistent with this teaching style. On the contrary, the mostly uni-directional flow of information from the instructor to the class effectively shields the instructor from interacting with the students. We describe various aspects of the self-paced, instructor-assisted method and show that it has a major positive impact on the students. Students find it extremely helpful being able to digest the material at their own pace. They also get a lot more practice compared to the standard lecture ++ homework model, develop good programming habits, and become skilled and experienced programmers. This method makes students better independent learners, problem solvers, and communicators. Our findings are based on 10+ years of teaching Python programming courses to diverse audiences in this way. Results of a sample student survey and student testimonials are presented.

In this paper we explain why we decided to abandon the traditional classroom instruction of Linear Algebra and switched to a self-paced, instructor-assisted course format. We also describe our experience with creating a self-paced course in NCLab, and with teaching it at a tier-1 research university for three consecutive semesters. The results of a student survey and student comments are presented.

This paper proposes a novel adaptive higher-order finite element (hp-FEM) method for solving elliptic eigenvalue problems, where n eigenpairs are calculated simultaneously, but on individual higher-order finite element meshes. The meshes are automatically hp-refined independently of each other, with the goal to use an optimal mesh sequence for each eigenfunction. The method and the adaptive algorithm are described in detail. Numerical examples clearly demonstrate the superiority of the novel method over the standard approach where all eigenfunctions are approximated on the same finite element mesh.

Numerical solutions of convection-dominated problems are known to exhibit spurious oscillations whose suppression requires the use of numerical stabilization. Stabilized methods which involve heuristic parameters are often applied. The parameters influence the quality of the solution but their optimal values are unknown. In this paper, we consider a stabilization method which adds numerical diffusion adaptively, based on minimizing a functional. The novelty of our approach consists in combining an error indicator with reduced residuals with a nonlinear SOLD method adding artificial diffusion in the crosswind direction. We demonstrate that this approach can lead to more physically meaningful solutions than techniques considered before.

We present a novel hp-adaptive space-time discontinuous Galerkin (hp-STDG) method for the numerical solution of the nonstationary Richards equation equipped with Dirichlet, Neumann and seepage face boundary conditions. The hp-STDG method presented in this paper is a generalization of a hp-STDG method which was developed for time dependent non-linear convective-diffusive problems. We describe the method and the single ring experiment, and then we present a numerical experiment which clearly demonstrates the superiority of the hp-STDG method over a discontinuous Galerkin method based on a static fine mesh.

Numerical solution of convection-dominated problems requires special techniques to suppress spurious oscillations in approximate solutions. Often, stabilized methods are applied which involve user-chosen parameters. These parameters significantly influence the quality of the solution but their optimal choice is usually not known. One possibility is to define them in an adaptive way by minimizing an error indicator characterizing the quality of the approximate solution. A non-trivial requirement on the error indicator is that its minimization with respect to the stabilization parameters should suppress spurious oscillations without smearing layers. In this paper, a new error indicator is introduced and its suitability is tested on two newly proposed benchmark problems for which previously proposed indicators do not provide satisfactory results.

This paper is concerned with the numerical simulation of time-dependent variably-saturated Darcian flow problems described by the Richards equation. We present the adaptive higher-order space-time discontinuous Galerkin (hp-STDG) method which optimizes accuracy and efficiency by balancing the errors that arise from the space and time discretizations and from the resulting nonlinear algebraic system. Convergence problems related to the transition between unsaturated flow and saturated flow are eliminated by regularizing the constitutive formulas. We also present an hp-anisotropic mesh adaptation technique capable of generating unstructured triangular elements with optimal sizes, shapes, and polynomial approximation degrees. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the numerical method presented here.

ESCO 2016 was the 5th event in a successful series of interdisciplinary meetings dedicated to modern methods and practices of scientific computing. It was held on June 5 - 10, 2016 in Pilsen, Czech Republic. It was attended by more than 350 participants from 30 countries. The conference included 13 minisymposia and two software workshops. Main thematic areas of the conference included Computational electromagnetics and coupled problems, Fluid-structure interaction and multi-phase flows, Computational chemistry and quantum physics, Computational civil and structural engineering, Computational biology and bioinformatics, Computational geometry and topology, Mathematics Education and Outreach, Petascale and exascale computing, Hydrology and porous media flows, Wave propagation and acoustics, Climate and weather modeling, GPU and cloud computing, Uncertainty quantification, Open source software. The conference featured the following keynote speakers: Christian Bucher (Vienna University of Technology, Austria), Volker John (WIAS, Berlin, Germany), Anders Logg (Chalmers University of Technology, Gothenburg, Sweden), Eugenio Roanes-Lozano (Universidad Complutense de Madrid, Spain), Padmanabhan Seshaiyer (George Mason University, Fairfax, U.S.A., NSF program director), Andrew Siegel (Argonne National Laboratory, U.S.A.) An indivisible part of each ESCO conference is a software afternoon featuring computational software projects of participants. Presented can be any computational software that has reached certain level of maturity, i.e., it is used outside of the author’s institution, and it has a web page and a user documentation. In closing, ESCO 2016 was successful in achieving its goal to bring together leading applied mathematicians and engineers working in the area of scientific computing, identify emerging research directions, and establish new collaborations.

We develop a new mesh adaptive technique for the numerical solution of partial differential equations (PDEs) using the -version of the finite element method ( -FEM). The technique uses a combination of approximation and interpolation error estimates to generate anisotropic triangular elements as well as appropriate polynomial approximation degrees. We present a -version of the continuous mesh model as well as the continuous error model which are used for the formulation of a mesh optimization problem. Solving the optimization problem leads to -mesh with the smallest number of degrees of freedom, under the constraint that the approximate solution has an error estimate below a given tolerance. Further, we propose an iterative algorithm to find a suitable anisotropic -mesh in the sense of the mesh optimization problem. Several numerical examples demonstrating the efficiency and applicability of the new method are presented.

We present a new adaptive higher-order finite element method (hp-FEM) for the solution of boundary value problems formulated in terms of partial differential equations (PDEs). The method does not use any information about the problem to be solved which makes it robust and equation-independent. It employs a higher-order reconstruction scheme over local element patches which makes it faster and easier to parallelize compared to hp-adaptive methods that are based on the solution of a reference problem on a globally hp-refined mesh. The method can be used for the solution of linear as well as nonlinear problems discretized by conforming or non-conforming finite element methods, and it can be combined with arbitrary a posteriori error estimators. The performance of the method is demonstrated by several examples carried out by the discontinuous Galerkin method.