# Numerical Analysis Seminar

## Fall 2017

Date: | September 20, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Martin Licht |

Title: | Smooth commuting projections in rough settings: Weakly Lipschitz domains and mixed boundary conditions |

Date: | September 28, 2017 |

Time: | 3:00pm |

Location: | BLOC 506A |

Speaker: | Rob Stevenson, University of Amsterdam |

Title: | An optimal adaptive Fictitious Domain Method |

Abstract: | We consider a Fictitious Domain formulation of an elliptic PDE, and solve the arising saddle-point problem by an inexact preconditioned Uzawa iteration. Solving the arising `inner' elliptic problems with an adaptive finite element method, we prove that the overall method converges with the best possible rate. So far our results apply to two-dimensional domains and lowest order finite elements (continuous piecewise linears on the fictitious domain, and piecewise constants on the boundary of the physical domain). Joint work with S. Berrone (Torino), A. Bonito (Texas A&M), and M. Verani (Milano). |

Date: | October 18, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Peter Jantsch, TAMU |

Title: | The Lebesgue Constant for Leja Points on Unbounded Domains |

Abstract: | The standard Leja points are a nested sequence of points defined on a compact subset of the real line, and can be extended to unbounded domains with the introduction of a weight function $w : R \rightarrow [0, 1]$. Due to a simple recursive formulation, such abcissas show promise as a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares, and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes. |

Date: | October 25, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Giorgio Bornia, Texas Tech |

Title: | Some questions arising in PDE-constrained optimal control and numerical linear algebra |

Abstract: | The talk will be divided in two parts. In the first one, we address some issues arising in a certain class of boundary optimization problems. Common boundary optimal control problems yield a mismatch between the regularity of solutions on the domain and their boundary data. We discuss a reformulation of these problems with a lifting approach whose goal is to fix this regularity mismatch. Moreover, this approach provides additional benefits when applied to constraints characterized by compatibility conditions on the boundary data, such as those arising for the incompressible Navier-Stokes equations. In the second part of the talk, we discuss an analysis of preconditioning schemes for the numerical solution of Rayleigh-Bénard convection problems discretized with inf-sup stable finite element spaces. The analysis is carried out using a notion of field-of-values (FOV) equivalence between the preconditioner and the system matrix. Numerical results are discussed for both topics. |

Date: | November 1, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Natalia Kopteva, University of Limerick, Ireland |

Title: | Fully computable a posteriori error estimators on anisotropic meshes |

Abstract: | It is well known that anisotropic meshes offer an efficient way of computing reliable numerical approximations of solutions that exhibit sharp boundary and interior layers. Our goal is to give explicitly and fully computable a posteriori error estimates on reasonably general anisotropic meshes in the energy norm. This goal is achieved by a certain combination of explicit flux reconstruction and flux equilibration. Our approach differs from the previous work, mostly done for shape-regular meshes, in a few ways. The fluxes are equilibrated within a local patch using anisotropic weights depending on the local, possibly anisotropic, mesh geometry. Prior to the flux equilibration, divergence-free corrections are introduced for pairs of anisotropic triangles sharing a short edge. We shall also give an upper bound for the constructed estimator, in which the error constant is independent of the diameters and the aspect ratios of mesh elements, and discuss the efficiency of a posteriori error estimators on anisotropic meshes. |

Date: | November 8, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Diane Guignard, TAMU |

Title: | A posteriori error estimation for PDEs with random input data |

Abstract: | In this talk, we perform a posteriori error analysis for partial differential equations with uncertain input data characterized using random variables. Considering first small uncertainties, we use a perturbation approach expanding the solution of the problem with respect to a parameter ε that controls the amount of uncertainty. We derive residual-based a posteriori error estimates that control the two sources of error: the finite element discretization and the truncation in the expansion. The methodology is presented first on an elliptic equation with random coefficients and then on the steady-state Navier-Stokes equations on random domains. In the case of large uncertainties, we use instead the stochastic collocation method for the random space approximation. We present a residual-based a posteriori error estimate that provides an upper bound for the total error, which is composed of the finite element and the stochastic collocation errors. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm. Several numerical examples are presented to illustrate the theoretical findings. |

Date: | November 15, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Paul Hand, Rice University |

Title: | Deep Compressed Sensing |

Abstract: | Combining principles of compressed sensing with deep neural network-based generative image priors has recently been empirically shown to require 10X fewer measurements than traditional compressed sensing in certain scenarios. As deep generative priors (such as those obtained via generative adversarial training) improve, analogous improvements in the performance of compressed sensing and other inverse problems may be realized across the imaging sciences. In joint work with Vladislav Voroninski, we provide a theoretical framework for studying inverse problems subject to deep generative priors. In particular, we prove that with high probability, the non-convex empirical risk objective for enforcing random deep generative priors subject to compressive random linear observations of the last layer of the generator has no spurious local minima, and that for a fixed network depth, these guarantees hold at order-optimal sample complexity. |

Date: | November 29, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Laura Saavedra, Universidad Politécnica de Madrid |

Title: | TBA |

Date: | December 6, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Chris Kees, Coastal and Hydraulics Laboratory, US Army ERDC |

Title: | A two-scale computational framework for air-water-sediment dynamics |

Abstract: | A better understanding of sediment erosion and deposition processes is critical to the mission of the US Army Corps of Engineers. Long-term engineering of the Mississippi river, beach nourishment projects for coastal communities, and design of levees, breakwaters, and dunes for flood and storm protection all turn on the interaction of fluids with granular materials. Hunter Rouse, ``the father of modern hydraulics'' wrote in 1939 that, ``neither mathematical tools nor physical understanding of their use can be considered sufficiently far advanced to cope with so intricate a problem at the present time''. Today, the state of practice in computational modeling of sediment dynamics still relies heavily on empirical relationships. In recent decades, however, much progress has been made on the development of numerical methods capable of obtaining qualitatively correct solutions of fluid-grain dynamics at the microscale and on thermodynamically correct averaging methods to obtain practical computational models at the macroscale. This presentation will describe a combination of level set and immersed boundary methods for simulating microscale air-water-solid dynamics, averaging methods for deriving a 3D air-water-sediment model, and an incremental projection scheme for the three-phase Navier-Stokes system that arises at the macroscale. |

Date: | December 12, 2017 |

Time: | 3:00pm |

Location: | BLOC 506A |

Speaker: | Prof. Kai Diethelm, Technische University at Braunschweig, Germany |

Title: | On the Principle of ``Fractionalization'' in Mathematical Modeling |

Abstract: | Traditional mathematical models for many phenomena in various different fields of science and engineering are based on the use of classical differential equations, i.e. on equations containing integer order derivatives. These models are usually well understood from an analytic point of view, in particular regarding the qualitative behavior of their solutions. The availability of such information is important for evaluating whether the mathematical model really reflects the actual properties that the process in question has, and thus for showing that the set of equations is indeed a suitable model for the concrete process. In many cases, it has been observed that a generalization of the classical models obtained by replacing the integer order derivative(s) by a derivative of fractional (i.e., non-integer) order leads to better quantitative agreement between the mathematical model and experimental data, but the knowledge about qualitative properties is frequently lacking. Thus, the question whether the fractional order model is in fact able to correctly reproduce the behavior that the underlying process must exhibit frequently remains unanswered. In this talk, specific examples from the life sciences are used to demonstrate potential approaches to handle such issues and point out possible pitfalls in this ``fractionalization'' procedure for differential equation based mathematical models. Parts of the work described in this presentation are based on results of the project READEX that has received funding from the European Union's Horizon 2020 research and innovation program under Grant Agreement No.\ 671657. |