# Brendan KeithBrown University · Division of Applied Mathematics

Brendan Keith

PhD

## About

37

Publications

5,111

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

290

Citations

Citations since 2017

Introduction

**Skills and Expertise**

Additional affiliations

February 2021 - June 2022

September 2020 - December 2021

September 2018 - August 2020

Education

August 2013 - August 2018

September 2011 - August 2013

September 2007 - August 2011

## Publications

Publications (37)

We develop a novel data-driven approach to modeling the atmospheric boundary layer. This approach leads to a nonlocal, anisotropic synthetic turbulence model which we refer to as the deep rapid distortion (DRD) model. Our approach relies on an operator regression problem that characterizes the best fitting candidate in a general family of nonlocal...

We introduce a gravitational waveform inversion strategy that discovers mechanical models of binary black hole (BBH) systems. We show that only a single time series of (possibly noisy) waveform data is necessary to construct the equations of motion for a BBH system. Starting with a class of universal differential equations parameterized by feed-for...

In this work, we revisit the marking decisions made in the standard adaptive finite element method (AFEM). Experience shows that a na\"{i}ve marking policy leads to inefficient use of computational resources for adaptive mesh refinement (AMR). Consequently, using AFEM in practice often involves ad-hoc or time-consuming offline parameter tuning to s...

Reducing the intensity of wind excitation via aerodynamic shape modification is a major strategy to mitigate the reaction forces on supertall buildings, reduce construction and maintenance costs, and improve the comfort of future occupants. To this end, computational fluid dynamics (CFD) combined with state-of-the-art stochastic optimization algori...

We introduce adaptive sampling methods for stochastic programs with deterministic constraints. First, we propose and analyze a variant of the stochastic projected gradient method, where the sample size used to approximate the reduced gradient is determined on-the-fly and updated adaptively. This method is applicable to a broad class of expectation-...

Adaptive mesh refinement (AMR) is necessary for efficient finite element simulations of complex physical phenomenon, as it allocates limited computational budget based on the need for higher or lower resolution, which varies over space and time. We present a novel formulation of AMR as a fully-cooperative Markov game, in which each element is an in...

Reducing the intensity of wind excitation via aerodynamic shape modification is a major strategy to mitigate the reaction forces on supertall buildings, reduce construction and maintenance costs, and improve the comfort of future occupants. To this end, computational fluid dynamics (CFD) combined with state-of-the-art stochastic optimization algori...

A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are LL⁎ methods. In this work, we argue that all high-order methods in this class should be expected to deliver substandard uniform h-refinement conv...

We develop a novel data-driven approach to modeling the atmospheric boundary layer. This approach leads to a nonlocal, anisotropic synthetic turbulence model which we refer to as the deep rapid distortion (DRD) model. Our approach relies on an operator regression problem which characterizes the best fitting candidate in a general family of nonlocal...

Nowadays, we are facing the increasing necessity of dealing with uncertainties in many different engineering fields, together with the constant development of computational capabilities. Current solvers must satisfy both statistical convergence and computational efficiency. This work aims to apply a specific Monte Carlo algorithm, developed to run...

We introduce a gravitational waveform inversion strategy that discovers mechanical models of binary black hole (BBH) systems. We show that only a single time series of (possibly noisy) waveform data is necessary to construct the equations of motion for a BBH system. Starting with a class of universal differential equations parameterized by feed-for...

A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are LL* methods. In this work, we argue that all high-order methods in this class should be expected to deliver substandard uniform h-refinement conv...

We introduce adaptive sampling methods for risk-neutral and risk-averse stochastic programs with deterministic constraints. In particular, we propose a variant of the stochastic projected gradient method where the sample size used to approximate the reduced gradient is determined a posteriori and updated adaptively. We also propose an SQP-type meth...

The surrogate matrix methodology delivers low-cost approximations of matrices (i.e., surrogate matrices) which are normally computed in Galerkin methods via element-scale quadrature formulas. In this paper, the methodology is applied to a number of model problems in wave mechanics treated in the Galerkin isogeometric setting. Herein, the resulting...

Nowadays, we are facing the increasing necessity of dealing with uncertainties in many different engineering fields, together with the constant development of computational capabilities. Current solvers must satisfy both statistical convergence and computational efficiency. This work aims to apply a specific Monte Carlo algorithm, developed to run...

This paper presents a class of turbulence models written in terms of fractional partial differential equations (FPDEs) with stochastic loads. Every solution of these FPDE models is an incompressible velocity field and the distribution of solutions is Gaussian. Interaction of the turbulence with solid walls is incorporated through the enforcement of...

The surrogate matrix methodology delivers low-cost approximations of matrices (i.e., surrogate matrices) which are normally computed in Galerkin methods via element-scale quadrature formulas. In this paper, the methodology is applied to a number of model problems in wave mechanics treated in the Galerkin isogeometic setting. Herein, the resulting s...

A new methodology in isogeometric analysis (IGA) is presented. This methodology delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed from element-scale quadrature formulas. To generate surrogate matrices, quadrature must only be performed on certain elements in the computation...

This article introduces the DPG-star (from now on, denoted DPG*) finite element method. It is a method that is in some sense dual to the discontinuous Petrov–Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG* methodology is a means to...

A reference implementation of a new method in isogeometric analysis (IGA) is presented. It delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed by element-scale quadrature. To generate surrogate matrices, quadrature must only be performed on a fraction of the elements in the c...

A reference implementation of a new method in isogeometric analysis (IGA) is presented. It delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed by element-scale quadrature. To generate surrogate matrices, quadrature must only be performed on a fraction of the elements in the c...

In this article, several discontinuous Petrov-Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov-Galerkin methods, with so-cal...

A new methodology in isogeometric analysis (IGA) is presented. This methodology delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed from element-scale quadrature formulas. To generate surrogate matrices, quadrature must only be performed on certain elements in the computation...

We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 2017]), which we hereby refer to as the surrogate matrix methodology. This methodology is based on the piece-wise smooth approximation of the matrices involved in a standard finite element discretizatio...

This dissertation presents a novel framework for the construction and analysis of finite element methods with trial and test spaces of unequal dimension. At the heart of this work is a new duality theory suitable for variational formulations with non-symmetric functional settings. The primary application of this theory, in this dissertation, is the...

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a mea...

In this article, a Petrov-Galerkin duality theory is developed. This theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of steady boundary value problems, however, many of the idea...

We introduce a cousin of the DPG method - the DPG* method - discuss their relationship and compare the two methods through numerical experiments.

A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the s...

We explore a vexing benchmark problem for viscoelastic fluid flows with the discontinuous Petrov-Galerkin (DPG) finite element method of Demkowicz and Gopalakrishnan [1,2]. In our analysis, we develop an intrinsic a posteriori error indicator which we use for adaptive mesh generation. The DPG method is useful for the problem we consider because the...

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually well-posed family of broken variational formulations of the original PDE. It can be exploited to solve challenging pro...

The flexibility of the DPG methodology is exposed by solving the linear elasticity equations under different variational formulations, including some with non-symmetric functional settings (different infinite-dimensional trial and test spaces). The family of formulations presented are proved to be mutually ill or well-posed when using traditional e...

A unified construction of high order shape functions is given for all four
classical energy spaces ($H^1$, $H(\mathrm{curl})$, $H(\mathrm{div})$ and
$L^2$) and for elements of "all" shapes (segment, quadrilateral, triangle,
hexahedron, tetrahedron, triangular prism and pyramid). The discrete spaces
spanned by the shape functions satisfy the commuti...

The subject of this thesis is the detection of Lagrangian Coherent Structures (LCS) in three-dimensional steady flows. LCS are influential material surfaces that act as skeletons of observed mixing patterns in a dynamical system. We review recent variational results for the construction of LCS in two-dimensional flows, and develop our own numerical...