All Outputs (19)
Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach (2024)
Journal Article
The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric PDE. The... Read More about Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach.
A unified framework for Navier-Stokes Cahn-Hilliard models with non-matching densities (2023)
Journal Article
Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard (NSCH) models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. T... Read More about A unified framework for Navier-Stokes Cahn-Hilliard models with non-matching densities.
Neural Control of Discrete Weak Formulations: Galerkin, Least Squares & Minimal-Residual Methods with Quasi-Optimal Weights (2022)
Journal Article
There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyse a framework for the neural optimization of discrete weak formulations, suitable for finite element methods. The main idea of... Read More about Neural Control of Discrete Weak Formulations: Galerkin, Least Squares & Minimal-Residual Methods with Quasi-Optimal Weights.
Linearization of the Travel Time Functional in Porous Media Flows (2022)
Journal Article
The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the postclosure safety assessment of deep geological storage facilitie... Read More about Linearization of the Travel Time Functional in Porous Media Flows.
Projection in negative norms and the regularization of rough linear functionals (2022)
Journal Article
In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as W01,q(Ω), where 1 < q< ∞ and Ωis a Lipschitz domain, we propose a projection method in negative Sobolev spacesW-1,p(Ω) , pbeing the conjugate expo... Read More about Projection in negative norms and the regularization of rough linear functionals.
Discretization of linear problems in banach spaces: Residual minimization, nonlinear petrov-galerkin, and monotone mixed methods (2020)
Journal Article
This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms and its inexact version using discrete d... Read More about Discretization of linear problems in banach spaces: Residual minimization, nonlinear petrov-galerkin, and monotone mixed methods.
A Mechanistic Investigation into Ischemia-Driven Distal Recurrence of Glioblastoma (2020)
Journal Article
Glioblastoma (GBM) is the most aggressive primary brain tumor with a short median survival. Tumor recurrence is a clinical expectation of this disease and usually occurs along the resection cavity wall. However, previous clinical observations have su... Read More about A Mechanistic Investigation into Ischemia-Driven Distal Recurrence of Glioblastoma.
A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations (2020)
Journal Article
We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The... Read More about A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations.
Speed Switch in Glioblastoma Growth Rate due to Enhanced Hypoxia-Induced Migration (2020)
Journal Article
We analyze the wave-speed of the Proliferation Invasion Hypoxia Necro-sis Angiogenesis (PIHNA) model that was previously created and applied to simulate the growth and spread of glioblastoma (GBM), a particularly aggressive primary brain tumor. We ex... Read More about Speed Switch in Glioblastoma Growth Rate due to Enhanced Hypoxia-Induced Migration.
The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces (2019)
Journal Article
© 2019 Walter de Gruyter GmbH, Berlin/Boston 2019. We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows f... Read More about The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces.
The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method (2019)
Journal Article
While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H10(Ω), the Banach Sobolev space W1,q0(Ω), 1 less than ∞ , is more general allowing more irregular solutions. In this paper we present a... Read More about The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method.
Explicit-in-time goal-oriented adaptivity (2018)
Journal Article
Goal-oriented adaptivity is a powerful tool to accurately approximate physically relevant solution features for Partial Differential Equations. In time dependent problems, we seek to represent the error in the quantity of interest as an integral over... Read More about Explicit-in-time goal-oriented adaptivity.
A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations (2018)
Journal Article
While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology for dualit... Read More about A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations.
Parallel-in-space-time, adaptive finite element framework for non-linear parabolic equations (2018)
Journal Article
We present an adaptive methodology for the solution of (linear and) non-linear time dependent problems that is especially tailored for massively parallel computations. The basic concept is to solve for large blocks of space-time unknowns instead of m... Read More about Parallel-in-space-time, adaptive finite element framework for non-linear parabolic equations.
Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method (2018)
Journal Article
© 2018 World Scientific Publishing Company. While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerica... Read More about Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method.
An abstract analysis of optimal goal-oriented adaptivity (2016)
Journal Article
We provide an abstract framework for optimal goal-oriented adaptivity for finite element methods and boundary element methods in the spirit of (Carstensen, Feischl, Page, and Praetorius, Axioms of adaptivity, Comput. Math. Appl., 67 (2014), pp. 1195–... Read More about An abstract analysis of optimal goal-oriented adaptivity.
Formal asymptotic limit of a diffuse-interface tumor-growth model (2015)
Journal Article
We consider a diffuse-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coefficient of the reaction term tends to infinity, the s... Read More about Formal asymptotic limit of a diffuse-interface tumor-growth model.
Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations (2014)
Journal Article
We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed... Read More about Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations.