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We generalize some of the functional (hyper-circle) a posteriori estimates from finite element settings to general graphs or Hilbert space settings. We provide several theoretical results in regard to the generalized a posteriori error…
We propose and explore a new, general-purpose method for the implicit time integration of elastica. Key to our approach is the use of a mixed variational principle. In turn its finite element discretization leads to an efficient alternating…
This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for…
We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the…
We propose a combined nodal integration and virtual element method for compressible and nearly incompressible elasticity, wherein the strain is averaged at the nodes from the strain of surrounding virtual elements. For the strain averaging…
We combine a systematic approach for deriving general a posteriori error estimates for convex minimization problems based on convex duality relations with a recently derived generalized Marini formula. The a posteriori error estimates are…
We introduce a new technique for gradient normalization during neural network training. The gradients are rescaled during the backward pass using normalization layers introduced at certain points within the network architecture. These…
We describe a framework for reformulating and solving optimization problems that generalizes the well-known framework originally introduced by Benders. We discuss details of the application of the procedures to several classes of…
We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly-singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error…
For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local…
We introduced and analyzed robust recovery-based a posteriori error estimators for various lower order finite element approximations to interface problems in [9, 10], where the recoveries of the flux and/or gradient are implicit (i.e.,…
In this work we present an extension of the Virtual Element Method with curved edges for the numerical approximation of the second order wave equation in a bidimensional setting. Curved elements are used to describe the domain boundary, as…
This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov-Maxwell system. The SD scheme yields a weak formulation, that corresponds to an…
The Cartesian reverse derivative is a categorical generalization of reverse-mode automatic differentiation. We use this operator to generalize several optimization algorithms, including a straightforward generalization of gradient descent…
Virtual element methods (VEMs) without extrinsic stabilization in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to…
Variational inference has become an increasingly attractive fast alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, a major obstacle to the widespread use of variational methods is the lack of…
The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain is analyzed under reduced elliptic regularity. The…
Generalized additive index models (GAIMs) offer a flexible semiparametric framework for capturing complex data relationships, balancing the interpretability of parametric models with the flexibility of nonparametric approaches. However,…
We introduce the Neural Approximated Virtual Element Method, a novel polygonal method that relies on neural networks to eliminate the need for projection and stabilization operators in the Virtual Element Method. In this paper, we discuss…
Purpose: To allow fast and high-quality reconstruction of clinical accelerated multi-coil MR data by learning a variational network that combines the mathematical structure of variational models with deep learning. Theory and Methods:…