149 citations · 414 across the 23 of their papers we have counts for
24 papers
Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries
Charalambos Makridakis, Aaron Pim, Tristan Pryer +1
We propose a neural network framework for solving stationary linear transport equations with inflow boundary conditions. The method represents the solution using a neural network a…
PINN-DG: Residual neural network methods trained with Finite Elements
Georgios Grekas, Charalambos G. Makridakis, Tristan Pryer
Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integ…
Neural Measures for learning distributions of Random PDEs
Georgios Arampatzis, Stylianos Katsarakis, Charalambos Makridakis
The integration of Scientific Machine Learning (SciML) techniques with uncertainty quantification (UQ) represents a rapidly evolving frontier in computational science. This work ad…
Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems
Georgios Grekas, Konstantinos Koumatos, Charalambos Makridakis +1
In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular,…
Runge-Kutta Physics Informed Neural Networks: Formulation and Analysis
Georgios Akrivis, Charalambos G. Makridakis, Costas Smaragdakis
In this paper we consider time-dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge--Kutta and related…
Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems
Omar Lakkis, Charalambos Makridakis
We derive aposteriori error estimates for fully discrete approximations to solutions of linear parabolic equations on the space-time domain. The space discretization uses finite el…
A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods
Charalambos G. Makridakis, Aaron Pim, Tristan Pryer
We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics…
Deep Uzawa for PDE constrained optimisation
Charalambos G. Makridakis, Aaron Pim, Tristan Pryer
In this work, we present a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs. The approach is based on recasting the pr…
Deep Ritz-Finite Element methods: Neural Network Methods trained with Finite Elements
Georgios Grekas, Charalambos G. Makridakis
While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains $Ω\su…
A class of Discontinuous Galerkin methods for nonlinear variational problems
Georgios Grekas, Konstantinos Koumatos, Charalambos Makridakis +1
In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems associated with convex energies. We propose element-wise nonconforming f…
On the Stability and Convergence of Physics Informed Neural Networks
Dimitrios Gazoulis, Ioannis Gkanis, Charalambos G. Makridakis
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and…
A new approach to generalisation error of machine learning algorithms: Estimates and convergence
Michail Loulakis, Charalambos G. Makridakis
In this work we consider a model problem of deep neural learning, namely the learning of a given function when it is assumed that we have access to its point values on a finite set…
Compressive Instabilities Enable Cell-Induced Extreme Densification Patterns in the Fibrous Extracellular Matrix: Discrete Model Predictions
Chrysovalantou Kalaitzidou, Georgios Grekas, Andreas Zilian +2
Through modelling and simulations we show that material instabilities play a dominant role in the mechanical behavior of the fibrous collagen Extracellular Matrix (ECM), as observe…
A posteriori error analysis for approximations of time-fractional subdiffusion problems
Lehel Banjai, Charalambos G. Makridakis
In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpre…
Approximations of energy minimization in cell-induced phase transitions of fibrous biomaterials: -convergence analysis
Georgios Grekas, Konstantinos Koumatos, Charalambos Makridakis +1
We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is b…
Cells exploit a phase transition to mechanically remodel the fibrous extracellular matrix
Georgios Grekas, Maria Proestaki, Phoebus Rosakis +3
Through mechanical forces, biological cells remodel the surrounding collagen network, generating striking deformation patterns. Tethers-tracts of high densification and fiber align…
Babuška-Osborn techniques in discontinuous Galerkin methods: -norm error estimates for unstructured meshes
Emmanuil Georgoulis, Charalambos Makridakis, Tristan Pryer
We prove the inf-sup stability of the interior penalty class of discontinuous Galerkin schemes in unbalanced mesh-dependent norms, under a mesh condition allowing for a general cla…
A posteriori error estimates for leap-frog and cosine methods for second order evolution problems
Emmanuil H. Georgoulis, Omar Lakkis, Charalambos Makridakis +1
We consider second order explicit and implicit two-step time-discrete schemes for wave-type equations. We derive optimal order aposteriori estimates controlling the time discretiza…
A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws
Jan Giesselmann, Charalambos Makridakis, Tristan Pryer
In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appro…
On atomistic-to-continuum couplings without ghost forces in three dimensions
Charalambos Makridakis, Dimitrios Mitsoudis, Phoebus Rosakis
In this paper we construct energy based numerical methods free of ghost forces in three dimensional lattices arising in crystalline materials. The analysis hinges on establishing a…
Energy consistent DG methods for the Navier-Stokes-Korteweg system
Jan Giesselmann, Charalambos Makridakis, Tristan Pryer
We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for t…
A posteriori -error bounds in finite element approximation of the wave equation
Emmanuil H. Georgoulis, Omar Lakkis, Charalambos Makridakis
We address the error control of Galerkin discretization (in space) of linear second order hyperbolic problems. More specifically, we derive a posteriori error bounds in the L\infty…
A posteriori error estimates in the maximum norm for parabolic problems
Alan Demlow, Omar Lakkis, Charalambos Makridakis
We derive a posteriori error estimates in the norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction t…
A comparison of duality and energy aposteriori estimates for L?(0,T;L2(Ω)) in parabolic problems
Omar Lakkis, Charalambos Makridakis, Tristan Pryer
We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parab…