Charalambos Makridakis

Director of the Institute of Applied and Computational Mathematics

My research focuses on computational methods for phenomena described by evolutionary differential equations. Currently, I am interested on self adjusted methods for linear and nonlinear PDEs, for models across scales and for data-assisted computational modelling. 
  • Charalambos Makridakis holds the post of Professor of Numerical Analysis at the University of Crete and of Professor of Mathematics at the University of Sussex (on leave). He was founding member of the Department of Applied Mathematics and of the Archimedes Center for Modelling, Analysis and Computation in Crete. He had active role in leading European network grants related to Nonlinear Partial Differential Equations and their Applications funded by the European Commission. He received his PhD from the University of Crete in 1990; subsequently he was a post-doc at the University of Maryland, College Park and at the University of Tennessee. He held short term visiting posts at several universities and centres, including, Institute for Pure and Applied Mathematics at UCLA, University of Oxford, École Normale Superieure-Paris, CIRM-France, Institut Mittag-Leffler-Sweden and University of Rennes. Professor Makridakis is member of the Editorial Board of IMA Journal of Numerical Analysis and coordinator of the International Network 'Modelling and Computations for Shocks and Interfaces’. 
  • The prediction of various phenomena using models and computation is essential in addressing numerous real-life problems. The computation of singular phenomena, such as interfaces in phase separation, shocks, defects, and cracks, arises in many complex systems, presenting significant challenges.
    These phenomena are intriguing not only for their importance and applicability but also for the difficulties they pose in scientific and mathematical research. From a computational standpoint, there is a need to develop algorithms that are not only fast but also reliable.
    Nonlinear partial differential equations or discrete microscopic systems typically model such phenomena. They exhibit sensitivity to small perturbations, and it is often the case   for "natural ad hoc" computational methods to predict irrelevant solutions. Numerical methods introduce non-obvious perturbations to the mathematical model, underscoring the necessity of mathematical analysis. Analysis serves two crucial purposes: (a) ensuring that our computational methods approximate physically relevant solutions and (b) facilitating the innovative design of reliable computational methods.
    The integration of modern AI approaches with novel, problem-dependent algorithmic design has the potential to drive significant advancements in computing nonlinear phenomena. This is particularly valuable in applications where the computational cost of traditional methods is prohibitive.

    For a recent review paper see here (preprint version).
    For recent studies of mathematical aspects of ML algorithms see 'publications' and also here.

    CURRENT RESEARCH INTERESTS

    • Machine Learning for Scientific Applications
    – Stability and convergence of machine learning algorithms
    – Machine learning for nonlinear phenomena
    – Hybrid AI - classical scientific computing algorithms

    • Computational Energy Minimisation
    – DG schemes in mathematical materials science
    – Convergence and error control for nonlinear variational problems

    • Multiscale Adaptive Modeling and Algorithms
    – Static atomistic/continuum coupling
    – Dynamic multiscale coupling
    – Coarse grained time integrators
    – Energy based multiscale methods

    • Adaptive Methods for Evolutionary Problems
    – Error control driven algorithms, a posteriori estimates
    – Geometric/Anisotropic adaptivity
    – Convergence of adaptive algorithms
    – Space-time adaptivity

    • Wave Propagation Problems
    – Shock-wave propagation problems
    – Methods for linear transport and kinetic equations
    – Error control for wave equations
    – Computational methods for dispersive wave equations
    – Computational methods for high frequency wave propagation problems

    • Problems in Fluid Mechanics.
    –  Error control for Navier-Stokes
    –  Adaptive methods for turbulent models
    –  Isothermal Navier-Stokes Korteweg system

    • Material Defects
    – Computational methods for material defects for static and dynamic problems

  • Selected  Articles

    G Grekas, CG Makridakis (2025) Deep Ritz - Finite element methods: Neural network methods trained with finite elements, Computer Methods in Applied Mechanics and Engineering, Vol. 437, 117798, https://doi.org/10.1016/j.cma.2025.117798

    G Grekas, K Koumatos, C Makridakis, A Vikelis (2025) Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems, arXiv preprint arXiv:2501.11944

    G Grekas, K Koumatos, C Makridakis, A Vikelis (2025) A class of discontinuous Galerkin methods for nonlinear variational problems, Mathematics of Computation (pdf)

    G Akrivis, CG Makridakis, C Smaragdakis (2024) Runge-Kutta Physics Informed Neural Networks: Formulation and Analysis, arXiv preprint arXiv:2412.20575

    EH Georgoulis, EJC Hall, CG Makridakis (2024) On a posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems, Studies in Applied Mathematics 153 (4), e12772 (pdf file)

    G Grekas, CG Makridakis (2024) Deep Ritz-Finite Element methods: Neural Network Methods trained with Finite Elements, arXiv preprint arXiv:2409.08362

    C Kalaitzidou, G Grekas, A Zilian, C Makridakis, P Rosakis (2024) Compressive instabilities enable cell-induced extreme densification patterns in the fibrous extracellular matrix: Discrete model predictions, PLOS Computational Biology 20 (7), e1012238 (pdf file)

    O. Lakkis, C. Makridakis (2024) Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, arXiv preprint arXiv:2412.05913v1

    CG Makridakis, A Pim, T Pryer (2024) A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods, arXiv preprint arXiv:2411.08702

    CG Makridakis, A Pim, T Pryer (2024) Deep Uzawa for PDE constrained optimisation, arXiv preprint arXiv:2410.17359

    DA Mitsoudis, M Plexousakis, GN Makrakis, C Makridakis (2024) Approximations of the Helmholtz equation with variable wave number in one dimension, Studies in Applied Mathematics, e12756 (pdf file)

    G Grekas, K Koumatos, C Makridakis, A Vikelis (2023) A class of Discontinuous Galerkin methods for nonlinear variational problems, arXiv preprint arXiv:2308.12891

    D Gazoulis, I Gkanis, CG Makridakis (2023) On the Stability and Convergence of Physics Informed Neural Networks, arXiv preprint arXiv:2308.05423

    M Loulakis, CG Makridakis (2023) A new approach to generalisation error of machine learning algorithms: Estimates and convergence, arXiv preprint arXiv:2306.13784

    EH Georgoulis, CG Makridakis (2023) Lower bounds, elliptic reconstruction and a posteriori error control of parabolic problems, IMA Journal of Numerical Analysis, 43 (6), 3212-3242. (html file)

    C Kalaitzidou, G Grekas, A Zilian, C Makridakis, P Rosakis (2022) Compressive Instabilities Enable Cell-Induced Extreme Densification Patterns in the Fibrous Extracellular Matrix: Discrete Model Predictions, arXiv preprint arXiv:2212.01213

    L Banjai, C Makridakis (2022) A posteriori error analysis for approximations of time-fractional subdiffusion problems, Mathematics of Computation 91 (336), 1711-1737 (    pdf file)

    G Akrivis, CG Makridakis (2022) A posteriori error estimates for Radau IIA methods via maximal parabolic regularity, Numerische Mathematik 150 (3), 691-717 (html file)

    G Akrivis, C Makridakis (2022) On maximal regularity estimates for discontinuous Galerkin time-discrete methods, SIAM Journal on Numerical Analysis 60 (1), 180-194 (html file)

    G Grekas, K Koumatos, C Makridakis, P Rosakis (2022) Approximations of Energy Minimization in Cell-Induced Phase Transitions of Fibrous Biomaterials: Gamma-Convergence Analysis, SIAM Journal on Numerical Analysis 60 (2), 715-750 (pdf file)

    I Gkanis, C Makridakis (2021) A new class of entropy stable schemes for hyperbolic systems: Finite element methods, Mathematics of Computation 90 (330), 1663-1699 (pdf file)

    G Grekas, M Proestaki, P Rosakis, J Notbohm, C Makridakis, G Ravichandran (2021) Cells exploit a phase transition to mechanically remodel the fibrous extracellular matrix, Journal of the Royal Society Interface 18 (175), 20200823 (pdf file)

    I Gkanis, G Grekas, E Karnessis, CG Makridakis (2020) Mathematics of computational modelling: some challenges of computing nonlinear phenomena, First Congress of Greek Mathematicians: Proceedings of the Congress held in Athens, Greece, June 25–30, 2018 (html file)

    G Grekas, M Proestaki, P Rosakis, J Notbohm, C Makridakis, G Ravichandran (2019) Cells exploit a phase transition to establish interconnections in fibrous extracellular matrices, arXiv preprint arXiv:1905.11246

    Grekas G, Koumatos K, Makridakis C, Rosakis P (2019) Approximations of energy minimization in cell-induced phase transitions of fibrous biomaterials: Gamma-convergence analysis, arXiv preprint arXiv:1907.01382

    EH Georgoulis, E Hall, C Makridakis (2019) An a posteriori error bound for discontinuous Galerkin approximations of convection–diffusion problems, IMA Journal of Numerical Analysis 39 (1), 34-60 (html file) (pdf file)

    E Baensch, F Karakatsani, CG Makridakis (2018) A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem, Calcolo 55, 1-32 (html file)

    Makridakis, CG (2018) On the Babuška-Osborn approach to finite element analysis: L2 estimates for unstructured meshes.  Numerische Mathematik, 139 (4). pp. 831-844. ISSN 0029-599X (pdf file )

    Georgoulis, EH, Hall, E and Makridakis, C (2017) An a posteriori error bound for discontinuous Galerkin approximations of convection-diffusion problems. IMA Journal of Numerical Analysis. ISSN 0272-4979 (pdf file)

    Georgoulis, EH, Lakkis, O, Makridakis, CG and Virtanen, JM (2016) A posteriori error estimates for leap-frog and cosine methods for second order evolution problems. SIAM Journal on Numerical Analysis (SINUM), 54 (1). pp. 120-136. ISSN 0036-1429 (pdf file)

    Lakkis, O, Makridakis, C and Pryer, T (2015) A comparison of duality and energy a posteriori estimates for L∞(0,T;L2(Ω)) in parabolic problems. Mathematics of Computation, 84 (294). pp. 1537-1569. ISSN 0025-5718 (pdf file)

    Giesselmann, J, Makridakis, C and Pryer, T (2015) A posteriori analysis of discontinuous galerkin schemes for systems of hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 53 (3). pp. 1280-1303. ISSN 0036-1429 (pdf file)

    Karakashian, O and Makridakis, C (2014) A posteriori error estimates for discontinuous Galerkin Methods for the Generalised Korteweg-de Vries Equation. Mathematics of Computation, 84. pp. 1145-1167. ISSN 0025-5718 (pdf file)

    Giesselmann, J, Makridakis, C and Pryer, T (2014) Energy consistent DG methods for the Navier-Stokes-Korteweg system. Mathematics of Computation, 83 (289). pp. 2071-2099. ISSN 0025-5718 (pdf file)

    Vairaktaris, E and Makridakis, C (2013) Control estimates and well-posedness of an inverse subsidence problem in geomechanics. Inverse Problems in Science and Engineering, 21 (5). pp. 823-839. ISSN 1741-5977 (pdf file)

    Makridakis, C and Süli, E (2013) Finite element analysis of Cauchy-Born approximations to atomistic models. Archive for Rational Mechanics and Analysis, 207 (3). pp. 813-843. ISSN 0003-9527 (pdf file)

    Bänsch, E, Karakatsani, F and Makridakis, C (2012) A posteriori error control for fully discrete Crank–Nicolson schemes. SIAM Journal on Numerical Analysis, 50 (6). pp. 2845-2872. ISSN 0036-1429 (pdf file)

    Kyza, I and Makridakis, C (2011) Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations. SIAM Journal on Numerical Analysis, 49 (1). pp. 405-426. ISSN 0036-1429 (pdf file)

    Akrivis, G, Makridakis, C and Nochetto, R H (2009) Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods.  Numerische Mathematik, 114 (1). pp. 133-160. ISSN 0029-599X (pdf file)