Publications

I  graduated in the field of dynamical Hamiltonian systems, symplectic geometry, and algebraic topology – studying the Floer and Morse homology groups associated with such systems. My dissertation, published as part of the Yvonne Choquet-Bruhat Festschrift, contributes to this foundational exploration. A fulfilling aspect of my subsequent research endeavors lies in bridging theoretical mathematics with the practical, empirical world of the sciences. I want to highlight our recent contribution:

The Fast Newton Transform (FNT): The FNT is a novel algorithm for multivariate polynomial interpolation with a runtime of nearly Nlog(N), where N scales only sub-exponentially with spatial dimension, surpassing the runtime of the tensorial Fast Fourier Transform (FFT). We have proven and demonstrated the optimal geometric approximation rates for a class of analytic functions—termed Bos–Levenberg–Trefethen functions—to be reached by the FNT and to be maintained for the derivatives of the interpolants. This establishes the FNT as a new standard in spectral methods, particularly suitable for high-dimensional, non-periodic PDE problems, and interpolation tasks, arising as the computational bottleneck in solving e.g. 6D Boltzmann, Fokker-Planck, or Vlaslov equations, multi-body Hamiltonian systems, and the inference of governing equations in complex self-organizing systems. Further applications can be reached by the Newton Neural Operator, realizing fast (de-)convolution in machine learning tasks. 

A preprint can be found here: https://arxiv.org/abs/2505.14909

Further synthesis of theoretical and applied mathematics is reflected in the list of publications below, covering diverse topics such as approximation theory, numerical differential geometry, machine learning and numerical methods, graph theory, optimization, computer science, and quantum physics.

Hofmann, P. A., D. Wicaksono, and M. Hecht, 2025: The Fast Newton Transform: Interpolation in downward closed polynomial spaces. https://arxiv.org/abs/2505.14909

Hecht, M., Wicaksono, Acosta, U. H., D., Gonciarz, K., Michelfeit, J., Sivkin, V., & Sbalzarini, I. F.,  Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions. IMA Journal on Numerical Analysis,  https://arxiv.org/abs/2504.17899

A. B. Robles, P.-A. Hofmann, T. Chuna, T. Dornheim, and M. Hecht, PyLIT: Reformulation and implementation of the analytic continuation problem using kernel representation methods,Comput. Phys. Commun. 335 (2025), 109904. https://doi.org/10.1016/j.cpc.2025.109904

S. B. Jain, Z. Shao, and M. Hecht, Automated detection of potential artifacts in machine learning based bio-image segmentation, Mach. Learn.: Sci. Technol. 6 (2025), https://doi.org/10.1088/2632-2153/ae1546

G. Volpe, C. Wählby, L. Tian, M. Hecht, A. Yakimovich, K. Monakhova, L. Waller, I. F. Sbalzarini, C. A. Metzler, M. Xie, et al., Roadmap on deep learning for microscopy, J. Phys.: Photonics (2025). https://doi.org/10.1088/2515-7647/ae0fd1

Chuna, T. M., Vorberger, J., Tolias, P., Benedix Robles, A., Hecht, M., Hofmann, P. A., ... & Dornheim, T. (2025). Second roton feature in the strongly coupled electron liquid. The Journal of Chemical Physics, 163(3). https://doi.org/10.1063/5.0281085

Wicaksono, D., U. H. Acosta, S. K. T. Veettil, J. Kissinger, and M. Hecht, 2025: Minterpy: multivariate polynomial interpolation in Python. J. Open Source Software, 10(109), 7702, https://doi.org/10.21105/joss.07702

Zavalani, G., Sander, O., and Hecht, M. (2025) High-order integration on regular triangulated manifolds reaches Super-Algebraic Approximation Rates through Cubical Re-parameterizations. accepted in SIAM Journal on Numerical Analysis, https://arxiv.org/abs/2311.13909

Zavalani, G., Hecht, M. (2025). High-Order Numerical Integration on Regular Embedded Surfaces. In: Sequeira, A., Silvestre, A., Valtchev, S.S., Janela, J. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2023, Volume 2. ENUMATH 2023. Lecture Notes in Computational Science and Engineering, vol 154. Springer, Cham. https://doi.org/10.1007/978-3-031-86169-7_51

Ramanaik CK, Willmann A, Suarez Cardona J-E, Hanfeld P, Hoffmann N, Hecht M. Ensuring Topological Data-Structure Preservation under Autoencoder Compression Due to Latent Space Regularization in Gauss–Legendre Nodes. Axioms. 2024; 13(8):535. https://doi.org/10.3390/axioms13080535

Suarez Cardona J.-E., Hofmann P.A. & Hecht M. Negative order Sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness. Machine Learning: Science and Technology. 2024;5(3):035029. https://doi.org/10.1088/2632-2153/ad62ac.

Zavalani, G., Shehu, E., & Hecht, M. (2024). A note on the rate of convergence of integration schemes for closed surfaces. Computational and Applied Mathematics, 43(2). https://doi.org/10.1007/s40314-024-02611-y

Suarez Cardona, J.-E., & Hecht, M. (2023). Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power. Machine Learning: Science and Technology, 4(4). https://doi.org/10.1088/2632-2153/acf97a

Wicaksono, D., & Hecht, M. (2023). UQTestFuns: A Python3 library of uncertainty quantification (UQ) test functions. Journal of Open Source Software, 8(90). https://doi.org/10.21105/joss.05671

Veettil, S. K. T., Zavalani, G., Acosta, U. H., Sbalzarini, I. F., & Hecht, M. (2023). Global Polynomial Level Sets for Numerical Differential Geometry of Smooth Closed Surfaces. SIAM Journal on Scientific Computing, 45(4), A1995–A2018. https://doi.org/10.1137/22m1536510

Schreiber, J., Wicaksono, D., & Hecht, M. (2023). Minimizing Black Boxes due to Polynomial-Model-Based Optimization. In Proceedings of the Companion Conference on Genetic and Evolutionary Computation (GECCO ’23 Companion). ACM. https://doi.org/10.1145/3583133.3590743

Schreiber, J., Wicaksono, D., & Hecht, M. (2023). Polynomial-Model-Based Optimization for Blackbox Objectives. OLA2023, Malaga Spain, arXiv:2309.00663.

Dornheim, T., Wicaksono, D. C., Suarez-Cardona, J. E., Tolias, P., Böhme, M. P., Moldabekov, Z. A., Hecht, M., Vorberger, J. (2023). Extraction of the frequency moments of spectral densities from imaginary-time correlation function data. Physical Review B, 107(15). https://doi.org/10.1103/physrevb.107.155148

Jain, S. B., Zongru, S., Veettil, S. K., & Hecht, M. (2022). Adversarial attacks for machine learning denoisers and how to resist them. In G. Volpe, J. B. Pereira, D. Brunner, & A. Ozcan (Eds.), Emerging Topics in Artificial Intelligence (ETAI) 2022. SPIE. https://doi.org/10.1117/12.2632954

Hecht, M., Gonciarz, K., & Horvát, S. (2021). Tight Localizations of Feedback Sets. ACM Journal of Experimental Algorithmics, 26. https://doi.org/10.1145/3447652

Hecht, M., & Sbalzarini, I. F. (2021). Biggs Theorem for Directed Cycles and Topological Invariants of Digraphs. Advances in Pure Mathematics, 11(06). https://doi.org/10.4236/apm.2021.116037

Hecht, M., & Sbalzarini, I. F. (2018). Fast Interpolation and Fourier Transform in High-Dimensional Spaces. In Intelligent Computing (pp. 53–75). Springer International Publishing. https://doi.org/10.1007/978-3-030-01177-2_5

Hecht, M. (2017). Exact Localisations of Feedback Sets. Theory of Computing Systems, 62(5), 1048–1084. https://doi.org/10.1007/s00224-017-9777-6

Hecht, M. (2017). A generalization of the most common subgraph distance and its application to graph editing. Pattern Recognition Letters, 87, 71–78. https://doi.org/10.1016/j.patrec.2016.09.008

Hecht, M.(2013). Isomorphic chain complexes of Hamiltonian dynamics on tori. Journal of Fixed Point Theory and Applications, 14(1), 165–221. (The Yvonne Choquet-Bruhat Festschrift) https://doi.org/10.1007/s11784-013-0149-9

J.-E. S. Cardona, S. Reddy, and M. Hecht (2024). Hybrid surrogate models: circumventing Gibbs phenomenon for partial differential equations with finite shock-type discontinuities, arXiv  https://arxiv.org/abs/2408.02497

Schreiber, J., Batlle, P., Wicaksono, D., & Hecht, M. (2024). PMBO: Enhancing Black-Box Optimization through Multivariate Polynomial Surrogates. arXiv. https://arxiv.org/abs/2403.07485

Cardona, J.-E. S., Hofmann, P.-A., & Hecht, M. (2023). Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces. arXiv. https://arxiv.org/abs/2301.04887

Veettil, S. K. T., Zheng, Y., Acosta, U. H., Wicaksono, D., & Hecht, M. (2022). Multivariate Polynomial Regression of Euclidean Degree Extends the Stability for Fast Approximations of Trefethen Functions. arXiv. https://arxiv.org/abs/2212.11706

Hecht, M., Gonciarz, K., Michelfeit, J., Sivkin, V., & Sbalzarini, I. F. (2020). Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality. arXiv. https://arxiv.org/abs/2010.10824

Hecht, M., Hoffmann, K. B., Cheeseman, B. L., & Sbalzarini, I. F. (2018). Multivariate Newton Interpolation. arXiv. https://arxiv.org/abs/1812.04256

Hecht, M., Cheeseman, B. L., Hoffmann, K. B., & Sbalzarini, I. F. (2017). A Quadratic-Time Algorithm for General Multivariate Polynomial Interpolation. arXiv. https://arxiv.org/abs/1710.10846

Hernandez Acosta, U., Wicaksono, D. C., Thekke Veettil, S. K., Michelfeit, J., & Hecht, M. (2023). Minterpy - multivariate polynomial interpolation (Version 0.3.1-alpha). GitHub: https://github.com/minterpy-project/minterpy  Rodare:  http://doi.org/10.14278/rodare.2062,

Wicaksono, D. C., & Hecht, M. (2023). UQTestFuns: A Python3 Library of Uncertainty Quantification (UQ) Test Functions.GitHub: https://github.com/casus/uqtestfuns Rodare. https://rodare.hzdr.de/record/3354

Veettil, S. K. T., Zavalani, G., Acosta, U. H., Sbalzarini, I. F., & Hecht, M. (2023). Minterpy - levelsets, Python library https://github.com/minterpy-project/minterpy-levelsets

Zavalani, G., and Hecht, M. (2024) Spectral methods and integration on surfaces, Python library  https://github.com/casus/surfgeopy https://github.com/casus/surfpy

Ramanaik CK, Willmann A, Suarez Cardona J-E, Hanfeld P, Hoffmann N, Hecht M. (2023) A Topological Data-Structure Preserving Autoencoder https://github.com/casus/autoencoder-regularisation

Suarez Cardona J.-E., Hofmann P.A. & Hecht M. (2023) Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces https://github.com/casus/PDE-Learning

Suarez Cardona, J.-E., & Hecht, M. (2023). Sobolev PINNs (SC-PINNS) https://github.com/casus/Sobolev-PINNs