The field of Applied Analysis brings together many mathematical topics, such differential equations, dynamical systems, variational calculus, functional analysis, geometry, and approximation theory. The Applied Analysis Group focuses on these mathematical disciplines and their application to the real world around us.

Partial differential equations, especially the nonlinear ones, are as different as animals in the zoo. We study them: do they have solutions, are these solutions unique? How do the solutions behave? How do they depend on parameters? How should we calculate solutions numerically? How do all these properties relate to the real-world situations that generated the equations in the first place? Tools such as functional analysis and variational calculus allow us to create order in the zoo.

EXPLORING THE ZOO OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Connecting PDEs with stochastics

Since the invention of Wasserstein gradient flows, exciting new connections have arisen between optimal transport, (generalized) gradient flows and large deviations. Optimal transport provides tools for analyzing PDEs, large deviations give us the gradient-flow structure underlying many PDEs, and the connection between PDEs and stochastic processes ties all these concepts together.

Dynamical-Variational Transport Costs

We develop a theory of dynamical-variational transport costs (DVTs), a class of large-deviation inspired functionals that provide a variational generalisation of a zoo of existing transport distances. DVTs generate non-homogeneous generalisations of length spaces that will be used to extend metric-space techniques to general length spaces, thereby allowing a variational framework for generalised gradient flows to be rigorously investigated, and the multiscale analysis of such evolutions used in the development of numerical schemes.

  1. Chaudhuri N, Choi YP, Tse O, Zatorska E. Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming. arXiv.org. 2024 Feb 11. doi: 10.48550/arXiv.2402.07130
  2. Hoeksema J, Holding T, Maurelli M, Tse O. Large deviations for singularly interacting diffusions. Annales de l'institut Henri Poincare (B) Probability and Statistics. 2024 Feb;60(1):492-548. doi: 10.1214/22-AIHP1319
  3. Koop SM, Peletier MA, Menkovski V, Portegies JW. Neural Langevin Dynamics: Towards Interpretable Neural Stochastic Differential Equations. In NLDL. 2024. p. 130-137
  4. de Keijzer R, Visser L, Tse O, Kokkelmans S. Qubit fidelity under stochastic Schrödinger equations driven by colored noise. arXiv.org. 2024 Jan 22. doi: 10.48550/arXiv.2401.11758
  5. Poot MM, Portegies J, Kostic D, Oomen T. Rational Basis Functions in Iterative Learning Control for Multivariable Systems. In 2023 62nd IEEE Conference on Decision and Control, CDC 2023. Institute of Electrical and Electronics Engineers. 2024. p. 4644-4649. 10383932 doi: 10.1109/CDC49753.2023.10383932
  6. Menkovski V, Portegies JW, Ravelonanosy M. Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold. Applied and Computational Harmonic Analysis. 2024 Jul;71:101642. doi: 10.1016/j.acha.2024.101642
  7. Chen S, Li Q, Tse O, Wright SJ. Accelerating optimization over the space of probability measures. arXiv.org. 2023 Oct 9, p. 1-35. doi: 10.48550/arXiv.2310.04006
  8. Bozkuş Z, Dinçer AE, Tijsseling AS, van de Ven AAF. Air gun with water bullet. In Jones SEL, editor, Proc. of the 14th Int. Conf. on Pressure Surges. Technische Universiteit Eindhoven. 2023. p. 295-309
  9. van Hee KM, van Berkel CH, de Graaf J. An Appropriate Probability Model for the Bell Experiment. arXiv. 2023 Feb 13;2023:2302.05174. doi: 10.48550/arXiv.2302.05174
  10. Peletier MA, Schlichting A. Cosh gradient systems and tilting. Nonlinear Analysis : Theory, Methods and Applications. 2023 Jun;231:113094. doi: 10.1016/j.na.2022.113094

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