1. Data assimilation - the collaborative research centre SFB1294
  2. Research
  3. Research Area A
  4. A07

A07 – Data-based model order reduction for stochastic dynamics

This project started with the second funding period in July 2021.

Model order reduction (MOR) constructs computationally efficient reduced-order models (ROMs)
of large-scale dynamical systems by approximating high-dimensional with low-dimensional states.
Such reduced models are beneficial in optimisation, uncertainty quantification, inverse problems
and control, where models are evaluated many times for different inputs.
A central task in this setting is to identify solution manifolds and approximate them using
low-dimensional linear subspaces. By projecting the system dynamics onto such a subspace, one
can define a reduced state variable that closely approximates the behaviour of the full system.
Numerous MOR techniques have been developed to tackle such problems in the deterministic
setting.
The need for model reduction becomes even more pronounced in stochastic contexts, where
the computational cost of system evaluations can become prohibitive. In particular, Markov
Chain Monte Carlo (MCMC) simulations, which require repeated application of possibly high-
dimensional systems, are too costly.
Most MOR methods require full knowledge of both structure and coefficients of the underlying
system of equations. Often, this information is not available. Therefore, approaches based on data
are practically more useful.
As a concrete example, in pharmacology, it is crucial to understand treatment response vari-
ability, which models based on stochastic differential equations (SDEs) can predict based on
patient information (covariates) and drug concentration or biomarker measurements. However,
estimating the coefficients in a full SDE model is challenging and can even be computationally
infeasible. Data-based MOR approaches can be used to overcome this issue.
The goal of this project is to develop and analyse projection- and data-based MOR methods for
time-dependent linear and nonlinear stochastic systems of the form:
dx(t) = [ f (x(t)) + g(x(t))u(t)]dt + h(x(t))dB(t),
y(t) = c(x(t)), (1)
with u an L2-control, B a Brownian motion, x a high-dimensional state vector and y the quantity
of interest. Evaluating (1) for many controls u and samples of B is computational expensive,
sometimes infeasible. It is even more challenging to solve optimal control or stopping problems
with (1). The goal of this project is to learn the dominant subspace of (1) from (noisy) data, to
compute an associated projection onto a low-dimensional subspace resulting in a ROM that is
computationally cheap, and to apply them to pharmacological models.

  • König, J., Cheng Lie, H., (2026). Posterior error bounds for prior-driven balancing in linear Gaussian inverse problems, arXiv 2601.03971

  • Josie König, Elizabeth Qian, Melina A. Freitag (2025). Dimension and model reduction approaches for linear Bayesian inverse problems with rank-deficient prior covariances, arXiv 2506.23892

  • Carere, G. and Lie, H. C. (2024). Optimal low-rank approximations of posteriors for linear Gaussian inverse problems on Hilbert spaces, arXiv 2411.01112

  • Siobhán Correnty, Melina A. Freitag, Kirk M. Soodhalter (2023). Chebyshev HOPGD with sparse grid sampling for parameterized linear systems. arXiv:2309.14178

  • Carere, G., Cheng Lie, H. (2027). Optimal low-rank posterior mean and distribution approximation in linear Gaussian inverse problems on Hilbert spaces. Inverse Problems and Imaging, Volume 27. https://doi.org/10.3934/ipi.2026031)

  • Altmann, R., Cortes Garcia, I., Paakkunainen, E., Schulze, P., Schöps, S. (2026). Energy-based modeling for field–circuit coupling. Applied Mathematical Modelling, Volume 155. https://doi.org/10.1016/j.apm.2025.116688

  • Kinon, P. L., Morandin, R., Schulze, P. (2026). Discrete gradient methods for port-Hamiltonian differential-algebraic equations. Applied Numerical Mathematics, Volume 223. https://doi.org/10.1016/j.apnum.2025.12.006

  • Carere, G., Cheng Lie, H. (2026). Generalized Rank-Constrained Approximations of Hilbert–Schmidt Operators on Separable Hilbert Spaces and Applications. Numerical Functional Analysis and Optimization, Volume 47, Issue 7–10. doi.org/10.1080/01630563.2025.2598467

  • Mach, T., & Freitag, M. A. (2025). Solving the parametric eigenvalue problem by Taylor series and Chebyshev expansion. SIAM Journal on Matrix Analysis and Applications. https://doi.org/10.1137/23M1551961

  • Koenig, J., Pfeffer, M. & Stoll, M. (2025): Efficient training of Gaussian processes with tensor product structure. Comput Optim Appl. https://doi.org/10.1007/s10589-025-00707-7

     

  • Freitag, M.A., König, J. and Qian, E. (2024): Inference-Oriented Balanced Truncation for Quadratic Dynamical Systems: Formulation for Bayesian Smoothing and Model Stability Analysis. Proc. Appl. Math. Mech., 24: e202400051. https://doi.org/10.1002/pamm.202400051

  • Quinn, P. D., Landmann, M. S., Davis, T., Freitag, M. A., Gazzola, S., and Dolgov, S. (2024): Optimal Sparse Energy Sampling for X-ray Spectro-Microscopy: Reducing the X-ray Dose and Experiment Time Using Model Order Reduction. Chem. Biomed. Imaging 2024. doi: 10.1021/cbmi.3c00116

  • Kaya, A. and Freitag, M. A. (2024). Low-rank solutions to the stochastic Helmholtz equation. Journal of Computational and Applied Mathematics. doi: 10.1016/j.cam.2024.115925

  • Freitag, M.A., Nicolaus, J.M., and Redmann, M. (2023). Model order reduction methods applied to neural network training. Proceedings in Applied Mathematics and Mechanics, e202300078. doi: 10.1002/pamm.202300078

  • Freitag, M.A., Kriz, P., Mach, T, and Nicolaus, J.M. (2023). Can one hear the depth of the water? Proceedings in Applied Mathematics and Mechanics, e202300122. doi: 10.1002/pamm.202300122

  • König, J. and Freitag, M.A. (2023). Time-Limited Balanced Truncation for Data Assimilation Problems. Journal of Scientific Computing, Volume 97, Number 47. doi: 10.1007/s10915-023-02358-4

  • König, J. and Freitag, M.A. (2023). Time-limited Balanced Truncation within Incremental Four-Dimensional Variational Data Assimilation. Proceedings in Applied Mathematics and Mechanics, e202300019. doi: 10.1002/pamm.202300019

  • Hijazi, S., Freitag, M. A., and Landwehr, N. (2023). POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier-Stokes equations. Adv. Model. Simul. Eng. Sci. doi: 10.1186/s40323-023-00242-2

  • Ayanbayev, B., Klebanov, I., Lie, H.C., and Sullivan, T.J. (2021). Gamma-convergence of Onsager–Machlup functionals: II. convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces. Inverse Problems, Volume 38, Number 2. doi:10.1088/1361-6420/ac3f82.

  • Redmann, M. and Freitag, M.A. (2021). Optimization based model order reduction for stochastic systems. Appl. Math. Comput., 398. doi: 10.1016/j.amc.2020.125783

  • Freitag, M.A. and Reich, S. (2022). Datenassimilation: Die nahtlose Verschmelzung von Daten und Modellen. Mitteilungen der Deutschen Mathematiker-VereinigungVerlag, De GruyterSeiten, 108‒112, Band 30. doi: 10.1515/dmvm-2022-0037

  • Lie, H. C. and Stahn, M. and Sullivan, T.J. (2022). Randomised one-step time integration methods for deterministic operator differential equations. Calcolo, Volume 59, Number 13, ArXiv 2103.16506, doi: 10.1007/s10092-022-00457-6.

  • Alqahtani, A., Mach, T., and Reichel, L. (2023). Solution of Ill-posed Problems with Chebfun. Numerical Algorithms (2023). doi:10.1007/s11075-022-01390-z​​​​​​​, arXiv 2007.16137

  • Mach, T., Reichel, L., and Van Barel, M. (2023). Adaptive cross approximation for Tikhonov regularization in general form. Numerical Algorithms. doi:10.1007/s11075-022-01395-8, arXiv 2204.05740

  • Birzhan Ayanbayev, Ilja Klebanov, Han Cheng Lie and T J Sullivan (2021). Gamma-convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems. Inverse Problems, Volume 38, Number 2, doi:10.1088/1361-6420/ac3f81.