{
  "abstract": "This paper studies the numerical approximation of evolution equations by nonlinear parametrizations $$u(t)=\\varPhi ({{\\theta }}(t))$$ u ( t ) = Φ ( θ ( t ) ) with time-dependent parameters $${{\\theta }}(t)$$ θ ( t ) , which are to be determined in the computation. The motivation comes from approximations in quantum dynamics by multiple Gaussians and approximations of various dynamical problems by tensor networks and neural networks. In all these cases, the parametrization is typically irregular: the derivative $$\\varPhi '({{\\theta }})$$ Φ ′ ( θ ) can have arbitrarily small singular values and may have varying rank. We derive approximation results for a regularized approach in the time-continuous case as well as in time-discretized cases. For the latter, there is a nontrivial interplay between the regularization parameter and the time stepsize, depending also on the defect size and local bounds of the second derivative of the parametrization map $$\\varPhi $$ Φ . When this is appropriately taken into account, the approach can be successfully applied in irregular situations, even though it runs counter to the basic principle in numerical analysis to avoid solving ill-posed subproblems when aiming for a stable algorithm. The paper also includes two theoretical case studies: regularized parametric steepest descent for optimization and regularized parametric Strang splitting for the time-dependent Schrödinger equation. Numerical experiments with sums of Gaussians for approximating quantum dynamics and with neural networks for approximating the flow map of a system of ordinary differential equations illustrate and complement the theoretical results.",
  "abstract_shown": true,
  "affiliations": [
    "Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040, Wien, Austria",
    "Department of Mathematics, TU München, Boltzmannstraße 3, 85748, Garching bei München, Germany",
    "Mathematisches Institut, University Tübingen, Auf der Morgenstelle 10, 72076, Tubingen, Germany",
    "Seminar für Angewandte Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Michael Feischl",
      "orcid": "0000-0001-7206-1652"
    },
    {
      "affs": [
        1
      ],
      "name": "Caroline Lasser",
      "orcid": "0000-0001-7272-2510"
    },
    {
      "affs": [
        2
      ],
      "name": "Christian Lubich",
      "orcid": "0000-0003-0105-4927"
    },
    {
      "affs": [
        3
      ],
      "name": "Jörg Nick",
      "orcid": "0000-0001-5524-5310"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d45ak5jg6ret7.bib",
  "cited_by": 18,
  "doi": "10.1007/s00211-026-01551-5",
  "fulltext_url": null,
  "id": "d45ak5jg6ret7",
  "issns": [
    "0029-599X",
    "0945-3245"
  ],
  "landing_url": "https://doi.org/10.1007/s00211-026-01551-5",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2612|Numerical Solvers for Ordinary Differential Equations"
    ],
    [
      "openalex",
      "2612|Numerical methods for differential equations"
    ]
  ],
  "openalex": "W4393335799",
  "pdf_url": "https://link.springer.com/content/pdf/10.1007/s00211-026-01551-5.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-15",
  "publisher": "Springer Science+Business Media",
  "ris": "https://research.draconitic.com/paper/d45ak5jg6ret7.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T13:20:59Z",
      "native_id": "W4393335799",
      "source": "openalex",
      "url": "https://doi.org/10.1007/s00211-026-01551-5"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2612|Numerical Solvers for Ordinary Differential Equations",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Regularized dynamical parametric approximation",
  "title_text": "Regularized dynamical parametric approximation",
  "type": "peer-reviewed",
  "updated_at": "2026-11-08T13:20:35Z",
  "url": "https://research.draconitic.com/paper/d45ak5jg6ret7",
  "venue": "Numerische Mathematik",
  "versions": []
}