{
  "abstract": "We develop a tensor reduced-order modeling (TROM) framework for optimization-based inverse problems governed by parameter-dependent dynamical systems. The approach approximates the parameter-to-observation map directly in tensor-train (TT) format using TT-SVD or TT-Cross and integrates it into a regularized nonlinear least-squares formulation. Beyond accelerating forward evaluations, the low-rank tensor structure reformulates the inverse problem in reduced coordinates, assembles Gauss--Newton quantities without forming the full observation-space Jacobian, and minimizes the TROM objective over the discrete parameter grid. This tensor optimization provides either a stand-alone approximate solution or a data-informed initialization for a subsequent Gauss--Newton solve. We study an inverse heat-transfer problem in a heterogeneous medium, where the parameters describe the locations and radii of low-conductivity inclusions, and a FitzHugh--Nagumo parameter-estimation problem with a highly nonconvex landscape. Numerical experiments assess reduced-order model error, measurement noise, regularization, initialization, spatial discretization, and increasing parameter dimension. The results show that TROM reproduces full-order inversion at substantially reduced online cost. They also demonstrate that reduced-coordinate inversion, tensor-based optimization, and appropriate regularization improve robustness in higher-dimensional, noisy, and strongly nonconvex regimes. For the continuous TROM inverse problem, we develop an error-to-inversion analysis. Under local strong convexity of the regularized FOM objective and parametric smoothness of the FOM observation map, the error between the parameters recovered with the full-order model and TROM is bounded by controlled uniform errors in the surrogate map and its Jacobian, together with local FOM and curvature quantities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.12613",
  "author_total": 3,
  "authors": [
    {
      "name": "Sahidul Islam"
    },
    {
      "name": "Andreas Mang"
    },
    {
      "name": "Maxim Olshanskii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vpq9851b2ad0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.12613",
  "fulltext_url": null,
  "id": "2vpq9851b2ad0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.12613",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7168337926",
  "pdf_url": "https://arxiv.org/pdf/2607.12613",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vpq9851b2ad0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.12613",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.12613"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Tensor-Based Reduced-Order Modeling for Optimization-Based Inverse Problems",
  "title_text": "Tensor-Based Reduced-Order Modeling for Optimization-Based Inverse Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2vpq9851b2ad0",
  "venue": "arXiv",
  "versions": []
}