A preprint reports a hybrid structural solver that combines a reduced-order model, or ROM, with a high-dimensional model, or HDM, representing the full finite-element calculation. In a rectangular test, one ROM construction matched the HDM exactly. But the result depended on the snapshot basis, the stored solutions used to build the ROM: a smaller CG-derived basis had maximum relative displacement errors of 2.24% in the HDM region and 10.30% in the ROM region, while a plant-model plane basis reached 66.69% and 55.28% in the corresponding regions.
A solver assembled from different model pieces
The design splits a structure into non-overlapping subdomains and lets ROM and HDM regions work together through an iterative domain-decomposition method. The HDM part uses a non-substructuring domain-decomposition method, while ROM regions use a substructuring-based method with ROM-approximated static condensation. Conjugate gradient, or CG, supplies the iterative solve. The study also derives a diagonal preconditioner and reports good convergence with fewer required iterations when approximate diagonal scaling is used.
The study varied how the ROM basis was generated. It tested three snapshot routes: unit interface displacements with other degrees of freedom fixed, planar interface tilts, and intermediate full-HDM solutions from CG. In the rectangular model, the ROM interface had 16 nodes. Unit snapshots produced 48 snapshots, while CG-based collection produced 53. Energy thresholds of 0.9999 and 0.99999 yielded 20 and 24 reduced-order basis vectors, respectively.
The rectangular test was a warning about basis design
The first model was a 208-node, 108-element rectangular solid. Its bottom was fixed and its top received a horizontal prescribed displacement. The bottom cubic region was one ROM subdomain, while the HDM remainder was divided into six. Here, HDM-ROM_point exactly matched the full HDM. HDM-ROM_cg_V24 was nearly identical. HDM-ROM_plane was approximate, and HDM-ROM_cg_V20 produced maximum relative errors of 2.24% in the HDM region and 10.30% in the ROM region.
The load test made the limitation clearer. CG snapshots gathered only under an x-direction load did not reproduce the HDM solution under a y-direction load. Point snapshots reproduced the HDM solution across various loads, except when the load was applied inside the ROM region. Plane snapshots were approximate across various loads. The test therefore did not support a blanket claim of exact reproduction across different loading conditions.
Plant models exposed a sharper accuracy tradeoff
The nuclear examples used two configurations. SMR4 contained four SMRs and was divided into eight subdomains. SMR1 contained one SMR and was divided into five. Both configurations included four HDM subdomains.
The accuracy comparison was stark. HDM-ROM_point_100% exactly matched the full HDM. With the point_80% model, maximum displacement errors were 0.25% in HDM regions and 0.63% in ROM regions. The plane model had much larger maximum errors, 66.69% in HDM regions and 55.28% in ROM regions.
GPU timing changed the picture
On CPU, iteration counts and total times did not move together. With diagonal scaling in SMR4, the full HDM took 5,426 iterations and 76.7 seconds. Point_100% used 1,452 iterations but took 371.3 seconds, while point_80% used 1,451 iterations and took 191.3 seconds. The plane model used 1,435 iterations and took 4.6 seconds. In SMR1, the full HDM took 3,481 iterations and 48.5 seconds; point_100%, point_80% and plane took 404.1, 206.3 and 4.4 seconds, respectively.
The reported GPU times for SMR1 were much shorter for the point-based models. Point_100% fell from 404.1 seconds on CPU to 13.0 seconds on GPU, with per-iteration time dropping from 0.2079 to 0.0067 seconds. Point_80% fell from 206.3 to 8.4 seconds, while its per-iteration time dropped from 0.1063 to 0.0043 seconds. For the plane model, the change was smaller, from 4.4 to 3.2 seconds overall and from 0.0023 to 0.0017 seconds per iteration. Dense ROM operations and recovery of the HDM solution were implemented with cuSOLVER and cuBLAS on GPUs, and with LAPACKE in Intel oneAPI MKL on CPUs.
A promising numerical result with a narrow reach
The comparison covers one rectangular solid and the SMR1 and SMR4 nuclear-plant configurations, under the reported snapshot and load choices. Its clearest warning is that a CG basis collected for one load direction did not carry over to another, while the plane basis remained approximate across various loads. The work demonstrates a way to combine ROM and HDM subdomains in an iterative solver, but it does not establish exact reproduction for arbitrary loads or a universal runtime advantage.
Paper data and sources
Original title: Combined High-Dimensional and Reduced-Order Modeling Based on an Iterative Domain Decomposition Method
Authors: Taiji Saito, Tomoshi Miyamura, Yasunori Yusa
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text