Tensor-Train Methods for 3D Linear Elasticity: Block and Global Operator Representations with Solver Performance Analysis

Implicit Tensor-Train Cross Integration of High-Dimensional Nonlinear PDEs via Fiber-Dependency Elimination

A principled fiber-dependency elimination framework is introduced that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities, which produces a closed collocation system while preserving the principal advantages of TT-cross methods.

Alternating Iterative Methods for Solving the Inverse Eigenvalue Problems on the Manifold of Tensors with Fixed TT-Ranks

Projection-based low-rank assembly in IgA

Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS.

Transform-Based Multilinear Algebra via Tensor Decomposition

This work develops TTD- and HTD-based formulations for the T-product and its associated key algebra by operating directly on the factor matrices or tensors of the decompositions of the decompositions.

Plasolver: Physics-Informed Neural Operators for Elastoplasticity

Plasolver, a physics-informed neural operator framework that combines the efficiency of operator learning with the accuracy and robustness of classical numerical solvers, provides an efficient, accurate, and discretization-invariant computational framework for nonlinear, path-dependent elastoplastic problems.

Iterative tensor network transformations for element-wise evaluation of elementary and filtering functions

Tensor networks are powerful formats for compressing large-scale data. However, their application to general data processing has been limited by the difficulty of performing nonlinear operations. Here, we introduce iterative tensor network transformations (ITNTs), a general algorithmic framework for the element-wise evaluation of elementary and nonlinear filtering functions on data encoded as tensor trains (TTs), a class of tensor networks. Our approach operates entirely in the compressed domain, enabling efficient computation on exponentially large datasets while maintaining a controlled computational cost. We demonstrate its power in two key areas: (I) evaluating highly nonlinear elementary and filtering functions on a 3D reactive flow field, enabling high-fidelity reaction rate computation and region filtering, and (II) finding extrema in complex optimization problems, such as solving Max-SAT instances on spaces up to $2^{70}$ configurations. These results establish ITNT as a foundational tool that provides tensor network methods with the capability for general-purpose data science and large-scale optimization.