arXiv subjects

Behzad Ghahremani

Publications and source records attributed to Behzad Ghahremani.

3 recordsLinked to original sources

arXiv2026

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

Tensor-train (TT) representations have emerged as an effective framework for mitigating the curse of dimensionality in the numerical solution of high-dimensional tensor differential equations. Among existing approaches, TT-cross methods are particularly attractive because they require only pointwise evaluations of the governing equations, naturally accommodate arbitrary nonlinearities, and avoid tangent-space projections and the numerical difficulties associated with nearly singular low-rank factors. However, existing TT-cross rank-truncation methods have been restricted to explicit time integration. Extending TT-cross methods to implicit schemes presents an obstacle: the collocation equations associated with the cross-selected fibers depend on neighboring fibers that are not part of the unknown set. Consequently, the resulting nonlinear system is not closed, preventing the direct application of standard implicit solvers. In this work, we introduce a principled fiber-dependency elimination framework that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities. The resulting formulation produces a closed collocation system while preserving the principal advantages of TT-cross methods. The proposed framework applies to both linear and nonlinear high-dimensional partial differential equations and is naturally combined with Newton iterations and rank adaptivity. Numerical experiments demonstrate rapid convergence of the dependency-elimination iterations, preservation of the temporal accuracy of implicit multistep schemes, and efficient implicit integration of high-dimensional nonlinear problems with full-order discretizations containing up to $10^{55}$ degrees of freedom.

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arXiv2024

Cross Interpolation for Solving High-Dimensional Dynamical Systems on Low-Rank Tucker and Tensor Train Manifolds

We present a novel tensor interpolation algorithm for the time integration of nonlinear tensor differential equations (TDEs) on the tensor train and Tucker tensor low-rank manifolds, which are the building blocks of many tensor network decompositions. This paper builds upon our previous work (Donello et al., Proceedings of the Royal Society A, Vol. 479, 2023) on solving nonlinear matrix differential equations on low-rank matrix manifolds using CUR decompositions. The methodology we present offers multiple advantages: (i) It delivers near-optimal computational savings both in terms of memory and floating-point operations by leveraging cross algorithms based on the discrete empirical interpolation method to strategically sample sparse entries of the time-discrete TDEs to advance the solution in low-rank form. (ii) Numerical demonstrations show that the time integration is robust in the presence of small singular values. (iii) High-order explicit Runge-Kutta time integration schemes are developed. (iv) The algorithm is easy to implement, as it requires the evaluation of the full-order model at strategically selected entries and does not use tangent space projections, whose efficient implementation is intrusive. We demonstrate the efficiency of the presented algorithm for several test cases, including a nonlinear 100-dimensional TDE for the evolution of a tensor of size $70^{100} \approx 3.2 \times 10^{184}$ and a stochastic advection-diffusion-reaction equation with a tensor of size $4.7 \times 10^9$.

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arXiv2024

A DEIM Tucker Tensor Cross Algorithm and its Application to Dynamical Low-Rank Approximation

We introduce a Tucker tensor cross approximation method that constructs a low-rank representation of a $d$-dimensional tensor by sparsely sampling its fibers. These fibers are selected using the discrete empirical interpolation method (DEIM). Our proposed algorithm is referred to as DEIM fiber sampling (DEIM-FS). For a rank-$r$ approximation of an $\mathcal{O}(N^d)$ tensor, DEIM-FS requires access to only $dNr^{d-1}$ tensor entries, a requirement that scales linearly with the tensor size along each mode. We demonstrate that DEIM-FS achieves an approximation accuracy close to the Tucker-tensor approximation obtained via higher-order singular value decomposition at a significantly reduced cost. We also present DEIM-FS (iterative) that does not require access to singular vectors of the target tensor unfolding and can be viewed as a black-box Tucker tensor algorithm. We employ DEIM-FS to reduce the computational cost associated with solving nonlinear tensor differential equations (TDEs) using dynamical low-rank approximation (DLRA). The computational cost of solving DLRA equations can become prohibitive when the exact rank of the right-hand side tensor is large. This issue arises in many TDEs, especially in cases involving non-polynomial nonlinearities, where the right-hand side tensor has full rank. This necessitates the storage and computation of tensors of size $\mathcal{O}(N^d)$. We show that DEIM-FS results in significant computational savings for DLRA by constructing a low-rank Tucker approximation of the right-hand side tensor on the fly. Another advantage of using DEIM-FS is to significantly simplify the implementation of DLRA equations, irrespective of the type of TDEs. We demonstrate the efficiency of the algorithm through several examples including solving high-dimensional partial differential equations.

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