Related papers: Adaptive Patching for Tensor Train Computations
Quantized tensor trains (QTTs) are a multiscale computational framework that can potentially reduce the computational cost of solving partial differential equations and initial value problems by making low-rank approximations. However, its…
Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often…
Approximating a tensor in the tensor train (TT) format has many important applications in scientific computing. Rounding a TT tensor involves further compressing a tensor that is already in the TT format. This paper proposes new randomized…
We present the first application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the…
We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical…
Correlation functions of quantum systems -- central objects in quantum field theories -- are defined in high-dimensional space-time domains. Their numerical treatment thus suffers from the curse of dimensionality, which hinders the…
The tensor-train (TT) format is a data-sparse tensor representation commonly used in high dimensional function approximations arising from computational and data sciences. Various sequential and parallel TT decomposition algorithms have…
Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial…
We propose a method for approximating the contraction of a tensor network by partitioning the network into a sum of computationally cheaper networks. This method, which we call a partitioned network expansion (PNE), builds upon recent work…
Tensor train (TT) format is a common approach for computationally efficient work with multidimensional arrays, vectors, matrices, and discretized functions in a wide range of applications, including computational mathematics and machine…
In the last two decades, increased need for high-fidelity simulations of the time evolution and propagation of forces in granular media has spurred renewed interest in discrete element method (DEM) modeling of frictional contact. Force…
In the Quantum-Train (QT) framework, mapping quantum state measurements to classical neural network weights is a critical challenge that affects the scalability and efficiency of hybrid quantum-classical models. The traditional QT framework…
Tensor networks have proven to be a valuable tool, for instance, in the classical simulation of (strongly correlated) quantum systems. As the size of the systems increases, contracting larger tensor networks becomes computationally…
We present the Tensor Train Multiplication (TTM) algorithm for the elementwise multiplication of two tensor trains with bond dimension $\chi$. The computational complexity and memory requirements of the TTM algorithm scale as $\chi^3$ and…
We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion partial differential equations (PDEs) in one dimension. Specifically, we show that, independently of the scale of…
Tensor network contraction is a fundamental mathematical operation that generalizes the dot product and matrix multiplication. It finds applications in numerous domains, such as database systems, graph theory, machine learning, probability…
We present a quantum-inspired solver for the one-dimensional Gross-Pitaevskii equation in the Quantics Tensor-Train (QTT) representation. By evolving the system entirely within a low-rank tensor manifold, the method sidesteps the memory and…
Quantized tensor trains (QTTs) have recently emerged as a framework for the numerical discretization of continuous functions, with the potential for widespread applications in numerical analysis. However, the theory of QTT approximation is…
The recent surge in 3D data acquisition has spurred the development of geometric deep learning models for point cloud processing, boosted by the remarkable success of transformers in natural language processing. While point cloud…
We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new…