Related papers: A hybrid Alternating Least Squares -- TT Cross alg…
The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation…
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large…
In this study, we introduce a tensor-train (TT) finite difference WENO method for solving compressible Euler equations. In a step-by-step manner, the tensorization of the governing equations is demonstrated. We also introduce…
Due to the COVID-19 pandemic, there is an increasing demand for portable CT machines worldwide in order to diagnose patients in a variety of settings. This has led to a need for CT image reconstruction algorithms that can produce high…
The family of temporal difference (TD) methods span a spectrum from computationally frugal linear methods like TD({\lambda}) to data efficient least squares methods. Least square methods make the best use of available data directly…
This paper studies the traffic state estimation (TSE) problem using sparse observations from mobile sensors. Most existing TSE methods either rely on well-defined physical traffic flow models or require large amounts of simulation data as…
Various tensor decomposition methods have been proposed for data compression. In real world applications of the tensor decomposition, selecting the tensor shape for the given data poses a challenge and the shape of the tensor may affect the…
There are many practical applications based on the Least Square Error (LSE) or Total Least Square Error (TLSE) methods. Usually the standard least square error is used due to its simplicity, but it is not an optimal solution, as it does not…
We study sparse solutions of optimal control problems governed by PDEs with uncertain coefficients. We propose two formulations, one where the solution is a deterministic control optimizing the mean objective, and a formulation aiming at…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
We propose a guaranteed and fully computable upper bound on the energy norm of the error in low-rank Tensor Train (TT) approximate solutions of (possibly) high dimensional reaction-diffusion problems. The error bound is obtained from…
In this paper, we propose a Tensor Train Neighborhood Preserving Embedding (TTNPE) to embed multi-dimensional tensor data into low dimensional tensor subspace. Novel approaches to solve the optimization problem in TTNPE are proposed. For…
It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with…
Tensor hypercontraction is a method that allows the representation of a high-rank tensor as a product of lower-rank tensors. In this paper, we show how tensor hypercontraction can be applied to both the electron repulsion integral (ERI)…
We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation…
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…
This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism…
This paper presents a novel hybrid algorithm for minimizing the sum of a continuously differentiable loss function and a nonsmooth, possibly nonconvex, sparse regularization function. The proposed method alternates between solving a…
Stochastic Alternating Least Squares (SALS) is a method that approximates the canonical decomposition of averages of sampled random tensors. Its simplicity and efficient memory usage make SALS an ideal tool for decomposing tensors in an…
By combining a certain approximation property in the spatial domain, and weighted $\ell_2$-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in [M. Bachmayr, A. Cohen, R. DeVore and G.…