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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 find that imposing the crossing symmetry in the iteration process considerably extends the range of convergence for solutions of the parquet equations for the Hubbard model. When the crossing symmetry is not imposed, the convergence of…
We use tensor network techniques to obtain high order perturbative diagrammatic expansions for the quantum many-body problem at very high precision. The approach is based on a tensor train parsimonious representation of the sum of all…
In this work we propose an efficient black-box solver for two-dimensional stationary diffusion equations, which is based on a new robust discretization scheme. The idea is to formulate an equation in a certain form without derivatives with…
Numerical methods capable of handling nonequilibrium impurity models are essential for the study of transport problems and the solution of the nonequilibrium dynamical mean field theory (DMFT) equations. In the strong correlation regime,…
One of the challenges in diagrammatic simulations of nonequilibrium phenomena in lattice models is the large memory demand for storing momentum-dependent two-time correlation functions. This problem can be overcome with the recently…
Understanding entanglement remains one of the most intriguing problems in physics. While particle and site entanglement have been studied extensively, the investigation of length or energy scale entanglement, quantifying the information…
The nonequilibrium Green's function formalism provides a versatile and powerful framework for numerical studies of nonequilibrium phenomena in correlated many-body systems. For calculations starting from an equilibrium initial state, a…
Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for…
Quantized tensor trains (QTTs) are a low-rank and multiscale framework that allows for efficient approximation and manipulation of multi-dimensional, high resolution data. One area of active research is their use in numerical simulation of…
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 tensor-train (TT) decomposition expresses a tensor in a data-sparse format used in molecular simulations, high-order correlation functions, and optimization. In this paper, we propose four parallelizable algorithms that compute the TT…
In this paper, we consider the tensor completion problem representing the solution in the tensor train (TT) format. It is assumed that tensor is high-dimensional, and tensor values are generated by an unknown smooth function. The assumption…
Pricing multi-asset options via the Black-Scholes PDE is limited by the curse of dimensionality: classical full-grid solvers scale exponentially in the number of underlyings and are effectively restricted to three assets. Practitioners…
There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that…
The Hadamard product of two tensors in the tensor-train (TT) format is a fundamental operation across various applications, such as TT-based function multiplication for nonlinear differential equations or convolutions. However, conventional…
We present an efficient and robust numerical algorithm for solving the two-dimensional linear elasticity problem that combines the Quantized Tensor Train format and a domain partitioning strategy. This approach makes it possible to solve…
Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of…
We consider sequential state and parameter learning in state-space models with intractable state transition and observation processes. By exploiting low-rank tensor train (TT) decompositions, we propose new sequential learning methods for…
We consider the approximate solution of parametric PDEs using the low-rank Tensor Train (TT) decomposition. Such parametric PDEs arise for example in uncertainty quantification problems in engineering applications. We propose an algorithm…