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We propose a projection-based model order reduction method for the solution of parameter-dependent dynamical systems. The proposed method relies on the construction of time-dependent reduced spaces generated from evaluations of the solution…
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…
Greedy algorithms, particularly the orthogonal greedy algorithm (OGA), have proven effective in training shallow neural networks for fitting functions and solving partial differential equations (PDEs). In this paper, we extend the…
A general class of stochastic Runge-Kutta methods for the weak approximation of It\^o and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an…
We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies…
In this paper, we propose a new adaptive cross algorithm for computing a low tubal rank approximation of third-order tensors, with less memory and lower computational complexity than the truncated tensor SVD (t-SVD). This makes it…
Tensor robust principal component analysis (TRPCA) is a fundamental model in machine learning and computer vision. Recently, tensor train (TT) decomposition has been verified effective to capture the global low-rank correlation for tensor…
We present a novel tensor network algorithm to solve the time-dependent, gray thermal radiation transport equation. The method invokes a tensor train (TT) decomposition for the specific intensity. The efficiency of this approach is dictated…
High-dimensional tensor-valued predictors arise in modern applications, increasingly as learned representations from neural networks. Existing tensor classification methods rely on sparsity or Tucker structures and often lack theoretical…
Low-rank tensor recovery problems have been widely studied in many applications of signal processing and machine learning. Tucker decomposition is known as one of the most popular decompositions in the tensor framework. In recent years,…
Dimensionality reduction is a main step in the learning process which plays an essential role in many applications. The most popular methods in this field like SVD, PCA, and LDA, only can be applied to data with vector format. This means…
Tile low rank representations of dense matrices partition them into blocks of roughly uniform size, where each off-diagonal tile is compressed and stored as its own low rank factorization. They offer an attractive representation for many…
We derive a method to efficiently compute the Green function of on arbitrary Hamiltonians defined on semi-infinite and periodic quasi-one-dimensional lattices. Computing the Green function is the backbone of quantum transport, electronic…
Applications of reduced basis method emulators are increasing in low-energy nuclear physics because they enable fast and accurate sampling of high-fidelity calculations, enabling robust uncertainty quantification. In this paper, we develop,…
This paper proposes a general framework for structure-preserving model reduction of a secondorder network system based on graph clustering. In this approach, vertex dynamics are captured by the transfer functions from inputs to individual…
Semilocal density functionals for the exchange-correlation energy are needed for large electronic systems. The Tao-Perdew-Staroverov-Scuseria (TPSS) meta-generalized gradient approximation (meta-GGA) is semilocal and usefully accurate, but…
This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank…
This paper proposes a novel formulation of the tensor completion problem to impute missing entries of data represented by tensors. The formulation is introduced in terms of tensor train (TT) rank which can effectively capture global…
Tensor train (TT) factorization and corresponding TT rank, which can well express the low-rankness and mode correlations of higher-order tensors, have attracted much attention in recent years. However, TT factorization based methods are…
In this work, we present the tree tensor network Nystr\"om (TTNN), an algorithm that extends recent research on streamable tensor approximation, such as for Tucker and tensor-train formats, to the more general tree tensor network format,…