arXiv subjects

Kim Ø. Rasmussen

Publications and source records attributed to Kim Ø. Rasmussen.

18 recordsLinked to original sources

arXiv2026

Generated Query Expansion Still Helps Strong Sparse Retrieval: A Controlled Study with SPLADE-v3

Scientific queries are often brief, while relevant papers use specialized vocabulary. Generated query expansion can bridge this mismatch, but earlier work suggests that its value shrinks as the underlying retriever becomes stronger. We test the four generated formats of term lists, a pseudo-document, multiple pseudo-references, and corpus-steered text all together with SPLADE-v3 on NFCorpus, TREC-COVID, and SciDocs. Every condition searches the same frozen document index and follows the same query-side integration rule and 256-dimension budget, isolating the effect of the added content. All twelve method-collection comparisons improve aggregate nDCG@10, with best relative gains of 4.81%, 8.92%, and 9.47%. Eleven remain significant after Holm correction. The gain persists in 103 of 114 interpolation settings, including every setting that assigns at least 30% of the mixture weight to the original query. Shuffled-text and non-contextual lexical-bag controls also remain above baseline in all 24 aggregate comparisons, showing that the added vocabulary carries most of the benefit. A corpus-induced typed concept graph, by contrast, produces no consistent gain, and its relation, depth, validation, random, and gating controls do not rescue it. Generated vocabulary can therefore complement a strong learned sparse retriever, provided that the original query remains strongly represented.

cs.IR↗

arXiv2026

Tensor-Train Methods for 3D Linear Elasticity: Block and Global Operator Representations with Solver Performance Analysis

This work develops tensor-train (TT) formulations for solving large-scale three-dimensional linear elasticity problems discretized by isogeometric analysis. By exploiting the tensor-product structure of the basis functions and the low-rank structure of geometry-dependent coefficient fields, the stiffness operator, mass operator, force vector, and displacement solution are represented in TT format. Two solution strategies are investigated: a block-operator formulation, in which the coupled elasticity operator is stored as separated TT blocks, and a single-operator formulation, in which the full coupled system is stored as one monolithic TT operator. A matrix-free three-field TT conjugate-gradient solver is introduced for the block formulation, while AMEn is used for the single-operator formulation. Numerical examples demonstrate substantial compression of both operators and solutions compared with conventional sparse full-grid representations, showing that TT-based formulations provide an efficient and scalable approach for large-scale three-dimensional elasticity simulations.

math.NA↗

arXiv2026

A Tensor-Train Discontinuous Galerkin Method for the Vlasov-Maxwell System

We present a tensor-train discontinuous Galerkin (TT-DG) formulation for the Vlasov--Maxwell system that combines a modal DG discretization with low-rank tensor representations of the phase-space solution and discrete operators. The formulation exploits the tensor-product structure of the DG discretization to perform quadrature, differentiation, nonlinear upwind flux evaluation, and time integration directly in compressed form. The method is evaluated on several standard 1D2V Vlasov--Maxwell benchmark problems, including the streaming Weibel instability, weak Landau damping, and two-stream instability problems. Across these problems, the TT formulation reproduces the accuracy and conservation behavior of the underlying full-grid DG discretization while substantially reducing memory usage and runtime. For weakly nonlinear problems, compression ratios exceeding $10^4$ are obtained together with significant speedups relative to the full-grid solver. For the strongly nonlinear two-stream instability problem, the TT formulation remains effective despite reduced compressibility caused by fine-scale phase-space filamentation. These results demonstrate that tensor-train representations provide an effective approach for reducing the computational cost of deterministic DG-based kinetic plasma simulations while retaining the favorable numerical properties of the underlying discretization.

math.NA↗

arXiv2025

Space-Time Spectral Collocation Tensor-Network Approach for Maxwell's Equations

In this work, we develop a space--time Chebyshev spectral collocation method for three-dimensional Maxwell's equations and combine it with tensor-network techniques in Tensor-Train (TT) format. Under constant material parameters, the Maxwell system is reduced to a vector wave equation for the electric field, which we discretize globally in space and time using a staggered spectral collocation scheme. The staggered polynomial spaces are designed so that the discrete curl and divergence operators preserve the divergence-free constraint on the magnetic field. The magnetic field is then recovered in a space--time post-processing step via a discrete version of Faraday's law. The global space--time formulation yields a large but highly structured linear system, which we approximate in low-rank TT-format directly from the operator and data, without assuming that the forcing is separable in space and time. We derive condition-number bounds for the resulting operator and prove spectral convergence estimates for both the electric and magnetic fields. Numerical experiments for three-dimensional electromagnetic test problems confirm the theoretical convergence rates and show that the TT-based solver maintains accuracy with approximately linear complexity in the number of grid points in space and time.

math.NA↗

arXiv2025

A Tensor Train-Based Isogeometric Solver for Large-Scale 3D Poisson Problems on Complex Geometries

We introduce a three-dimensional (3D) fully tensor train (TT)-assembled isogeometric analysis (IGA) framework, TT-IGA, for solving partial differential equations (PDEs) on complex geometries. Our method reformulates IGA discrete operators into TT format, enabling efficient compression and computation while retaining geometric flexibility and accuracy. Unlike previous low-rank approaches that typically rely on structured domains, our framework accommodates general 3D geometries through low-rank TT representations of both the geometry mapping and the PDE discretization. We demonstrate the effectiveness of the proposed TT-IGA framework on the 3D Poisson equation, achieving substantial reductions in memory usage and computational cost without compromising solution quality.

math.NA↗

arXiv2025

Breaking the Curse of Dimensionality: Solving Configurational Integrals for Crystalline Solids by Tensor Networks

Accurately evaluating configurational integrals for dense solids remains a central and difficult challenge in the statistical mechanics of condensed systems. Here, we present a novel tensor network approach that reformulates the high-dimensional configurational integral for identical-particle crystals into a sequence of computationally efficient summations. We represent the integrand as a high-dimensional tensor and apply tensor-train (TT) decomposition together with a custom TT-cross interpolation scheme. This approach avoids the need to explicitly construct the full tensor, which would otherwise be computationally intractable. We introduce tailored rank-1 and rank-2 schemes optimized for sharply peaked Boltzmann probability densities, typical in crystalline solids. When applied to the calculation of internal energy and pressure-temperature curves for crystalline copper (Cu) and argon (Ar), as well as the alpha-to-beta phase transition in tin (Sn), our method accurately reproduces molecular dynamics simulation results using tight-binding, machine learning (HIP-NN), and MEAM potentials, all within seconds of computation time.

cond-mat.stat-mech↗

arXiv2024

Space-Time Spectral Element Tensor Network Approach for Time Dependent Convection Diffusion Reaction Equation with Variable Coefficients

In this paper, we present a new space-time Petrov-Galerkin-like method. This method utilizes a mixed formulation of Tensor Train (TT) and Quantized Tensor Train (QTT), designed for the spectral element discretization (Q1-SEM) of the time-dependent convection-diffusion-reaction (CDR) equation. We reformulate the assembly process of the spectral element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the spectral element operators. Recognizing the banded structure inherent in the spectral element framework's discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.

math.NA↗

arXiv2024

Domain-Specific Retrieval-Augmented Generation Using Vector Stores, Knowledge Graphs, and Tensor Factorization

Large Language Models (LLMs) are pre-trained on large-scale corpora and excel in numerous general natural language processing (NLP) tasks, such as question answering (QA). Despite their advanced language capabilities, when it comes to domain-specific and knowledge-intensive tasks, LLMs suffer from hallucinations, knowledge cut-offs, and lack of knowledge attributions. Additionally, fine tuning LLMs' intrinsic knowledge to highly specific domains is an expensive and time consuming process. The retrieval-augmented generation (RAG) process has recently emerged as a method capable of optimization of LLM responses, by referencing them to a predetermined ontology. It was shown that using a Knowledge Graph (KG) ontology for RAG improves the QA accuracy, by taking into account relevant sub-graphs that preserve the information in a structured manner. In this paper, we introduce SMART-SLIC, a highly domain-specific LLM framework, that integrates RAG with KG and a vector store (VS) that store factual domain specific information. Importantly, to avoid hallucinations in the KG, we build these highly domain-specific KGs and VSs without the use of LLMs, but via NLP, data mining, and nonnegative tensor factorization with automatic model selection. Pairing our RAG with a domain-specific: (i) KG (containing structured information), and (ii) VS (containing unstructured information) enables the development of domain-specific chat-bots that attribute the source of information, mitigate hallucinations, lessen the need for fine-tuning, and excel in highly domain-specific question answering tasks. We pair SMART-SLIC with chain-of-thought prompting agents. The framework is designed to be generalizable to adapt to any specific or specialized domain. In this paper, we demonstrate the question answering capabilities of our framework on a corpus of scientific publications on malware analysis and anomaly detection.

cs.CL↗

arXiv2024

Tensor Network Space-Time Spectral Collocation Method for Solving the Nonlinear Convection Diffusion Equation

Spectral methods provide highly accurate numerical solutions for partial differential equations, exhibiting exponential convergence with the number of spectral nodes. Traditionally, in addressing time-dependent nonlinear problems, attention has been on low-order finite difference schemes for time discretization and spectral element schemes for spatial variables. However, our recent developments have resulted in the application of spectral methods to both space and time variables, preserving spectral convergence in both domains. Leveraging Tensor Train techniques, our approach tackles the curse of dimensionality inherent in space-time methods. Here, we extend this methodology to the nonlinear time-dependent convection-diffusion equation. Our discretization scheme exhibits a low-rank structure, facilitating translation to tensor-train (TT) format. Nevertheless, controlling the TT-rank across Newton's iterations, needed to deal with the nonlinearity, poses a challenge, leading us to devise the "Step Truncation TT-Newton" method. We demonstrate the exponential convergence of our methods through various benchmark examples. Importantly, our scheme offers significantly reduced memory requirement compared to the full-grid scheme.

math.NA↗

arXiv2024

Tensor Network Space-Time Spectral Collocation Method for Time Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of Partial Differential Equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultrafast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for solution of the time dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time dependent PDEs often use finite difference for time, and a spectral scheme for the spatial dimensions, which leads to slow linear convergence. Spectral collocation space-time methods show exponential convergence, however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to very high compression of linear operators from terabytes to kilobytes in TT-format, and tens of thousands times speedup when compared to full grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

math.NA↗

arXiv2023

Tensor Networks for Solving Realistic Time-independent Boltzmann Neutron Transport Equation

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 Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing, (ii) multigroup-in-energy, and (iii) discrete ordinate collocation leads to huge generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores and solving the tensorized generalized eigenvalue problem in a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying it to two realistic examples of 3D neutron transport problems, currently solved by the PARallel TIme-dependent SN (PARTISN) solver. We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to a yottabyte compression of the memory storage, and more than 7500 times speedup with a discrepancy of less than 1e-5 when compared to the PARTISN solution.

math.NA↗

arXiv2022

Patterns and Stability of Coupled Multi-Stable Nonlinear Oscillators

Nonlinear isolated and coupled oscillators are extensively studied as prototypical nonlinear dynamics models. Much attention has been devoted to oscillator synchronization or the lack thereof. Here, we study the synchronization and stability of coupled driven-damped Helmholtz-Duffing oscillators in bi-stability regimes. We find that despite the fact that the system parameters and the driving force are identical, the stability of the two states to spatially non-uniform perturbations is very different. Moreover, the final stable states, resulting from these spatial perturbations, are not solely dictated by the wavelength of the perturbing mode and take different spatial configurations in terms of the coupled oscillator phases.

nlin.PS↗

arXiv2020

Identification of Anomalous Diffusion Sources by Unsupervised Learning

Fractional Brownian motion (fBm) is a ubiquitous diffusion process in which the memory effects of the stochastic transport result in the mean squared particle displacement following a power law, $\langle {Δr}^2 \rangle \sim t^α$, where the diffusion exponent $α$ characterizes whether the transport is subdiffusive, ($α<1$), diffusive ($α= 1$), or superdiffusive, ($α>1$). Due to the abundance of fBm processes in nature, significant efforts have been devoted to the identification and characterization of fBm sources in various phenomena. In practice, the identification of the fBm sources often relies on solving a complex and ill-posed inverse problem based on limited observed data. In the general case, the detected signals are formed by an unknown number of release sources, located at different locations and with different strengths, that act simultaneously. This means that the observed data is composed of mixtures of releases from an unknown number of sources, which makes the traditional inverse modeling approaches unreliable. Here, we report an unsupervised learning method, based on Nonnegative Matrix Factorization, that enables the identification of the unknown number of release sources as well the anomalous diffusion characteristics based on limited observed data and the general form of the corresponding fBm Green's function. We show that our method performs accurately for different types of sources and configurations with a predetermined number of sources with specific characteristics and introduced noise.

physics.app-ph↗

arXiv2019

Dynamics and escape of active particles in a harmonic trap

The dynamics of active particles is of interest at many levels and is the focus of theoretical and experimental research. There have been many attempts to describe the dynamics of particles affected by random active forces in terms of an effective temperature. This kind of description is tempting due to the similarities (or lack thereof) with systems in or near thermal equilibrium. However, the generality and validity of the effective temperature is not yet fully understood. Here, we studied the dynamics of trapped particles subjected to both thermal and active forces. The particles were not overdamped. Expressions for the effective temperature due to the potential and kinetic energies were derived, and they differ from each other. A third possible effective temperature can be derived from the escape time of the particle from the trap, using a Kramers-like expression for the mean escape time. We found that over a large fraction of the parameter space, the potential energy effective temperature is in agreement with the escape temperature, while the kinetic effective temperature only agrees with the former two in the overdamped limit. Moreover, we show that the specific implementation of the random active force, and not only its first two moments and the two point auto-correlation function, affects the escape time distribution.

cond-mat.stat-mech↗

arXiv2013

Statistical Mechanics of a Discrete Schrödinger Equation with Saturable Nonlinearity

We study the statistical mechanics of the one-dimensional discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearity. Our study represents an extension of earlier work [Phys. Rev. Lett. {\bf 84}, 3740 (2000)] regarding the statistical mechanics of the one-dimensional DNLS equation with a cubic nonlinearity. As in this earlier study we identify the spontaneous creation of localized excitations with a discontinuity in the partition function. The fact that this phenomenon is retained in the saturable DNLS is non-trivial, since in contrast to the cubic DNLS whose nonlinear character is enhanced as the excitation amplitude increases, the saturable DNLS in fact becomes increasingly linear as the excitation amplitude increases. We explore the nonlinear dynamics of this phenomenon by direct numerical simulations.

nlin.PS↗

arXiv2010

Staggered and short period solutions of the Saturable Discrete Nonlinear Schrödinger Equation

We point out that the nonlinear Schr{ö}dinger lattice with a saturable nonlinearity also admits staggered periodic as well as localized pulse-like solutions. Further, the same model also admits solutions with a short period. We examine the stability of these solutions and find that the staggered as well as the short period solutions are stable in most cases. We also show that the effective Peierls-Nabarro barrier for the pulse-like soliton solutions is zero.

nlin.SI↗

arXiv2010

Exact Solutions of the Two-Dimensional Discrete Nonlinear Schrödinger Equation with Saturable Nonlinearity

We show that the two-dimensional, nonlinear Schrödinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero.

nlin.SI↗

arXiv2006

Discrete Nonlinear Schrodinger Equations with arbitrarily high order nonlinearities

A class of discrete nonlinear Schrodinger equations with arbitrarily high order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrodinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers and moving solutions, are investigated.

nlin.PS↗