Jul 2026· SIAM Journal on Scientific Computing· Vol 48, pp. 2027-· 0 citations· 33 references
Computer Science
TL;DR
Numerical experiments demonstrate that the TT-accelerated SL methods achieve their theoretical convergence rates, exhibit modest growth in memory usage and runtime with dimension, and significantly outperform grid-based SL in accuracy per CPU-second.
Abstract
Abstract.
We introduce a fully discrete scheme to solve a class of high-dimensional mean field games systems. Our approach couples semi-Lagrangian (SL) time discretizations with Tensor-Train (TT) decompositions to tame the curse of dimensionality. By reformulating the classical Hamilton–Jacobi–Bellman and Fokker–Planck equations as a sequence of advection–diffusion–reaction subproblems within a smoothed policy iteration, we construct both first and second order in time SL schemes. The TT format and appropriate quadrature rules reduce storage and computational cost from exponential to polynomial in the dimension. Numerical experiments demonstrate that our TT-accelerated SL methods achieve their theoretical convergence rates, exhibit modest growth in memory usage and runtime with dimension, and significantly outperform grid-based SL in accuracy per CPU-second.
A particle-based workflow for approximating the time-dependent law of finite-volume discretizations of the Dean–Kawasaki model is proposed, which accurately captures the site-wise correlations and other observables of the true model.
This paper presents a high-order rank-adaptive implicit integrator for the tensor solution of high-dimensional diffusion equations. We extend the 3D version of this method from the Tucker decomposition to higher dimensions using the hierarchical Tucker (HT) decomposition, since the storage complexity for the Tucker decomposition increases exponentially with the number of dimensions $d>3$. The HT format avoids this issue by decomposing the solution according to a binary tree consisting of bases for each dimension and core tensors which connect the bases. Spectral methods are considered for spatial discretization, and diagonally implicit Runge-Kutta methods are considered for time discretization. At each stage of the Runge-Kutta method, the bases computed at the previous stages are augmented to predict the upcoming basis and construct projection subspaces. By projecting onto these enriched subspaces, the bases and cores can be updated in a sequential manner going up the tree from leaf-to-root. Unlike the 3D Tucker method which has a single core tensor, the HT method also updates the intermediate core tensors. Numerical experiments demonstrate that the method observes high-order accuracy, and test how well the integrator captures the solution rank for various sets of time-dependent diffusion coefficients.
A new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies is introduced, and it is found that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.
A principled fiber-dependency elimination framework is introduced that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities, which produces a closed collocation system while preserving the principal advantages of TT-cross methods.
Behzad Ghahremani, H. Babaee· arXiv.org· 1 citation
Multi-agent interaction in linear–quadratic (LQ) differential games gives rise to open-loop Nash equilibria that rarely admit closed-form expressions, motivating the development of reliable numerical solvers. Classical approaches such as shooting and spectral collocation are sensitive to the initial guess on the unknown boundary values and accumulate discretisation error over long horizons, while deep-learning alternatives require iterative gradient-based training with architecture- and convergence-specific overhead. To overcome these limitations, we recast the LQ nonzero-sum game as a linear two-point boundary value problem (TPBVP) via the Pontryagin maximum principle (PMP) and solve it with a single-layer feedforward neural network (SLFN) in which hidden-layer parameters are sampled once and fixed. The state and all player-specific costates are parameterised by random hidden features on a uniform time grid, the boundary conditions are appended as dedicated rows of the linear collocation system, and the output weights follow from a single Moore–Penrose pseudoinverse, entirely bypassing gradient-based iteration. For the scalar LQ optimal-control TPBVP, a residual-to-solution stability theorem converts the continuous equation and boundary residuals into uniform state, costate, control, and cost error bounds. Validation across two-player low- and high-dimensional benchmarks, a heterogeneous three-player game, and paired seed sweeps confirms high accuracy against analytical and matrix-exponential references, while revealing that no single activation function dominates across all problem types: tanh is most accurate in one-dimensional settings, and Gaussian RBF leads in multidimensional cases.
Text-to-image latent diffusion models produce unprecedented visual fidelity but remain severely bottlenecked by the computational latency of iterative sampling. While optimizing the discretization of the continuous-time variable offers a powerful, training-free acceleration pathway, the comparative tradeoffs of foundational spacing heuristics remain under-explored. In this paper, we systematically benchmark three primary timestep spacing strategies—leading, linspace, and trailing—across a broad spectrum of Number of Function Evaluations (NFEs) to explicitly isolate their impact on generation quality within the text-to-image task. Through extensive evaluations on MS-COCO assessing structural fidelity (FID, IS) and text-image semantic alignment (CLIP Score), we identify distinct solver-dependent behaviors. With first-order ODE solvers such as DDIM, linspace yields stable, marginally superior outcomes for high-step generation but suffers from noisy outputs under low-step constraints. Conversely, trailing significantly mitigates this low-step performance collapse, while leading consistently degrades image quality and restricts the brightness range. However, with high-order ODE solvers, exemplified by DEIS, the performance gap between linspace and trailing diminishes, though leading remains strictly inferior. Ultimately, our findings suggest leveraging trailing as a highly versatile timestep spacing strategy, particularly for optimizing fast generation in latency-sensitive applications.
Tan-Yan Bao, Huy-Tan Thai· International Conference on...· 0 citations
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