Large particle systems are often described by high-dimensional (linear) kinetic equations that are simulated using Monte Carlo methods for which the asymptotic convergence rate is independent of the dimensionality. Even though the asymptotic convergence rate is known, predicting the actual value of the statistical error remains a challenging problem. In this paper, we show how the statistical error of an analog particle tracing Monte Carlo method can be calculated (expensive) and predicted a priori (cheap) when estimating quantities of interest (QoI) on a histogram. We consider two types of QoI estimators: point estimators for which each particle provides one independent contribution to the QoI estimates, and analog estimators for which each particle provides multiple correlated contributions to the QoI estimates. The developed statistical error predictors can be applied to other QoI estimators and nonanalog simulation routines as well. The error analysis is based on interpreting the number of particle visits to a histogram bin as the result of a (correlated) binomial experiment. The resulting expressions can be used to optimize (non)analog particle tracing Monte Carlo methods and hybrid simulation methods involving a Monte Carlo component, as well as to select an optimal particle tracing Monte Carlo method from several available options. Additionally, the cheap statistical error predictors can be used to determine a priori the number of particles N that is needed to reach a desired accuracy. We illustrate the theory using a linear kinetic equation describing neutral particles in the plasma edge of a fusion device and show numerical results. The code used to perform the numerical experiments is openly available.
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We present a new algorithm for the rigorous integration of the variational equation (i.e. producing $\mathcal C^1$ estimates) for a class of dissipative PDEs on the torus. As an application for some parameter value for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions we prove the existence of infinite number of homo- and heteroclinic orbits to two periodic orbits. The proof is computer assisted.
Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variations of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton--Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift--Hohenberg equation, the Ohta--Kawasaki equation and the Kuramoto--Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.
This paper focuses on the numerical scheme for delay-type stochastic McKean-Vlasov equations (DSMVEs) driven by fractional Brownian motion with Hurst parameter $H\in (0,1/2)\cup (1/2,1)$. The existence and uniqueness of the solutions to such DSMVEs whose drift coefficients contain polynomial delay terms are proved by exploting the Banach fixed point theorem. Then the propagation of chaos between interacting particle system and non-interacting system in $\mathcal{L}^p$ sense is shown. We find that even if the delay term satisfies the polynomial growth condition, the unmodified classical Euler-Maruyama scheme still can approximate the corresponding interacting particle system without the particle corruption. The convergence rates are revealed for $H\in (0,1/2)\cup (1/2,1)$. Finally, as an example that closely fits the original equation, a stochastic opinion dynamics model with both extrinsic memory and intrinsic memory is simulated to illustrate the plausibility of the theoretical result.
Traditional low-rank approximation is a powerful tool to compress the huge data matrices that arise in simulations of partial differential equations (PDE), but suffers from high computational cost and requires several passes over the PDE data. The compressed data may also lack interpretability thus making it difficult to identify feature patterns from the original data. To address this issue, we present an online randomized algorithm to compute the interpolative decomposition (ID) of large-scale data matrices in situ. Compared to previous randomized IDs that used the QR decomposition to determine the column basis, we adopt a streaming ridge leverage score-based column subset selection algorithm that dynamically selects proper basis columns from the data and thus avoids an extra pass over the data to compute the coefficient matrix of the ID. In particular, we adopt a single-pass error estimator based on the non-adaptive Hutch++ algorithm to provide real-time error approximation for determining the best coefficients. As a result, our approach only needs a single pass over the original data and thus is suitable for large and high-dimensional matrices stored outside of core memory or generated in PDE simulations. We also provide numerical experiments on turbulent channel flow and ignition simulations, and on the NSTX Gas Puff Image dataset, comparing our algorithm with the offline ID algorithm to demonstrate its utility in real-world applications.
Softmax attention is the principle backbone of foundation models for various artificial intelligence applications, yet its quadratic complexity in sequence length can limit its inference throughput in long-context settings. To address this challenge, alternative architectures such as linear attention, State Space Models (SSMs), and Recurrent Neural Networks (RNNs) have been considered as more efficient alternatives. While connections between these approaches exist, such models are commonly developed in isolation and there is a lack of theoretical understanding of the shared principles underpinning these architectures and their subtle differences, greatly influencing performance and scalability. In this paper, we introduce the Dynamical Systems Framework (DSF), which allows a principled investigation of all these architectures in a common representation. Our framework facilitates rigorous comparisons, providing new insights on the distinctive characteristics of each model class. For instance, we compare linear attention and selective SSMs, detailing their differences and conditions under which both are equivalent. We also provide principled comparisons between softmax attention and other model classes, discussing the theoretical conditions under which softmax attention can be approximated. Additionally, we substantiate these new insights with empirical validations and mathematical arguments. This shows the DSF's potential to guide the systematic development of future more efficient and scalable foundation models.
Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over Wasserstein distance. In this paper we prove that the infinite dimensionality of the space of probabilities drastically deteriorates its sample complexity, which is slower than any polynomial rate in the sample size. We propose a new distance that preserves many desirable properties of the former while achieving a parametric rate of convergence. In particular, our distance 1) metrizes weak convergence; 2) can be estimated numerically through samples with low complexity; 3) can be bounded analytically from above and below. The main ingredient are integral probability metrics, which lead to the name hierarchical IPM.
Discrepancy is a well-known measure for the irregularity of the distribution of a point set. Point sets with small discrepancy are called low-discrepancy and are known to efficiently fill the space in a uniform manner. Low-discrepancy points play a central role in many problems in science and engineering, including numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. In this work, we present the first machine learning approach to generate a new class of low-discrepancy point sets named Message-Passing Monte Carlo (MPMC) points. Motivated by the geometric nature of generating low-discrepancy point sets, we leverage tools from Geometric Deep Learning and base our model on Graph Neural Networks. We further provide an extension of our framework to higher dimensions, which flexibly allows the generation of custom-made points that emphasize the uniformity in specific dimensions that are primarily important for the particular problem at hand. Finally, we demonstrate that our proposed model achieves state-of-the-art performance superior to previous methods by a significant margin. In fact, MPMC points are empirically shown to be either optimal or near-optimal with respect to the discrepancy for every dimension and the number of points for which the optimal discrepancy can be determined.
Estimation and goodness-of-fit testing for positive random variables with explicit Laplace transform
Many flexible families of positive random variables exhibit non-closed forms of the density and distribution functions and this feature is considered unappealing for modelling purposes. However, such families are often characterized by a simple expression of the corresponding Laplace transform. Relying on the Laplace transform, we propose to carry out parameter estimation and goodness-of-fit testing for a general class of non-standard laws. We suggest a novel data-driven inferential technique, providing parameter estimators and goodness-of-fit tests, whose large-sample properties are derived. The implementation of the method is specifically considered for the positive stable and Tweedie distributions. A Monte Carlo study shows good finite-sample performance of the proposed technique for such laws.
The computing resource needs of LHC experiments are expected to continue growing significantly during the Run 3 and into the HL-LHC era. The landscape of available resources will also evolve, as High Performance Computing (HPC) and Cloud resources will provide a comparable, or even dominant, fraction of the total compute capacity. The future years present a challenge for the experiments' resource provisioning models, both in terms of scalability and increasing complexity. The CMS Submission Infrastructure (SI) provisions computing resources for CMS workflows. This infrastructure is built on a set of federated HTCondor pools, currently aggregating 400k CPU cores distributed worldwide and supporting the simultaneous execution of over 200k computing tasks. Incorporating HPC resources into CMS computing represents firstly an integration challenge, as HPC centers are much more diverse compared to Grid sites. Secondly, evolving the present SI, dimensioned to harness the current CMS computing capacity, to reach the resource scales required for the HLLHC phase, while maintaining global flexibility and efficiency, will represent an additional challenge for the SI. To preventively address future potential scalability limits, the SI team regularly runs tests to explore the maximum reach of our infrastructure. In this note, the integration of HPC resources into CMS offline computing is summarized, the potential concerns for the SI derived from the increased scale of operations are described, and the most recent results of scalability test on the CMS SI are reported.
This work aims at improving the energy efficiency of decentralized learning by optimizing the mixing matrix, which controls the communication demands during the learning process. Through rigorous analysis based on a state-of-the-art decentralized learning algorithm, the problem is formulated as a bi-level optimization, with the lower level solved by graph sparsification. A solution with guaranteed performance is proposed for the special case of fully-connected base topology and a greedy heuristic is proposed for the general case. Simulations based on real topology and dataset show that the proposed solution can lower the energy consumption at the busiest node by 54%-76% while maintaining the quality of the trained model.
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