Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator -- with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics.
相關內容
Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
SIT：
We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This assumption is difficult to justify in many inverse problems, where the specification of the data generation process is not obvious. We adopt a Gibbs posterior framework that directly posits a regularized variational problem on the space of probability distributions of the parameter. We propose a novel model comparison framework that evaluates the optimality of a given loss based on its ''predictive performance''. We provide cross-validation procedures to calibrate the regularization parameter of the variational objective and compare multiple loss functions. Some novel theoretical properties of Gibbs posteriors are also presented. We illustrate the utility of our framework via a simulated example, motivated by dispersion-based wave models used to characterize arterial vessels in ultrasound vibrometry.
We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. Numerical experiments suggest that the presence of an interface seems to be of minor importance for the continuation of the solution beyond the data domain. On the other hand, certain convexity properties of the geometry are crucial as has already been observed for many other problems without interfaces.
Conventionally, piecewise polynomial basis functions (PBFs) are used in the boundary elements method (BEM) to approximate unknown functions. Since, smooth radial basis functions (RBFs) are more stable and accurate than the PBFs for two and three dimensional domains, the unknown functions are approximated by the RBFs in this paper. Therefore, a new formulation of BEM, called radial BEM, is proposed. There are some singular boundary integrals in BEM which mostly are calculated analytically. Analytical schemes are only applicable for PBFs defined on straight boundary element, and become more complicated for polynomials of higher degree. To overcome this difficulty, this paper proposes a distribution for boundary source points so that the boundary integrals can be calculated by Gaussian quadrature rule (GQR) with high precision. Using advantages of the proposed approach, boundary integrals of the radial BEM are calculated, easily and precisely. Several numerical examples are presented to show efficiency of the radial BEM versus standard BEM for solving partial differential equations (PDEs).
Traditional coverage grey-box fuzzers perform a breadth-first search of the state space of Program Under Test (PUT). This aimlessness wastes a lot of computing resources. Directed grey-box fuzzing focuses on the target of PUT and becomes one of the most popular topics of software testing. The early termination of unreachable test cases is a method to improve directed grey-box fuzzing. However, existing solutions have two problems: firstly, reachability analysis needs to introduce extra technologies (e.g., static analysis); secondly, the performance of reachability analysis and auxiliary technologies lack versatility. We propose FGo, a probabilistic exponential cut-the-loss directed grey-box fuzzer. FGo terminates unreachable test cases early with exponentially increasing probability. Compared to other technologies, FGo makes full use of the unreachable information contained in iCFG and doesn't generate any additional overhead caused by reachability analysis. Moreover, it is easy to generalize to all PUT. This strategy based on probability is perfectly adapted to the randomness of fuzzing. The experiment results show that FGo is 106% faster than AFLGo in reproducing crashes. We compare multiple parameters of probabilistic exponential cut-the-loss algorithm and analyze them in detail. In addition, for enhancing the inerpretability of FGo, this paper discusses the difference between the theoretical performance and the practical performance of probabilistic exponential cut-the-loss algorithm.
Smoothed Particle Hydrodynamics (SPH) is plagued by the phenomenon of tensile instability, which is the occurrence of short wavelength zero energy modes resulting in unphysical clustering of particles. The root cause of the instability is the shape of derivative of the compactly supported kernel function which may yield negative stiffness in the particle interaction under certain circumstances. In this work, an adaptive algorithm is developed to remove tensile instability in SPH for weakly compressible fluids. Herein, a B-spline function is used as the SPH kernel and the knots of the B-spline are adapted to change the shape of the kernel, thereby satisfying the condition associated with stability. The knot-shifting criterion is based on the particle movement within the influence domain. This enables the prevention of instability in fluid problems where excessive rearrangement of particle positions occurs. A 1D dispersion analysis of an Oldroyd B fluid material model is performed to show how the algorithm prevents instabilities for short wavelengths but ensures accuracy at large wavelengths. The efficacy of the approach is demonstrated through a few benchmark fluid dynamics simulations where a visco-elastic Oldroyd B material model and a non-viscous Eulerian fluid material model are considered.
The number of modes in a probability density function is representative of the model's complexity and can also be viewed as the number of existing subpopulations. Despite its relevance, little research has been devoted to its estimation. Focusing on the univariate setting, we propose a novel approach targeting prediction accuracy inspired by some overlooked aspects of the problem. We argue for the need for structure in the solutions, the subjective and uncertain nature of modes, and the convenience of a holistic view blending global and local density properties. Our method builds upon a combination of flexible kernel estimators and parsimonious compositional splines. Feature exploration, model selection and mode testing are implemented in the Bayesian inference paradigm, providing soft solutions and allowing to incorporate expert judgement in the process. The usefulness of our proposal is illustrated through a case study in sports analytics, showcasing multiple companion visualisation tools. A thorough simulation study demonstrates that traditional modality-driven approaches paradoxically struggle to provide accurate results. In this context, our method emerges as a top-tier alternative offering innovative solutions for analysts.
We present a flexible data-driven method for dynamical system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approximating the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically. Furthermore, the method is agnostic of whether data is generated through a deterministic or stochastic process, so its implementation requires no prior adjustments by the user to accommodate these different scenarios. Rigorous convergence results justify the applicability of the method, while also extending and uniting similar results from across the literature. Examples on discovering Lyapunov functions, performing ergodic optimization, and bounding extrema over attractors for both deterministic and stochastic dynamics exemplify these convergence results and demonstrate the performance of the method.
Recent advances in large language models have prompted researchers to examine their abilities across a variety of linguistic tasks, but little has been done to investigate how models handle the interactions in meaning across words and larger syntactic forms -- i.e. phenomena at the intersection of syntax and semantics. We present the semantic notion of agentivity as a case study for probing such interactions. We created a novel evaluation dataset by utilitizing the unique linguistic properties of a subset of optionally transitive English verbs. This dataset was used to prompt varying sizes of three model classes to see if they are sensitive to agentivity at the lexical level, and if they can appropriately employ these word-level priors given a specific syntactic context. Overall, GPT-3 text-davinci-003 performs extremely well across all experiments, outperforming all other models tested by far. In fact, the results are even better correlated with human judgements than both syntactic and semantic corpus statistics. This suggests that LMs may potentially serve as more useful tools for linguistic annotation, theory testing, and discovery than select corpora for certain tasks. Code is available at //github.com/lindiatjuatja/lm_sem
We study the complexity of producing $(\delta,\epsilon)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $\Omega(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(d\delta^{-1}\epsilon^{-3})$, which is optimal (up to numerical constants) with respect to $d$ and also optimal with respect to the accuracy parameters $\delta,\epsilon$, thus solving an open question due to Lin et al. (NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also optimal for smooth objectives, proving that in the nonconvex stochastic zero-order setting, nonsmooth optimization is as easy as smooth optimization. We provide algorithms that achieve the aforementioned convergence rate in expectation as well as with high probability. Our analysis is based on a simple yet powerful geometric lemma regarding the Goldstein-subdifferential set, which allows utilizing recent advancements in first-order nonsmooth nonconvex optimization.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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