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We derive an intuitionistic version of G\"odel-L\"ob modal logic ($\sf{GL}$) in the style of Simpson, via proof theoretic techniques. We recover a labelled system, $\sf{\ell IGL}$, by restricting a non-wellfounded labelled system for $\sf{GL}$ to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that $\sf{GL}$'s usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of $\sf{GL}$ are typically defined over only the box (and not the diamond), our presentation includes both modalities. Our main result is that $\sf{\ell IGL}$ coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic $\sf{IGL}$. While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of $\sf{IGL}$.

At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an $n^{O(d)}$-delay algorithm listing all minimal transversals of an $n$-vertex hypergraph of degeneracy $d$. Recently at IWOCA 2019, Conte, Kant\'e, Marino, and Uno asked whether this XP-delay algorithm parameterized by $d$ could be made FPT-delay parameterized by $d$ and the maximum degree $\Delta$, i.e., an algorithm with delay $f(d,\Delta)\cdot n^{O(1)}$ for some computable function $f$. Moreover, as a first step toward answering that question, they note that the same delay is open for the intimately related problem of listing all minimal dominating sets in graphs. In this paper, we answer the latter question in the affirmative.

Using diffusion models to solve inverse problems is a growing field of research. Current methods assume the degradation to be known and provide impressive results in terms of restoration quality and diversity. In this work, we leverage the efficiency of those models to jointly estimate the restored image and unknown parameters of the degradation model. In particular, we designed an algorithm based on the well-known Expectation-Minimization (EM) estimation method and diffusion models. Our method alternates between approximating the expected log-likelihood of the inverse problem using samples drawn from a diffusion model and a maximization step to estimate unknown model parameters. For the maximization step, we also introduce a novel blur kernel regularization based on a Plug \& Play denoiser. Diffusion models are long to run, thus we provide a fast version of our algorithm. Extensive experiments on blind image deblurring demonstrate the effectiveness of our method when compared to other state-of-the-art approaches.

A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.

Hawkes processes are often applied to model dependence and interaction phenomena in multivariate event data sets, such as neuronal spike trains, social interactions, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is generally a computationally expensive task, all the more with Bayesian estimation methods. In particular, for generalised nonlinear Hawkes processes, Monte-Carlo Markov Chain methods applied to compute the doubly intractable posterior distribution are not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we first unify existing variational Bayes approaches under a general nonparametric inference framework, and analyse the asymptotic properties of these methods under easily verifiable conditions on the prior, the variational class, and the nonlinear model. Secondly, we propose a novel sparsity-inducing procedure, and derive an adaptive mean-field variational algorithm for the popular sigmoid Hawkes processes. Our algorithm is parallelisable and therefore computationally efficient in high-dimensional setting. Through an extensive set of numerical simulations, we also demonstrate that our procedure is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to some type of model mis-specification.

We define game semantics for the constructive $\mu$-calculus and prove its correctness. We use these game semantics to prove that the $\mu$-calculus collapses to modal logic over $\mathsf{CS5}$ frames. Finally, we prove the completeness of $\mathsf{\mu CS5}$ over $\mathsf{CS5}$ frames.

This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its H\"older continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings.

We propose a new randomized method for solving systems of nonlinear equations, which can find sparse solutions or solutions under certain simple constraints. The scheme only takes gradients of component functions and uses Bregman projections onto the solution space of a Newton equation. In the special case of euclidean projections, the method is known as nonlinear Kaczmarz method. Furthermore, if the component functions are nonnegative, we are in the setting of optimization under the interpolation assumption and the method reduces to SGD with the recently proposed stochastic Polyak step size. For general Bregman projections, our method is a stochastic mirror descent with a novel adaptive step size. We prove that in the convex setting each iteration of our method results in a smaller Bregman distance to exact solutions as compared to the standard Polyak step. Our generalization to Bregman projections comes with the price that a convex one-dimensional optimization problem needs to be solved in each iteration. This can typically be done with globalized Newton iterations. Convergence is proved in two classical settings of nonlinearity: for convex nonnegative functions and locally for functions which fulfill the tangential cone condition. Finally, we show examples in which the proposed method outperforms similar methods with the same memory requirements.

Estimating the probability of the binomial distribution is a basic problem, which appears in almost all introductory statistics courses and is performed frequently in various studies. In some cases, the parameter of interest is a difference between two probabilities, and the current work studies the construction of confidence intervals for this parameter when the sample size is small. Our goal is to find the shortest confidence intervals under the constraint of coverage probability being larger than a predetermined level. For the two-sample case, there is no known algorithm that achieves this goal, but different heuristics procedures have been suggested, and the present work aims at finding optimal confidence intervals. In the one-sample case, there is a known algorithm that finds optimal confidence intervals presented by Blyth and Still (1983). It is based on solving small and local optimization problems and then using an inversion step to find the global optimum solution. We show that this approach fails in the two-sample case and therefore, in order to find optimal confidence intervals, one needs to solve a global optimization problem, rather than small and local ones, which is computationally much harder. We present and discuss the suitable global optimization problem. Using the Gurobi package we find near-optimal solutions when the sample sizes are smaller than 15, and we compare these solutions to some existing methods, both approximate and exact. We find that the improvement in terms of lengths with respect to the best competitor varies between 1.5\% and 5\% for different parameters of the problem. Therefore, we recommend the use of the new confidence intervals when both sample sizes are smaller than 15. Tables of the confidence intervals are given in the Excel file in this link.

We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits very good performances and robustness with respect to all parameters of interest.