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In this paper we present a new high order semi-implicit DG scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

It is well-known that decision-making problems from stochastic control can be formulated by means of forward-backward stochastic differential equation (FBSDE). Recently, the authors of Ji et al. 2022 proposed an efficient deep learning-based algorithm which was based on the stochastic maximum principle (SMP). In this paper, we provide a convergence result for this deep SMP-BSDE algorithm and compare its performance with other existing methods. In particular, by adopting a similar strategy as in Han and Long 2020, we derive a posteriori error estimate, and show that the total approximation error can be bounded by the value of the loss functional and the discretization error. We present numerical examples for high-dimensional stochastic control problems, both in case of drift- and diffusion control, which showcase superior performance compared to existing algorithms.

By abstracting over well-known properties of De Bruijn's representation with nameless dummies, we design a new theory of syntax with variable binding and capture-avoiding substitution. We propose it as a simpler alternative to Fiore, Plotkin, and Turi's approach, with which we establish a strong formal link. We also show that our theory easily incorporates simple types and equations between terms.

Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work we present a higher-order GNN model trained to predict the fourth-order stiffness tensor of periodic strut-based lattices. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate the benefits of the encoded equivariance and energy conservation in terms of predictive performance and reduced training requirements.

We propose a local modification of the standard subdiffusion model by introducing the initial Fickian diffusion, which results in a multiscale diffusion model. The developed model resolves the incompatibility between the nonlocal operators in subdiffusion and the local initial conditions and thus eliminates the initial singularity of the solutions of the subdiffusion, while retaining its heavy tail behavior away from the initial time. The well-posedness of the model and high-order regularity estimates of its solutions are analyzed by resolvent estimates, based on which the numerical discretization and analysis are performed. Numerical experiments are carried out to substantiate the theoretical findings.

In this paper, we develop a class of high-order conservative methods for simulating non-equilibrium radiation diffusion problems. Numerically, this system poses significant challenges due to strong nonlinearity within the stiff source terms and the degeneracy of nonlinear diffusion terms. Explicit methods require impractically small time steps, while implicit methods, which offer stability, come with the challenge to guarantee the convergence of nonlinear iterative solvers. To overcome these challenges, we propose a predictor-corrector approach and design proper implicit-explicit time discretizations. In the predictor step, the system is reformulated into a nonconservative form and linear diffusion terms are introduced as a penalization to mitigate strong nonlinearities. We then employ a Picard iteration to secure convergence in handling the nonlinear aspects. The corrector step guarantees the conservation of total energy, which is vital for accurately simulating the speeds of propagating sharp fronts in this system. For spatial approximations, we utilize local discontinuous Galerkin finite element methods, coupled with positive-preserving and TVB limiters. We validate the orders of accuracy, conservation properties, and suitability of using large time steps for our proposed methods, through numerical experiments conducted on one- and two-dimensional spatial problems. In both homogeneous and heterogeneous non-equilibrium radiation diffusion problems, we attain a time stability condition comparable to that of a fully implicit time discretization. Such an approach is also applicable to many other reaction-diffusion systems.

We propose a novel algorithm for the support estimation of partially known Gaussian graphical models that incorporates prior information about the underlying graph. In contrast to classical approaches that provide a point estimate based on a maximum likelihood or a maximum a posteriori criterion using (simple) priors on the precision matrix, we consider a prior on the graph and rely on annealed Langevin diffusion to generate samples from the posterior distribution. Since the Langevin sampler requires access to the score function of the underlying graph prior, we use graph neural networks to effectively estimate the score from a graph dataset (either available beforehand or generated from a known distribution). Numerical experiments demonstrate the benefits of our approach.

We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(\mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.

In this manuscript we propose and analyze an implicit two-point type method (or inertial method) for obtaining stable approximate solutions to linear ill-posed operator equations. The method is based on the iterated Tikhonov (iT) scheme. We establish convergence for exact data, and stability and semi-convergence for noisy data. Regarding numerical experiments we consider: i) a 2D Inverse Potential Problem, ii) an Image Deblurring Problem; the computational efficiency of the method is compared with standard implementations of the iT method.

In this article we consider an aggregate loss model with dependent losses. The losses occurrence process is governed by a two-state Markovian arrival process (MAP2), a Markov renewal process process that allows for (1) correlated inter-losses times, (2) non-exponentially distributed inter-losses times and, (3) overdisperse losses counts. Some quantities of interest to measure persistence in the loss occurrence process are obtained. Given a real operational risk database, the aggregate loss model is estimated by fitting separately the inter-losses times and severities. The MAP2 is estimated via direct maximization of the likelihood function, and severities are modeled by the heavy-tailed, double-Pareto Lognormal distribution. In comparison with the fit provided by the Poisson process, the results point out that taking into account the dependence and overdispersion in the inter-losses times distribution leads to higher capital charges.