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We propose a new simple and explicit numerical scheme for time-homogeneous stochastic differential equations. The scheme is based on sampling increments at each time step from a skew-symmetric probability distribution, with the level of skewness determined by the drift and volatility of the underlying process. We show that as the step-size decreases the scheme converges weakly to the diffusion of interest. We then consider the problem of simulating from the limiting distribution of an ergodic diffusion process using the numerical scheme with a fixed step-size. We establish conditions under which the numerical scheme converges to equilibrium at a geometric rate, and quantify the bias between the equilibrium distributions of the scheme and of the true diffusion process. Notably, our results do not require a global Lipschitz assumption on the drift, in contrast to those required for the Euler--Maruyama scheme for long-time simulation at fixed step-sizes. Our weak convergence result relies on an extension of the theory of Milstein \& Tretyakov to stochastic differential equations with non-Lipschitz drift, which could also be of independent interest. We support our theoretical results with numerical simulations.

Optimization constrained by high-fidelity computational models has potential for transformative impact. However, such optimization is frequently unattainable in practice due to the complexity and computational intensity of the model. An alternative is to optimize a low-fidelity model and use limited evaluations of the high-fidelity model to assess the quality of the solution. This article develops a framework to use limited high-fidelity simulations to update the optimization solution computed using the low-fidelity model. Building off a previous article [22], which introduced hyper-differential sensitivity analysis with respect to model discrepancy, this article provides novel extensions of the algorithm to enable uncertainty quantification of the optimal solution update via a Bayesian framework. Specifically, we formulate a Bayesian inverse problem to estimate the model discrepancy and propagate the posterior model discrepancy distribution through the post-optimality sensitivity operator for the low-fidelity optimization problem. We provide a rigorous treatment of the Bayesian formulation, a computationally efficient algorithm to compute posterior samples, a guide to specify and interpret the algorithm hyper-parameters, and a demonstration of the approach on three examples which highlight various types of discrepancy between low and high-fidelity models.

We study three systems of equations, together with a way to count the number of solutions. One of the results was used in the recent computation of D(9), the others have potential to speed up existing techniques in the future.

In this paper, to address the optimization problem on a compact matrix manifold, we introduce a novel algorithmic framework called the Transformed Gradient Projection (TGP) algorithm, using the projection onto this compact matrix manifold. Compared with the existing algorithms, the key innovation in our approach lies in the utilization of a new class of search directions and various stepsizes, including the Armijo, nonmonotone Armijo, and fixed stepsizes, to guide the selection of the next iterate. Our framework offers flexibility by encompassing the classical gradient projection algorithms as special cases, and intersecting the retraction-based line-search algorithms. Notably, our focus is on the Stiefel or Grassmann manifold, revealing that many existing algorithms in the literature can be seen as specific instances within our proposed framework, and this algorithmic framework also induces several new special cases. Then, we conduct a thorough exploration of the convergence properties of these algorithms, considering various search directions and stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, allowing us to extend classical inequalities related to retractions from the literature. Building upon these insights, we establish the weak convergence, convergence rate, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments, we observe that the TGP algorithms, owing to their increased flexibility in choosing search directions, outperform classical gradient projection and retraction-based line-search algorithms in several scenarios.

This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

The problems of optimal recovering univariate functions and their derivatives are studied. To solve these problems, two variants of the truncation method are constructed, which are order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. For numerical summation, it has been established how the parameters characterizing the problem being solved affect its stability.

The recent 1/2-equation model of turbulence is a simplification of the standard Kolmogorov-Prandtl 1-equation URANS model. Surprisingly, initial numerical tests indicated that the 1/2-equation model produces comparable velocity statistics at reduced cost. It is also a test problem and first step for developing numerical analysis to address a full 1-equation model. This report begins the numerical analysis of the 1/2 equation model. Stability, convergence and error estimates are proven for a semi-discrete and fully discrete approximation. Finally, numerical tests are conducted to validate our convergence theory.

We explore a class of splitting schemes employing implicit-explicit (IMEX) time-stepping to achieve accurate and energy-stable solutions for thin-film equations and Cahn-Hilliard models with variable mobility. This splitting method incorporates a linear, constant coefficient implicit step, facilitating efficient computational implementation. We investigate the influence of stabilizing splitting parameters on the numerical solution computationally, considering various initial conditions. Furthermore, we generate energy-stability plots for the proposed methods, examining different choices of splitting parameter values and timestep sizes. These methods enhance the accuracy of the original bi-harmonic-modified (BHM) approach, while preserving its energy-decreasing property and achieving second-order accuracy. We present numerical experiments to illustrate the performance of the proposed methods.

Temporal reasoning with conditionals is more complex than both classical temporal reasoning and reasoning with timeless conditionals, and can lead to some rather counter-intuitive conclusions. For instance, Aristotle's famous "Sea Battle Tomorrow" puzzle leads to a fatalistic conclusion: whether there will be a sea battle tomorrow or not, but that is necessarily the case now. We propose a branching-time logic LTC to formalise reasoning about temporal conditionals and provide that logic with adequate formal semantics. The logic LTC extends the Nexttime fragment of CTL*, with operators for model updates, restricting the domain to only future moments where antecedent is still possible to satisfy. We provide formal semantics for these operators that implements the restrictor interpretation of antecedents of temporalized conditionals, by suitably restricting the domain of discourse. As a motivating example, we demonstrate that a naturally formalised in our logic version of the `Sea Battle' argument renders it unsound, thereby providing a solution to the problem with fatalist conclusion that it entails, because its underlying reasoning per cases argument no longer applies when these cases are treated not as material implications but as temporal conditionals. On the technical side, we analyze the semantics of LTC and provide a series of reductions of LTC-formulae, first recursively eliminating the dynamic update operators and then the path quantifiers in such formulae. Using these reductions we obtain a sound and complete axiomatization for LTC, and reduce its decision problem to that of the modal logic KD.

The present paper is devoted to study the effect of connected and disconnected rotations of G\"odel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable G\"odel algebras endowed with modal operators.