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Studying unified model averaging estimation for situations with complicated data structures, we propose a novel model averaging method based on cross-validation (MACV). MACV unifies a large class of new and existing model averaging estimators and covers a very general class of loss functions. Furthermore, to reduce the computational burden caused by the conventional leave-subject/one-out cross validation, we propose a SEcond-order-Approximated Leave-one/subject-out (SEAL) cross validation, which largely improves the computation efficiency. In the context of non-independent and non-identically distributed random variables, we establish the unified theory for analyzing the asymptotic behaviors of the proposed MACV and SEAL methods, where the number of candidate models is allowed to diverge with sample size. To demonstrate the breadth of the proposed methodology, we exemplify four optimal model averaging estimators under four important situations, i.e., longitudinal data with discrete responses, within-cluster correlation structure modeling, conditional prediction in spatial data, and quantile regression with a potential correlation structure. We conduct extensive simulation studies and analyze real-data examples to illustrate the advantages of the proposed methods.

Parameter inference is essential when interpreting observational data using mathematical models. Standard inference methods for differential equation models typically rely on obtaining repeated numerical solutions of the differential equation(s). Recent results have explored how numerical truncation error can have major, detrimental, and sometimes hidden impacts on likelihood-based inference by introducing false local maxima into the log-likelihood function. We present a straightforward approach for inference that eliminates the need for solving the underlying differential equations, thereby completely avoiding the impact of truncation error. Open-access Jupyter notebooks, available on GitHub, allow others to implement this method for a broad class of widely-used models to interpret biological data.

We consider the problem of optimizing the parameter of a two-stage algorithm for approximate solution of a system of linear algebraic equations with a sparse $n\times n$-matrix, i.e., with one in which the number of nonzero elements is $m\!=\!O(n)$. The two-stage algorithm uses conjugate gradient method at its stages. At the 1st stage, an approximate solution with accuracy $\varepsilon_1$ is found for zero initial vector. All numerical values used at this stage are represented as single-precision numbers. The obtained solution is used as initial approximation for an approximate solution with a given accuracy $\varepsilon_2$ that we obtain at the 2nd stage, where double-precision numbers are used. Based on the values of some matrix parameters, computed in a time not exceeding $O(m)$, we need to determine the value $\varepsilon_1$ which minimizes the total computation time at two stages. Using single-precision numbers for computations at the 1st stage is advantageous, since the execution time of one iteration will be approximately half that of one iteration at the 2nd stage. At the same time, using machine numbers with half the mantissa length accelerates the growth of the rounding error per iteration of the conjugate gradient method at the 1st stage, which entails an increase in the number of iterations performed at 2nd stage. As parameters that allow us to determine $\varepsilon_1$ for the input matrix, we use $n$, $m$, an estimate of the diameter of the graph associated with the matrix, an estimate of the spread of the matrix' eigenvalues, and estimates of its maximum eigenvalue. The optimal or close to the optimal value of $\varepsilon_1$ can be determined for matrix with such a vector of parameters using the nearest neighbor regression or some other type of regression.

Two sequential estimators are proposed for the odds p/(1-p) and log odds log(p/(1-p)) respectively, using independent Bernoulli random variables with parameter p as inputs. The estimators are unbiased, and guarantee that the variance of the estimation error divided by the true value of the odds, or the variance of the estimation error of the log odds, are less than a target value for any p in (0,1). The estimators are close to optimal in the sense of Wolfowitz's bound.

An extremely schematic model of the forces acting an a sailing yacht equipped with a system of foils is here presented and discussed. The role of the foils is to raise the hull from the water in order to reduce the total resistance and then increase the speed. Some CFD simulations are providing the total resistance of the bare hull at some values of speed and displacement, as well as the characteristics (drag and lift coefficients) of the 2D foil sections used for the appendages. A parametric study has been performed for the characterization of a foil of finite dimensions. The equilibrium of the vertical forces and longitudinal moments, as well as a reduced displacement, is obtained by controlling the pitch angle of the foils. The value of the total resistance of the yacht with foils is then compared with the case without foils, evidencing the speed regime where an advantage is obtained, if any.

We propose a generalized free energy potential for active systems, including both stochastic master equations and deterministic nonlinear chemical reaction networks. Our generalized free energy is defined variationally as the "most irreversible" state observable. This variational principle is motivated from several perspectives, including large deviations theory, thermodynamic uncertainty relations, Onsager theory, and information-theoretic optimal transport. In passive systems, the most irreversible observable is the usual free energy potential and its irreversibility is the entropy production rate (EPR). In active systems, the most irreversible observable is the generalized free energy and its irreversibility gives the excess EPR, the nonstationary contribution to dissipation. The remaining "housekeeping" EPR is a genuine nonequilibrium contribution that quantifies the nonconservative nature of the forces. We derive far-from-equilibrium thermodynamic speed limits for excess EPR, applicable to both linear and nonlinear systems. Our approach overcomes several limitations of the steady-state potential and the Hatano-Sasa (adiabatic/nonadiabatic) decomposition, as we demonstrate in several examples.

We consider the discretization of a class of nonlinear parabolic equations by discontinuous Galerkin time-stepping methods and establish a priori as well as conditional a posteriori error estimates. Our approach is motivated by the error analysis in [9] for Runge-Kutta methods for nonlinear parabolic equations; in analogy to [9], the proofs are based on maximal regularity properties of discontinuous Galerkin methods for non-autonomous linear parabolic equations.

We present a novel computational framework to assess the structural integrity of welds. In the first stage of the simulation framework, local fractions of microstructural constituents within weld regions are predicted based on steel composition and welding parameters. The resulting phase fraction maps are used to define heterogeneous properties that are subsequently employed in structural integrity assessments using an elastoplastic phase field fracture model. The framework is particularised to predicting failure in hydrogen pipelines, demonstrating its potential to assess the feasibility of repurposing existing pipeline infrastructure to transport hydrogen. First, the process model is validated against experimental microhardness maps for vintage and modern pipeline welds. Additionally, the influence of welding conditions on hardness and residual stresses is investigated, demonstrating that variations in heat input, filler material composition, and weld bead order can significantly affect the properties within the weld region. Coupled hydrogen diffusion-fracture simulations are then conducted to determine the critical pressure at which hydrogen transport pipelines will fail. To this end, the model is enriched with a microstructure-sensitive description of hydrogen transport and hydrogen-dependent fracture resistance. The analysis of an X52 pipeline reveals that even 2 mm defects in a hard heat-affected zone can drastically reduce the critical failure pressure.

We consider the vorticity formulation of the Euler equations describing the flow of a two-dimensional incompressible ideal fluid on the sphere. Zeitlin's model provides a finite-dimensional approximation of the vorticity formulation that preserves the underlying geometric structure: it consists of an isospectral Lie--Poisson flow on the Lie algebra of skew-Hermitian matrices. We propose an approximation of Zeitlin's model based on a time-dependent low-rank factorization of the vorticity matrix and evolve a basis of eigenvectors according to the Euler equations. In particular, we show that the approximate flow remains isospectral and Lie--Poisson and that the error in the solution, in the approximation of the Hamiltonian and of the Casimir functions only depends on the approximation of the vorticity matrix at the initial time. The computational complexity of solving the approximate model is shown to scale quadratically with the order of the vorticity matrix and linearly if a further approximation of the stream function is introduced.