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Fairness holds a pivotal role in the realm of machine learning, particularly when it comes to addressing groups categorised by protected attributes, e.g., gender, race. Prevailing algorithms in fair learning predominantly hinge on accessibility or estimations of these protected attributes, at least in the training process. We design a single group-blind projection map that aligns the feature distributions of both groups in the source data, achieving (demographic) group parity, without requiring values of the protected attribute for individual samples in the computation of the map, as well as its use. Instead, our approach utilises the feature distributions of the privileged and unprivileged groups in a boarder population and the essential assumption that the source data are unbiased representation of the population. We present numerical results on synthetic data and real data.

In this paper, we propose high order numerical methods to solve a 2D advection diffusion equation, in the highly oscillatory regime. We use an integrator strategy that allows the construction of arbitrary high-order schemes {leading} to an accurate approximation of the solution without any time step-size restriction. This paper focuses on the multiscale challenges {in time} of the problem, that come from the velocity, an $\varepsilon-$periodic function, whose expression is explicitly known. $\varepsilon$-uniform third order in time numerical approximations are obtained. For the space discretization, this strategy is combined with high order finite difference schemes. Numerical experiments show that the proposed methods {achieve} the expected order of accuracy, and it is validated by several tests across diverse domains and boundary conditions. The novelty of the paper consists of introducing a numerical scheme that is high order accurate in space and time, with a particular attention to the dependency on a small parameter in the time scale. The high order in space is obtained enlarging the interpolation stencil already established in [44], and further refined in [46], with a special emphasis on the squared boundary, especially when a Dirichlet condition is assigned. In such case, we compute an \textit{ad hoc} Taylor expansion of the solution to ensure that there is no degradation of the accuracy order at the boundary. On the other hand, the high accuracy in time is obtained extending the work proposed in [19]. The combination of high-order accuracy in both space and time is particularly significant due to the presence of two small parameters-$\delta$ and $\varepsilon$-in space and time, respectively.

In this paper, we introduce a highly accurate and efficient numerical solver for the radial Kohn--Sham equation. The equation is discretized using a high-order finite element method, with its performance further improved by incorporating a parameter-free moving mesh technique. This approach greatly reduces the number of elements required to achieve the desired precision. In practice, the mesh redistribution involves no more than three steps, ensuring the algorithm remains computationally efficient. Remarkably, with a maximum of $13$ elements, we successfully reproduce the NIST database results for elements with atomic numbers ranging from $1$ to $92$.

In this paper, we propose a novel eigenpair-splitting method, inspired by the divide-and-conquer strategy, for solving the generalized eigenvalue problem arising from the Kohn-Sham equation. Unlike the commonly used domain decomposition approach in divide-and-conquer, which solves the problem on a series of subdomains, our eigenpair-splitting method focuses on solving a series of subequations defined on the entire domain. This method is realized through the integration of two key techniques: a multi-mesh technique for generating approximate spaces for the subequations, and a soft-locking technique that allows for the independent solution of eigenpairs. Numerical experiments show that the proposed eigenpair-splitting method can dramatically enhance simulation efficiency, and its potential towards practical applications is also demonstrated well through an example of the HOMO-LUMO gap calculation. Furthermore, the optimal strategy for grouping eigenpairs is discussed, and the possible improvements to the proposed method are also outlined.

In this work is considered an elliptic problem, referred to as the Ventcel problem, involvinga second order term on the domain boundary (the Laplace-Beltrami operator). A variationalformulation of the Ventcel problem is studied, leading to a finite element discretization. Thefocus is on the construction of high order curved meshes for the discretization of the physicaldomain and on the definition of the lift operator, which is aimed to transform a functiondefined on the mesh domain into a function defined on the physical one. This lift is definedin a way as to satisfy adapted properties on the boundary, relatively to the trace operator.The Ventcel problem approximation is investigated both in terms of geometrical error and offinite element approximation error. Error estimates are obtained both in terms of the meshorder r $\ge$ 1 and to the finite element degree k $\ge$ 1, whereas such estimates usually have beenconsidered in the isoparametric case so far, involving a single parameter k = r. The numericalexperiments we led, both in dimension 2 and 3, allow us to validate the results obtained andproved on the a priori error estimates depending on the two parameters k and r. A numericalcomparison is made between the errors using the former lift definition and the lift defined inthis work establishing an improvement in the convergence rate of the error in the latter case.

Nowadays, we have seen that dual quaternion algorithms are used in 3D coordinate transformation problems due to their advantages. 3D coordinate transformation problem is one of the important problems in geodesy. This transformation problem is encountered in many application areas other than geodesy. Although there are many coordinate transformation methods (similarity, affine, projective, etc.), similarity transformation is used because of its simplicity. The asymmetric transformation is preferred to the symmetric coordinate transformation because of its ease of use. In terms of error theory, the symmetric transformation should be preferred. In this study, the topic of symmetric similarity 3D coordinate transformation based on the dual quaternion algorithm is discussed, and the bottlenecks encountered in solving the problem and the solution method are discussed. A new iterative algorithm based on the dual quaternion is presented. The solution is implemented in two different models: with constraint equations and without constraint equations. The advantages and disadvantages of the two models compared to each other are also evaluated. Not only the transformation parameters but also the errors of the transformation parameters are determined. The detailed derivation of the formulas for estimating the symmetric similarity of 3D transformation parameters is presented step by step. Since symmetric transformation is the general form of asymmetric transformation, we can also obtain asymmetric transformation results with a simple modification of the model we developed for symmetric transformation. The proposed algorithm is capable of performing both 2D and 3D symmetric and asymmetric similarity transformations. For the 2D transformation, it is sufficient to replace the z and Z coordinates in both systems with zero.

In the present work, strong approximation errors are analyzed for both the spatial semi-discretization and the spatio-temporal fully discretization of stochastic wave equations (SWEs) with cubic polynomial nonlinearities and additive noises. The fully discretization is achieved by the standard Galerkin ffnite element method in space and a novel exponential time integrator combined with the averaged vector ffeld approach. The newly proposed scheme is proved to exactly satisfy a trace formula based on an energy functional. Recovering the convergence rates of the scheme, however, meets essential difffculties, due to the lack of the global monotonicity condition. To overcome this issue, we derive the exponential integrability property of the considered numerical approximations, by the energy functional. Armed with these properties, we obtain the strong convergence rates of the approximations in both spatial and temporal direction. Finally, numerical results are presented to verify the previously theoretical findings.

In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems associated with convex energies. We propose element-wise nonconforming finite element methods to discretize the continuous minimisation problem. Using $\Gamma$-convergence arguments we show that the discrete minimisers converge to the unique minimiser of the continuous problem as the mesh parameter tends to zero, under the additional contribution of appropriately defined penalty terms at the level of the discrete energies. We finally substantiate the feasibility of our methods by numerical examples.

We propose a novel, highly efficient, second-order accurate, long-time unconditionally stable numerical scheme for a class of finite-dimensional nonlinear models that are of importance in geophysical fluid dynamics. The scheme is highly efficient in the sense that only a (fixed) symmetric positive definite linear problem (with varying right hand sides) is involved at each time-step. The solutions to the scheme are uniformly bounded for all time. We show that the scheme is able to capture the long-time dynamics of the underlying geophysical model, with the global attractors as well as the invariant measures of the scheme converge to those of the original model as the step size approaches zero. In our numerical experiments, we take an indirect approach, using long-term statistics to approximate the invariant measures. Our results suggest that the convergence rate of the long-term statistics, as a function of terminal time, is approximately first order using the Jensen-Shannon metric and half-order using the L1 metric. This implies that very long time simulation is needed in order to capture a few significant digits of long time statistics (climate) correct. Nevertheless, the second order scheme's performance remains superior to that of the first order one, requiring significantly less time to reach a small neighborhood of statistical equilibrium for a given step size.