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We analyse an energy minimisation problem recently proposed for modelling smectic-A liquid crystals. The optimality conditions give a coupled nonlinear system of partial differential equations, with a second-order equation for the tensor-valued nematic order parameter $\mathbf{Q}$ and a fourth-order equation for the scalar-valued smectic density variation $u$. Our two main results are a proof of the existence of solutions to the minimisation problem, and the derivation of a priori error estimates for its discretisation using the $\mathcal{C}^0$ interior penalty method. More specifically, optimal rates in the $H^1$ and $L^2$ norms are obtained for $\mathbf{Q}$, while optimal rates in a mesh-dependent norm and $L^2$ norm are obtained for $u$. Numerical experiments confirm the rates of convergence.

We propose the Terminating-Knockoff (T-Knock) filter, a fast variable selection method for high-dimensional data. The T-Knock filter controls a user-defined target false discovery rate (FDR) while maximizing the number of selected true positives. This is achieved by fusing the solutions of multiple early terminated random experiments. The experiments are conducted on a combination of the original data and multiple sets of randomly generated knockoff variables. A finite sample proof based on martingale theory for the FDR control property is provided. Numerical simulations show that the FDR is controlled at the target level while allowing for a high power. We prove under mild conditions that the knockoffs can be sampled from any univariate distribution. The computational complexity of the proposed method is derived and it is demonstrated via numerical simulations that the sequential computation time is multiple orders of magnitude lower than that of the strongest benchmark methods in sparse high-dimensional settings. The T-Knock filter outperforms state-of-the-art methods for FDR control on a simulated genome-wide association study (GWAS), while its computation time is more than two orders of magnitude lower than that of the strongest benchmark methods.

In this paper, we propose a monotone approximation scheme for a class of fully nonlinear partial integro-differential equations (PIDEs) which characterize the nonlinear $\alpha$-stable L\'{e}vy processes under sublinear expectation space with $\alpha \in(1,2)$. Two main results are obtained: (i) the error bounds for the monotone approximation scheme of nonlinear PIDEs, and (ii) the convergence rates of a generalized central limit theorem of Bayraktar-Munk for $\alpha$-stable random variables under sublinear expectation. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen.

We propose and study a temporal, and spatio-temporal discretisation of the 2D stochastic Navier--Stokes equations in bounded domains supplemented with no-slip boundary conditions. Considering additive noise, we base its construction on the related nonlinear random PDE, which is solved by a transform of the solution of the stochastic Navier--Stokes equations. We show strong rate (up to) $1$ in probability for a corresponding discretisation in space and time (and space-time). Convergence of order (up to) 1 in time was previously only known for linear SPDEs.

This paper develops a robust dynamic mode decomposition (RDMD) method endowed with statistical and numerical robustness. Statistical robustness ensures estimation efficiency at the Gaussian and non-Gaussian probability distributions, including heavy-tailed distributions. The proposed RDMD is statistically robust because the outliers in the data set are flagged via projection statistics and suppressed using a Schweppe-type Huber generalized maximum-likelihood estimator that minimizes a convex Huber cost function. The latter is solved using the iteratively reweighted least-squares algorithm that is known to exhibit a better convergence property and numerical stability than the Newton algorithms. Several numerical simulations using canonical models of dynamical systems demonstrate the excellent performance of the proposed RDMD method. The results reveal that it outperforms several other methods proposed in the literature.

We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems.

This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor $\dsig$ and displacement $\du$ inside each element, and the modifications of these two variables $\hsig$ and $\hu$ on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as $H^1$-based and the other known as $H({\rm div})$-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.

In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional derivative in standard approximation schemes, existing numerical analysis techniques do not apply directly. By an approximate inversion technique, the proposed model is transformed as a second kind Volterra integral equation, based on which a collocation method under uniform or graded mesh is developed and analyzed. In particular, the error estimates improve the existing results by proving a consistent and sharper mesh grading parameter and characterizing the convergence rates in terms of the initial value of the variable order, which demonstrates its critical role in determining the smoothness of the solutions and thus the numerical accuracy.

In this study, we focus on the modelling of coupled systems of shallow water flows and solute transport with source terms due to variable topography and friction effect. Our aim is to propose efficient and accurate numerical techniques for modelling these systems using unstructured triangular grids. We used a Riemann-solver free method for the hyperbolic shallow water system and a suitable discretization technique for the bottom topography. The friction source term is discretized using the techniques proposed by (Xia and Liang 2018). Our approach performs very well for stationary flow in the presence of variable topography, and it is well-balanced for the concentration in the presence of wet and dry zones. In our techniques, we used linear piecewise reconstructions for the variables of the coupled system. The proposed method is well-balanced, and we prove that it exactly preserves the nontrivial steady-state solutions of the coupled system. Numerical experiments are carried out to validate the performance and robustness of the proposed numerical method. Our numerical results show that the method is stable, well-balanced and accurate to model the coupled systems of shallow water flows and solute transport.