We present a Rayleigh–Ritz method for the approximation of fluid flow in a curved duct, including the secondary cross-flow, which is well known to develop for nonzero Dean numbers. Having a straightforward method to estimate the cross-flow for ducts with a variety of cross-sectional shapes is important for many applications. One particular example is in microfluidics where curved ducts with low aspect ratio are common, and there is an increasing interest in nonrectangular duct shapes for the purpose of size-based cell separation. We describe functionals which are minimized by the axial flow velocity and cross-flow stream function which solve an expansion of the Navier–Stokes model of the flow. A Rayleigh–Ritz method is then obtained by computing the coefficients of an appropriate polynomial basis, taking into account the duct shape, such that the corresponding functionals are stationary. Whilst the method itself is quite general, we describe an implementation for a particular family of duct shapes in which the top and bottom walls are described by a polynomial with respect to the lateral coordinate. Solutions for a rectangular duct and two nonstandard duct shapes are examined in detail. A comparison with solutions obtained using a finite-element method demonstrates the rate of convergence with respect to the size of the basis. An implementation for circular cross-sections is also described, and results are found to be consistent with previous studies.

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

Explicit solutions are rarely available for water wave scattering problems. An analytical procedure is presented here to solve the boundary value problem associated with wave scattering by a complete vertical porous barrier with two gaps in it. The original problem is decomposed into four problems involving vertical solid barriers. The decomposed problems are solved analytically by using a weakly singular integral equation. Explicit expressions are obtained for the scattering amplitudes and numerical results are presented. The results obtained can be used as a benchmark for other wave scattering problems involving complex geometrical structures.

We study the problem of choosing the best subset of $p$ features in linear regression, given $n$ observations. This problem naturally contains two objective functions including minimizing the amount of bias and minimizing the number of predictors. The existing approaches transform the problem into a single-objective optimization problem. We explain the main weaknesses of existing approaches and, to overcome their drawbacks, we propose a bi-objective mixed integer linear programming approach. A computational study shows the efficacy of the proposed approach.

We propose a new adaptive and composite Barzilai–Borwein (BB) step size by integrating the advantages of such existing step sizes. Particularly, the proposed step size is an optimal weighted mean of two classical BB step sizes and the weights are updated at each iteration in accordance with the quality of the classical BB step sizes. Combined with the steepest descent direction, the adaptive and composite BB step size is incorporated into the development of an algorithm such that it is efficient to solve large-scale optimization problems. We prove that the developed algorithm is globally convergent and it R-linearly converges when applied to solve strictly convex quadratic minimization problems. Compared with the state-of-the-art algorithms available in the literature, the proposed step size is more efficient in solving ill-posed or large-scale benchmark test problems.

We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results.