This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.

We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65 and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder wants to maximize total withdrawals and minimize expected shortfall at the end of the retirement horizon of 30 years (i.e., it is assumed that the holder survives to age 95). The holder controls the amount withdrawn each year and the fraction of the retirement portfolio invested in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and the solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay dramatically outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.

We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation (Waring and Siegel (2015) Financial Analysts Journal, 71(1), 91–107) to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combination of expected total withdrawals (EW) and expected shortfall (ES), defined as the average of the worst 5% of the outcomes of real terminal wealth. We compare the performance of our dynamic strategy to simpler alternatives which maintain constant asset allocation weights over time accompanied by either our same modified ARVA spending rule or withdrawals that are constant over time in real terms. Tests are carried out using both a parametric model of historical asset returns as well as bootstrap resampling of historical data. Consistent with previous literature that has used different measures of reward and risk than EW and ES, we find that allowing some variability in withdrawals leads to large improvements in efficiency. However, unlike the prior literature, we also demonstrate that further significant enhancements are possible through incorporating a dynamic asset allocation strategy rather than simply keeping asset allocation weights constant throughout retirement.

We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.

Incompressible flows with zero Reynolds number can be modeled by the Stokes equations. When numerically solving the Stokes flow in stream-vorticity formulation with high-order accuracy, it will be important to solve both the stream function and velocity components with the high-order accuracy simultaneously. In this work, we will develop a fifth-order spectral/combined compact difference (CCD) method for the Stokes equation in stream-vorticity formulation on the polar geometries, including a unit disk and an annular domain. We first use the truncated Fourier series to derive a coupled system of singular ordinary differential equations for the Fourier coefficients, then use a shifted grid to handle the coordinate singularity without pole condition. More importantly, a three-point CCD scheme is developed to solve the obtained system of differential equations. Numerical results are presented to show that the proposed spectral/CCD method can obtain all physical quantities in the Stokes flow, including the stream function and vorticity function as well as all velocity components, with fifth-order accuracy, which is much more accurate and efficient than low-order methods in the literature.

The numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the unbounded domain is thus reduced to an initial boundary value problem on a bounded computational domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.

In this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

Hexagonal grids are valuable in two-dimensional applications involving Laplacian. The methods and analysis are investigated in current work in both linear and nonlinear problems related to anisotropic Laplacian. Ordinary and compact hexagonal grid finite difference methods are developed by elementary arguments, and then analyzed by perturbation for standard Laplacian. In the anisotropic case, analysis is done through reduction to the standard one by using Fourier vectors of mixed types. These hexagonal seven-point methods, with established theoretic stabilities and accuracies, are numerically confirmed in linear and semi-linear anisotropic Poisson problems, and can be applied also in time-dependent problems and in many applications in two-dimensional irregular domains.

This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.

The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

In this paper, we consider the transform magnetic (TM) model of electromagnetic scattering in the cavity. By the Polynomial Preserving Recovery technique, we present superconvergence analysis for the vertex-edge-face type finite element. From the numerical example, we can see that the provided method is efficient and stable.

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do not involve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight post-shock oscillations, we design fifth order fixed-point sweeping WENO methods for efficient computation of steady state solution of hyperbolic conservation laws. Especially, we show that although the methods do not have linear computational complexity, they converge to steady state solutions much faster than regular time-marching approach by stability improvement for high order schemes with a forward Euler time-marching.

The differential capacitance of electric double-layer capacitors is studied by developing a generalized model of the self-consistent Gaussian field theory. This model includes many-body effects of particles near the interface such as ionic sizes, the order of water alignment and electrostatic correlations, and thus can present more accurate predictions of the electric double-layer structure and hence the capacitance than traditional continuum theories. Analytical simplification of the model and efficient numerical method are introduced, in particular, the approximation of the self-Green's function which describes the self energy of a mobile ion. We show that, when the applied voltage on interfaces is small the dielectric effect of the electrode materials plays an important role. For large voltage, this effect is screened, but the dielectric saturation due to the alignment of the nearby water is shown to be essential. For 2:1 electrolytes, abnormal enhancement on the capacitance due to the dielectric electrode is observed, which is due to the interplay of the image charge effect and Born solvation energy in the self energy of ions.

In this paper, we propose a novel and simple technique to construct effective difference schemes for solving systems of singularly perturbed convection-diffusion-reaction equations, whose solutions may display boundary or interior layers. We illustrate the technique by taking the Il'in-Allen-Southwell scheme for 1-D scalar equations as a basis to derive a formally second-order scheme for 1-D coupled systems and then extend the scheme to 2-D case by employing an alternating direction approach. Numerical examples are given to demonstrate the high performance of the obtained scheme on uniform meshes as well as piecewise-uniform Shishkin meshes.

The maximal return and optimal leverage of a constant proportion debt obligation with finite termination and two boundaries are analysed by numerically solving Hamilton–Jacobi–Bellman equations. We discuss the probabilities of the asset value reaching the upper or lower bound under the optimal control and the optimal control problem with a time-varying boundary. Furthermore, we also analyse the relationship between the optimal return, the optimal policy and different parameters.

Nonnegative directional splittings of anisotropic diffusion operators in the divergence form are investigated. Conditions are established for nonnegative directional splittings to hold in a neighborhood of an arbitrary interior point. The result is used to construct monotone finite difference schemes for the boundary value problem of anisotropic diffusion operators. It is shown that such a monotone scheme can be constructed if the underlying diffusion matrix is continuous on the closure of the physical domain and symmetric and uniformly positive definite on the domain, the mesh spacing is sufficiently small, and the size of finite difference stencil is sufficiently large. An upper bound for the stencil size is obtained, which is determined completely by the diffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the larger stencil is required. An exception is the situation with a strictly diagonally dominant diffusion matrix where a three-by-three stencil is sufficient for the construction of a monotone finite difference scheme. Numerical examples are presented to illustrate the theoretical findings.

In this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order