A log-robust optimization approach to portfolio management

We present a robust optimization approach to portfolio management under uncertainty that builds upon insights gained from the well-known Lognormal model for stock prices, while addressing the model’s limitations, in particular, the issue of fat tails being underestimated in the Gaussian framework and the active debate on the correct distribution to use. Our approach, which we call Log-robust in the spirit of the Lognormal model, does not require any probabilistic assumption, and incorporates the randomness on the continuously compounded rates of return by using range forecasts and a budget of uncertainty, thus capturing the decision-maker’s degree of risk aversion through a single, intuitive parameter. Our objective is to maximize the worst-case portfolio value (over a set of allowable deviations of the uncertain parameters from their nominal values) at the end of the time horizon in a one-period setting; short sales are not allowed. We formulate the robust problem as a linear programming problem and derive theoretical insights into the worst-case uncertainty and the optimal allocation. We then compare in numerical experiments the Log-robust approach with the traditional robust approach, where range forecasts are applied directly to the stock returns. Our results indicate that the Log-robust approach significantly outperforms the benchmark with respect to 95 or 99% Value-at-Risk. This is because the traditional robust approach leads to portfolios that are far less diversified.     

On Determination of the Stressed State of a Annular Orthotropic Plate

We propose a method of solving a plane problem for thin orthotropic plates using a parameter expansion technique. A solution to the respective isotropic problem serves as a null approximation. For particular materials, the results obtained by this method are shown to agree well with the known solution for an infinite plate with a hole. We study the stressed state of an annular orthotropic plate, derive the stress distribution functions, and compare the results obtained with those available for a similar isotropic plate.     

线性流形上复对称矩阵的最小二乘问题

本文主要讨论了线性流形上复对称矩阵的最小二乘问题。在推导出所给线性流形中任意矩阵的显式表达的基础上,利用奇异值分解和Frobenius范数的酉不变性得到了该最小二乘问题通解的一般表达式。此外,文章还考虑了任一给定矩阵对此最小二乘问题解集合的最佳逼近问题,证明了该最佳逼近问题存在唯一解,并利用酉矩阵的性质得到了最佳逼近解的表达式。     

A ‘best points’ interpolation method for efficient approximation of parametrized functions

We present an interpolation method for efficient approximation of parametrized functions. The method recognizes and exploits the low‐dimensional manifold structure of the parametrized functions to provide good approximation. Basic ingredients include a specific problem‐dependent basis set defining a low‐dimensional representation of the parametrized functions, and a set of ‘best interpolation points’ capturing the spatial‐parameter variation of the parametrized functions. The best interpolation points are defined as solution of a least‐squares minimization problem which can be solved efficiently using standard optimization algorithms. The approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure. In addition, an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost. Numerical results are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2007 John Wiley & Sons, Ltd.     

The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem

We investigate an application of the method of fundamental solutions (MFS) to the time-dependent two-dimensional Cauchy heat conduction problem, which is an inverse ill-posed problem. Data in the form of the solution and its normal derivative is given on a part of the boundary and no data is prescribed on the remaining part of the boundary of the solution domain. To generate a numerical approximation we generalize the work for the stationary case in Marin (2011) [23] to the time-dependent setting building on the MFS proposed in Johansson and Lesnic (2008) [15], for the one-dimensional heat conduction problem. We incorporate Tikhonov regularization to obtain stable results. The proposed approach is flexible and can be adjusted rather easily to various solution domains and data. An additional advantage is that the initial data does not need to be known a priori, but can be reconstructed as well.     

The MFS–MPS for two-dimensional steady-state thermoelasticity problems

We consider the numerical approximation of the boundary and internal thermoelastic fields in the case of two-dimensional isotropic linear thermoelastic solids by combining the method of fundamental solutions (MFS) with the method of particular solutions (MPS). A particular solution of the non-homogeneous equations of equilibrium associated with a planar isotropic linear thermoelastic material is derived from the MFS approximation of the boundary value problem for the heat conduction equation. Moreover, such a particular solution enables one to easily develop analytical solutions corresponding to any two-dimensional domain occupied by an isotropic linear thermoelastic solid. The accuracy and convergence of the proposed MFS–MPS procedure are validated by considering three numerical examples.     

Banach空间二阶积分-微分方程初值问题的唯一解

本文在一般Banach空间中利用半序方法和一个新的比较结果,研究了二阶积分-微分方程初值问题的唯一解。仅使用了一个上解或下解,在比较广泛的上控制条件下得到了显形式表达的逼近解的迭代序列及误差估计,本文没有使用任何紧性条件,改进并推广了最近的一些结果。     

Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs

In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.     

Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation

We investigate in this paper a Cauchy problem for the time-fractional diffusion equation (TFDE). Based on the idea of kernel-based approximation, we construct an efficient numerical scheme for obtaining the solution of a Cauchy problem of TFDE. The use of M-Wright functions as the kernel functions for the approximation space allows us to express the solution in terms of M-Wright functions, whose numerical evaluation can be accurately achieved by applying the inverse Laplace transform technique. To handle the ill-posedness of the resultant coefficient matrix due to the noisy Cauchy data, we adapt the standard Tikhonov regularization technique with the L-curve method for obtaining the optimal regularization parameter to give a stable numerical reconstruction of the solution. Numerical results indicate the efficiency and effectiveness of the proposed scheme.     

On discrete lot streaming in no-wait flow shops

Lot streaming is the process of splitting a job or lot to allow overlapping between successive operations in a multistage production system. This use of transfer lots usually results in a substantially shorter makespan for the corresponding schedule. In this paper, we study the discrete lot streaming problem for a single job in no-wait flow shops. We present a new linear programming formulation for the problem. We show that the optimal solutions are the same for the m ×2 case with or without no-wait constraints. We also present a fast, polynomial-time solution method for this case. For the general case, we prove that any solution which is 'close' to the continuous optimal solution will be a good approximation for the discrete problem. This property allows us to present two quickly obtainable approximations of very good quality.     

Parametric optimal control of uncertain systems under an optimistic value criterion

It is well known that the optimal control of a linear quadratic model is characterized by the solution of a Riccati differential equation. In many cases, the corresponding Riccati differential equation cannot be solved exactly such that the optimal feedback control may be a complex time-oriented function. In this article, a parametric optimal control problem of an uncertain linear quadratic model under an optimistic value criterion is considered for simplifying the expression of optimal control. Based on the equation of optimality for the uncertain optimal control problem, an approximation method is presented to solve it. As an application, a two-spool turbofan engine optimal control problem is given to show the utility of the proposed model and the efficiency of the presented approximation method.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

Developments of the method of difference potentials for linear elastic fracture mechanics problems

Recently, the method of difference potentials has been extended to linear elastic fracture mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the extended finite element method, were introduced to improve the approximation near the crack tip leading to near‐optimal convergence rates. Now, the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has been proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near‐optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the fast Fourier transform is provided with the aim of further increasing the efficiency of the method.     

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

The aim of this paper is to present a new semi‐analytic numerical method for strongly nonlinear steady‐state advection‐diffusion‐reaction equation (ADRE) in arbitrary 2‐D domains. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of an analytic basis function and a special correcting function which is chosen to satisfy the homogeneous boundary conditions of the problem. The polynomials, trigonometric functions, conical radial basis functions, and the multiquadric radial basis functions are used in approximation of the ADRE. This allows us to seek an approximate solution in the analytic form which satisfies the boundary conditions of the initial problem with any choice of free parameters. As a result, we separate the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain. The numerical examples confirm the high accuracy and efficiency of the proposed method in solving strongly nonlinear equations in an arbitrary domain.     

A stochastic programming approach to manufacturing flow control

This paper proposes and tests an approximation of the solution of a class of piecewise deterministic control problems, typically used in the modeling of manufacturing flow processes. This approximation uses a stochastic programming approach on a suitably discretized and sampled system. The method proceeds through two stages: (i) the Hamilton-Jacobi-Bellman (HJB) dynamic programming equations for the finite horizon continuous time stochastic control problem are discretized over a set of sampled times; this defines an associated discrete time stochastic control problem which, due to the finiteness of the sample path set for the Markov disturbance process, can be written as a stochastic programming problem; and (ii) the very large event tree representing the sample path set is replaced with a reduced tree obtained by randomly sampling over the set of all possible paths. It is shown that the solution of the stochastic program defined on the randomly sampled tree converges toward the solution of the discrete time control problem when the sample size increases to infinity. The discrete time control problem solution converges to the solution of the flow control problem when the discretization mesh tends to zero. A comparison with a direct numerical solution of the dynamic programming equations is made for a single part manufacturing flow control model in order to illustrate the convergence properties. Applications to larger models affected by the curse of dimensionality in a standard dynamic programming techniques show the possible advantages of the method.     

Thermal stressed state of a conducting ball under the action of pulsed electromagnetic fields

We pose the dynamic centrally symmetric problem of thermomechanics for a continuous conducting ball subjected to the nonstationary uniform electromagnetic action and propose a method for its solution based on the use of the cubic approximation of the azimuthal component of the magnetic vector and the radial component of the stress tensor in the radial coordinate. The solution of the problem is obtained and the thermal stressed state of the ball under the action of electromagnetic pulses is numerically investigated.     

Reconstructing a thin absorbing obstacle in a half-space of tissue

We solve direct and inverse obstacle-scattering problems in a half-space composed of a uniform absorbing and scattering medium. Scattering is sharply forward-peaked, so we use the modified Fokker-Planck approximation to the radiative transport equation. The obstacle is an absorbing inhomogeneity that is thin with respect to depth. Using the first Born approximation, we derive a method to recover the depth and shape of the absorbing obstacle. This method requires only plane-wave illumination at two incidence angles and a detector with a fixed numerical aperture. First we recover the depth of the obstacle through solution of a simple nonlinear least-squares problem. Using that depth, we compute a point-spread function explicitly. We use that point-spread function in a standard deconvolution algorithm to reconstruct the shape of the obstacle. Numerical results show the utility of this method even in the presence of measurement noise.     

Solution of the Helmholtz eigenvalue problem via the boundary element method

The numerical solution of the Helmholtz eigenvalue problem is considered. The application of the boundary element method reduces it to that of a non-linear eigenvalue problem. Through a polynomial approximation with respect to the wavenumber, the non-linear eigenvalue problem is reduced to a standard generalized eigenvalue problem. The method is applied to the test problems of a three-dimensional sphere with an axisymmetric boundary condition and a two-dimensional square.     

An exact solution of the first-exit time problem for a class of structural systems

The mean time to escape from a region of desired operations is one the basic reliability measures in stochastic dynamics. In general, a precise solution of the first-exit time problem is unavailable. This paper demonstrates an exact solution of the mean exit time problem for a multidimensional non-dissipative Lagrangian system excited by additive Gaussian white noise. We identify the Fokker–Planck equation whose solution characterizes the mean time needed to reach a critical energy and explicitly construct the solution. For illustration, we apply the developed theory to engineering examples. We calculate the mean time of the standard operation for a flexural nanotube with likely noise-induced buckling and analyze the mean time of the stable functioning for a gyroscope subjected to random and dissipation torques. It is demonstrated that the solution of the first-exit time problem for a non-dissipative system gives a quite good approximation to a numerical solution of a similar problem for a system with small dissipation.