Weak solutions of the forward problem in EEG for different conductivity values

The potential distribution on the scalp produced by current sources in the brain can be measured by an EEG recorder. The relationship between these sources and the scalp potential distribution may be described by a well-known mathematical model where some simplifications are usually introduced. The head is modeled as a multicompartment nested set and the conductivity of the different tissues is approximated by a positive piecewise constant function. This simplified model is used to solve the forward problem (FP), i.e., to calculate the scalp potential for a current source configuration. In this work, we prove that the weak solutions of the FP are continuous with respect to the conductivity values, that is, the difference between the scalp potentials is small if the conductivity values are closed enough. We present numerical examples that illustrates this property.     

The discrete forward–reserve problem – Allocating space,selecting products,and area sizing in forward order picking

To reduce labor-intensive and costly order picking activities, many distribution centers are subdivided into a forward area and a reserve (or bulk) area. The former is a small area where most popular stock keeping units (SKUs) can conveniently be picked, and the latter is applied for replenishing the forward area and storing SKUs that are not assigned to the forward area at all. Clearly, reducing SKUs stored in forward area enables a more compact forward area (with reduced picking effort) but requires a more frequent replenishment. To tackle this basic trade-off, different versions of forward–reserve problems determine the SKUs to be stored in forward area, the space allocated to each SKU, and the overall size of the forward area. As previous research mainly focuses on simplified problem versions (denoted as fluid models), where the forward area can continuously be subdivided, we investigate discrete forward–reserve problems. Important subproblems are defined and computation complexity is investigated. Furthermore, we experimentally analyze the model gaps between the different fluid models and their discrete counterparts.     

A coupled complex boundary method for an inverse conductivity problem with one measurement

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov regularization method can reconstruct the diffusion parameters stably and effectively.     

Study of a boundary value transmission problem for two-dimensional flows in a piecewise anisotropic inhomogeneous porous layer

We consider a boundary value transmission problem for two-dimensional filtration flows in an anisotropic porous layer consisting of adjacent domains in which the media have essentially different conductivities (permeability and thickness). In general, the layer conductivity is specified by a nonsymmetric second rank tensor whose components are modeled by continuously differentiable functions of coordinates. To study the problem, we use two complex planes, the physical plane and an auxiliary plane, which are related by a homeomorphic (one-to-one and continuous) transformation satisfying an equation of the Beltrami type. On the physical plane, we pose a transmission problem for a rather complicated elliptic system of equations. This problem is reduced on the auxiliary plane to canonical form, which dramatically simplifies the analysis of the problem. Then the problem is reduced to a system of boundary singular integral equations with generalized kernels of the Cauchy type, which are expressed via the fundamental solutions of the main equations. The boundary value transmission problem studied here can be used as a mathematical model of processes arising in the recovery of fluids (water and oil) from natural soil formations of complicated geological structure.     

Identification of an inclusion in multifrequency electric impedance tomography

The multifrequency electrical impedance tomography is considered to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation, we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem since in general infinitely many input currents are needed to obtain the uniqueness in the determination of the conductivity.     

The exterior magnetic field for the multilayer ellipsoidal model of the brain

The magnetic induction field in the exterior of an ellipsoidallyinhomogeneous, four-conducting-layer model of the human headis obtained analytically up to its quadrupole approximation.The interior ellipsoidal core represents the homogeneous brainwhile each one of the shells represents the cerebrospinal fluid,the skull and the scalp, all characterized by different conductivities.The inhomogeneities of these four domains, together with theanisotropy imposed by the use of the ellipsoidal geometry, providethe most realistic physical and geometrical model of the brainfor which an analytic solution of the biomagnetic forward problemis possible. It is shown that in contrast to the spherical model,where shells of different conductivity are magnetically invisible,the magnetic induction field in ellipsoidal geometry is stronglydependent on the conductivity supports. The fact that sphericalshells of different conductivity are invisible has enhancedthe common belief that the biomagnetic forward solution doesnot depend on the conductivity profiles. As we demonstrate inthe present work, this is not true. Hence, the proposed multilayeredellipsoidal model provides a qualitative improvement of therealistic interpretation of magnetoencephalography (MEG) measurements.We show that the presence of the shells of different conductivitycan be incorporated in the form of the dipole vector for thehomogeneous model. Numerical investigations show that the effectsof shell inhomogeneities are almost as sound as the level ofMEG measurements themselves. The degenerate cases, where eitherthe differences of the conductivities within the shells disappear,or the ellipsoidal geometry is reduced to the spherical one,are also considered.     

New exact multi-coated ellipsoidal inclusion model for anisotropic thermal conductivity of composite materials

The present study deals with a new micromechanical modeling of the thermal conductivity of multi-coated inclusion-reinforced composites. The proposed approach has been developed in the general frame of anisotropic thermal behavior per phase and arbitrary ellipsoidal inclusions. Based on the Green's function technique, a new formulation of the problem of multi-coated inclusion is proposed. This formulation consists in constructing a system of integral equations, each associated to the thermal conductivity of each coating and the reference medium. Thanks to the concept of interior- and exterior-point Eshelby's conduction tensors, the exact solution of the problem of multicoated inclusion is obtained. Analytical expressions of the intensity in each phase and the effective thermal conductivity of the composite, through homogenizations schemes such as Generalized self-consistent and Mori-Tanaka models are provided. Results of the present model are successfully compared with those issued from both analytical models and finite elements methods for composites with doubly coated inclusions. Moreover, the developed micromechanical model has been applied to a three phase composite materials in order to analyze combined effects of the aspect ratio and the volume fraction of the ellipsoidal inclusions, the anisotropy of the thermal conductivity of interphase, the thermal conductivity contrast between local phases on the predicted effective thermal conductivity.     

Dynamic mean-variance problem with constrained risk control for the insurers

In this paper, we study optimal reinsurance/new business and investment (no-shorting) strategy for the mean-variance problem in two risk models: a classical risk model and a diffusion model. The problem is firstly reduced to a stochastic linear-quadratic (LQ) control problem with constraints. Then, the efficient frontiers and efficient strategies are derived explicitly by a verification theorem with the viscosity solutions of Hamilton–Jacobi–Bellman (HJB) equations, which is different from that given in Zhou et al. (SIAM J Control Optim 35:243–253, 1997). Furthermore, by comparisons, we find that they are identical under the two risk models. This work was supported by National Basic Research Program of China (973 Program) 2007CB814905 and National Natural Science Foundation of China (10571092).     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

Averaging the electrical properties of fiber-reinforced metal composites

Using the regular structure model, we average the electrical properties of unidirectional fiber-reinforced metal composites and propose procedures for determination of the effective electrical conductivity tensor of these materials. For the general case of packing of fibers of arbitrary cross section, the problem is reduced to calculation of some functionals determined in solutions of the integral equation of the corresponding boundary current problem for the structure. In the special case of symmetric packing of fibers of circular cross section, the solution is written in the form of series in elliptic functions.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 533–539, July–August, 1995.     

Numerical analysis of quenching – Heat conduction in metallic materials

Metallic materials present a complex behavior during heat treatment processes. In a certain temperature range, change of temperature induces a phase transformation of metallic structure, which alters physical properties of the material. Indeed, measurements of specific heat and conductivity show strong temperature-dependence during processes such as quenching of steel. Several mathematical models, as solid mixtures and thermal–mechanical coupling, for problems of heat conduction in metallic materials, have been proposed. In this work, we take a simpler approach without thermal–mechanical coupling of deformation, by considering the nonlinear temperature-dependence of thermal parameters as the sole effect due to those complex behaviors. The above discussion of phase transformation of metallic materials serves only as a motivation for the strong temperature-dependence as material properties. In general, thermal properties of materials do depend on the temperature, and the present formulation of heat conduction problem may be served as a mathematical model when the temperature-dependence of material parameters becomes important. For this mathematical model we present the error estimate using the finite element method for the continuous-time case.     

通胀市场下的风险补偿在线租赁策略设计

考虑到实际租赁市场中设备的租赁和购买价格随着时间推移持续上涨的特征,研究了通货膨胀市场中的租赁问题在有利率情形下的在线竞争策略,并建立了相应的概率预期的风险补偿模型。首先运用在线算法和竞争分析理论设计了该问题的最优竞争策略,并给出最优竞争比。接着,在风险补偿竞争分析框架下,进一步讨论该问题,投资者可以控制风险,根据自己不同的风险容忍度和未来预期选择补偿最大的租赁策略;也可以根据给定的补偿收益约束选择风险最小的策略。最后通过数值分析,验证了相关结论的正确性,也表明了基于概率预期的风险补偿策略大大改善了该租赁问题的竞争比性能。     

Engineering graph-based models for dynamic timetable information systems

In the last years we have witnessed remarkable progress in providing efficient algorithmic solutions to the problem of computing best journeys (or routes) in schedule-based public transportation systems. We have now models to represent timetables that allow us to answer queries for optimal journeys in a few milliseconds, also at a very large scale. Such models can be classified into two types: those representing the timetable as an array, and those representing it as a graph. Array-based models have been shown to be very effective in terms of query time, while graph-based ones usually answer queries by computing shortest paths, and hence they are suitable to be combined with the speed-up techniques developed for road networks.In this paper, we study the behavior of graph-based models in the prominent case of dynamic scenarios, i.e., when delays might occur to the original timetable. In particular, we make the following contributions. First, we consider the graph-based reduced time-expanded model and give a simplified and optimized routine for handling delays, and a re-engineered and fine-tuned query algorithm. Second, we propose a new graph-based model, namely the dynamic timetable model, natively tailored to efficiently incorporate dynamic updates, along with a query algorithm and a routine for handling delays. Third, we show how to adapt the ALT algorithm to such graph-based models. We have chosen this speed-up technique since it supports dynamic changes, and a careful implementation of it can significantly boost its performance. Finally, we provide an experimental study to assess the effectiveness of all proposed models and algorithms, and to compare them with the array-based state of the art solution for the dynamic case. We evaluate both new and existing approaches by implementing and testing them on real-world timetables subject to synthetic delays.Our experimental results show that: (i) the dynamic timetable model is the best model for handling delays; (ii) graph-based models are competitive to array-based models with respect to query time in the dynamic case; (iii) the dynamic timetable model compares favorably with both the original and the reduced time-expanded model regarding space; (iv) combining the graph-based models with speed-up techniques designed for road networks, such as ALT, is a very promising approach.     

A penalized criterion for variable selection in classification

In this paper, the problem of variable selection in classification is considered. On the basis of recent developments in model selection theory, we provide a criterion based on penalized empirical risk, where the penalization explicitly takes into account the number of variables of the considered models. Moreover, we give an oracle-type inequality that non-asymptotically guarantees the performance of the resulting classification rule. We discuss the optimality of the proposed criterion and present an application of the main result to backward and forward selection procedures.     

Robust reward–risk ratio optimization with application in allocation of generation asset

In this article, we study reward–risk ratio models under partially known message of random variables, which is called robust (worst-case) performance ratio problem. Based on the positive homogenous and concave/convex measures of reward and risk, respectively, the new robust ratio model is reduced equivalently to convex optimization problems with a min–max optimization framework. Under some specially partial distribution situation, the convex optimization problem is converted into simple framework involving the expectation reward measure and conditional value-at-risk measure. Compared with the existing reward–risk portfolio research, the proposed ratio model has two characteristics. First, the addressed problem combines with two different aspects. One is to consider an incomplete information case in real-life uncertainty. The other is to focus on the performance ratio optimization problem, which can realize the best balance between the reward and risk. Second, the complicated optimization model is transferred into a simple convex optimization problem by the optimal dual theorem. This indeed improves the usability of models. The generation asset allocation in power systems is presented to validate the new models.     

Modeling a swimming fish with an initial boundary value problem: Unsteady maneuvers of an elastic plate with internal force generation

In order to model unsteady maneuvers in swimming fish, we develop an initial-boundary value problem for a fourth-order hyperbolic partial differential equation in which the fish's body is treated as an inhomogeneous elastic plate. The model is derived from the three-dimensional equations of elastic dynamics, and is essentially a simple variant of the classical Kirchhoff model for a dynamic plate. The model incorporates body forces generating moment to simulate muscle force generation in fish. The initial-boundary value problem is reduced to a beam model in one spatial dimension and formulated computationally using finite differences. Interaction with the surrounding water is represented by nonlinear viscous damping. Two example applications using simple but physically reasonable physiological parameters are presented and interpreted. One models the acceleration from rest to steady swimming, the other a rapid turn from rest.     

基于Logit离散选择模型的品类优化问题综述

品类优化问题(Assortment Optimization Problem)是收益管理的经典问题.它研究零售商在满足运营约束的前提下,应如何从给定产品集合中选择一个子集提供给消费者,以最大化预期收益.该问题的核心在于如何准确地刻画消费者在面对细分产品时的选择行为、建立相应的优化模型并设计高效率的求解算法.基于Logit离散选择模型的品类优化问题:首先,介绍了基于Multinomial Logit模型的品类优化问题.然后介绍了两个更复杂的变种:第一个是基于两层以及多层Nested Logit模型的品类优化问题,这类问题可合理刻画细分产品之间的"替代效应";第二个是基于Mixtures of Multinomial Logits模型的品类优化问题,这类问题可充分考虑消费者群体的异质性.随后,介绍了数据驱动的品类优化问题的相关进展.最后,指出该问题未来可能的若干研究方向.     

The inverse scattering problem for a partially coated penetrable cavity with interior measurements

In this paper we consider the inverse scattering problem for a cavity that is bounded by a partially coated penetrable inhomogeneous medium of compact support and recover the shape of the cavity and the surface conductivity from a knowledge of measured scattered waves due to point sources located on a curve or surface inside the cavity. First, we prove that both the shape of the cavity and the surface conductivity on the coated part can be uniquely determined from a knowledge of the measured data. Next, we establish a linear sampling method for determining both the shape of the cavity and the surface conductivity. A central role in our justification is played by an eigenvalue problem which we call the exterior transmission eigenvalue problem. Finally, we present some numerical examples to illustrate the validity of our method.     

On the thermodynamic limit for Hartree–Fock type models

We continue here our study [10–13] of the thermodynamic limit for various models of Quantum Chemistry, this time focusing on the Hartree–Fock type models. For the reduced Hartree–Fock models, we prove the existence of the thermodynamic limit for the energy per unit volume. We also define a periodic problem associated to the Hartree–Fock model, and prove that it is well-posed.     

Distributionally robust scheduling on parallel machines under moment uncertainty

This paper investigates a distributionally robust scheduling problem on identical parallel machines, where job processing times are stochastic without any exact distributional form. Based on a distributional set specified by the support and estimated moments information, we present a min-max distributionally robust model, which minimizes the worst-case expected total flow time out of all probability distributions in this set. Our model doesn’t require exact probability distributions which are the basis for many stochastic programming models, and utilizes more information compared to the interval-based robust optimization models. Although this problem originates from the manufacturing environment, it can be applied to many other fields when the machines and jobs are endowed with different meanings. By optimizing the inner maximization subproblem, the min-max formulation is reduced to an integer second-order cone program. We propose an exact algorithm to solve this problem via exploring all the solutions that satisfy the necessary optimality conditions. Computational experiments demonstrate the high efficiency of this algorithm since problem instances with 100 jobs are optimized in a few seconds. In addition, simulation results convincingly show that the proposed distributionally robust model can hedge against the bias of estimated moments and enhance the robustness of production systems.