Temporal Grid Research Articles

Robust Portfolio Choice with Sticky Wages

We present a robust version of the life-cycle optimal portfolio choice problem in the presence of labor income, as introduced in Biffis, Gozzi, and Prosdocimi [SIAM J. Control Optim., 58 (2020), pp. 1906--1938] and Dybvig and Liu [J. Econom. Theory, 145 (2010), pp. 885--907]. In particular, in Biffis, Gozzi, and Prosdocimi the influence of past wages on the future ones is modeled linearly in the evolution equation of labor income, through a given weight function. The optimization relies on the resolution of an infinite dimensional HJB equation. We improve the state of art in three ways. First, we allow the weight to be a Radon measure. This accommodates for more realistic weighting of the sticky wages, e.g., on a discrete temporal grid according to some periodic income. Second, there is a general correlation structure between labor income and stock markets. This naturally affects the optimal hedging demand, which may increase or decrease according to the correlation sign. Third, we allow the weight to change with time, possibly lacking perfect identification. The uncertainty is specified by a given set of Radon measures K, in which the weight process takes values. This renders the inevitable uncertainty on how the past affects the future, and includes the standard case of error bounds on a specific estimate for the weight. Under uncertainty averse preferences, the decision maker takes a maxmin approach to the problem. Our analysis confirms the intuition: in the infinite dimensional setting, the optimal policy remains the best investment strategy under the worst case weight.

The decay of stably stratified grid turbulence in a viscosity-affected stratified flow regime

The decay of stably stratified turbulence generated by a towed rake of vertical plates is investigated by direct numerical simulations (DNS) of temporally evolving grid turbulence in a linearly stratified fluid. The Reynolds number $Re_M=U_0M/\nu$ is 5000 or 10 000 while the Froude number $Fr_M=U_0/MN$ is between 0.1 and 6 ( $U_0$ : towing speed; $M$ : mesh size; $\nu$ : kinematic viscosity; $N$ : Brunt–Väisälä frequency). The DNS results are compared with the theory of stably stratified axisymmetric Saffman turbulence. Here, the theory is extended to a viscosity-affected stratified flow regime with low buoyancy Reynolds number $Re_b$ , and power laws are derived for the temporal variations of the horizontal velocity scale ( $U_H$ ) and the horizontal and vertical integral length scales ( $L_H$ and $L_V$ ). Temporal grid turbulence initialized with the mean velocity deficit of wakes exhibits a $k^{2}$ energy spectrum at a low-wavenumber range and invariance of $U_H^2L_H^2L_V$ , which are the signatures of axisymmetric Saffman turbulence. The decay of various quantities follows the power laws predicted for low- $Re_b$ Saffman turbulence when $Fr_M$ is sufficiently small. However, the decay of $U_H^2$ at $Fr_M=6$ is no longer expressed by a power law with a constant exponent. This behaviour is related to the scaling of kinetic energy dissipation rate $\varepsilon$ , for which $\alpha =\varepsilon /(U_H^3/L_H)$ is constant during the decay for $Fr_M\leq 1$ while it varies with time for $Fr_M=6$ . We also examine the experimental data of towed-grid experiments by Praud et al. (J. Fluid Mech., vol. 522, 2005, pp. 1–33), which is shown to agree with the theory of low- $Re_b$ Saffman turbulence.

CO2 emissions from electric flying cars: Impacts from battery specific energy and grid emission factor

Electric flying cars are attracting global attentions considering their ability to provide high-efficiency point-to-point transportation. However, there is a lack of understanding on the environmental impacts from electric flying cars. In this study, by modeling flying processes and sizing components of flying cars, the well-to-wheel CO 2 emissions from electric flying cars are estimated. The results show that when assuming battery specific energy of 200 Wh/kg and grid emission factor in China's context, CO 2 emissions from 100 km-range-capacity electric flying cars are 66 g CO 2 /passenger-ground km eq., 11% and 62% higher than those from on-road internal combustion engine vehicles and battery electric vehicles. With battery specific energy increasing to 400 Wh/kg and 600 Wh/kg, CO 2 emissions can be reduced by 39% and 45%. Reduction in grid emission factor further contributes to emissions reduction, with emissions from electric flying cars lower than those from on-road internal combustion engine vehicles in multiple regional and temporal grid contexts. For electric flying cars with longer range capacities, more aggressive battery and grid improvements are needed to find breakeven with on-ground vehicles. This study justifies the critical roles next-generation batteries and clean power grid play in leveraging the development of electric flying cars, and calls for joint efforts from the government, industry and academia. • Life cycle CO 2 emissions from electric flying cars are estimated. • The methods for sizing components of flying cars are proposed. • Currently CO 2 from electric flying cars significantly exceed on-road vehicles. • Improvements in battery specific energy substantially reduce CO 2 emissions. • Clean power grid is essential for reducing emissions from electric flying cars.

Convergence of a spectral Galerkin scheme for the dynamic elastica with periodical or hinged boundary

For the dynamic elastica with the quasi-periodic or the hinged boundary conditions, a spectral Galerkin scheme is presented for approximating the smooth solution. It is established that the scheme enjoys a spectral and second order accuracy with respect to the size of the spatial and temporal grid, respectively. Moreover, some results of numerical experiments are given to examine the convergence of discrete solutions to the corresponding exact solutions.

Efficient explicit numerical solutions of the time-dependent Schrödinger equation.

Explicit numerical solutions of the time-dependent Schrödinger equationare more efficient than those obtained by commonly used implicit approaches. They are more practical, especially for a system with higher spatial dimensions. To that end, we introduce a generalization of an explicit three-level method to obtain solutions with spatial and temporal errors of the order of O[(Δx)^{2r}] and O[(Δt)^{2M+3}], where Δx and Δt are the spatial and temporal grid elements, and r and M are positive integers. Sample calculations illustrate the efficacy and stability of the algorithm.

Numerical solutions of time fractional Burgers’ equation involving Atangana–Baleanu derivative via cubic B-spline functions

The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers’ equation with Atangana–Baleanu derivative. A non-singular kernel is involved in the Atangana–Baleanu fractional derivative. For discretization along temporal and spatial grids, the proposed numerical technique employs the finite difference approach and cubic B-spline functions, respectively. This scheme is unconditionally stable and second order convergent in spatial and temporal directions. The presented approach is endorsed by some numerical examples, which show that it is applicable and accurate.

Wave Packet Control and Simulation Protocol for Entangled Two-Photon Absorption of Molecules.

Quantum light spectroscopy, providing novel molecular information nonaccessible by classical light, necessitates new computational tools when applied to complex molecular systems. We introduce two computational protocols for the molecular nuclear wave packet dynamics interacting with an entangled photon pair to produce an entangled two-photon absorption signal. The first involves summing over transition pathways in a temporal grid defined by two light-matter interaction times accompanied by the field correlation functions of quantum light. The signal is obtained by averaging over the two time distribution characteristics of the entangled photon state. The other protocol involves a Schmidt decomposition of the entangled light and requires summing over the Schmidt modes. We demonstrate how photon entanglement can be used to control and manipulate the two-photon excited nuclear wave packets in a displaced harmonic oscillator model.

Py-fmas: A python package for ultrashort optical pulse propagation in terms of forward models for the analytic signal

We present a flexible, open-source Python package for the accurate simulation of the z-propagation dynamics of ultrashort optical pulses in nonlinear waveguides, especially valid for few-cycle pulses and their interaction. The simulation approach is based on unidirectional propagation equations for the analytic signal. The provided software allows to account for dispersion, attenuation, four-wave mixing processes including, e.g., third-harmonic generation, and features various models for the Raman response. The propagation equations are solved on a periodic temporal domain. For z-propagation, a selection of pseudospectral methods is available. Propagation scenarios for a custom propagation constant and initial field pulses can either be specified in terms of a HDF5 based input file format or by direct implementation using a python script. We demonstrate the functionality for a test-case for which an exact solution is available, by reproducing exemplary results documented in the scientific literature, and a complex propagation scenario involving multiple pulses. Program summaryProgram Title: py-fmasCPC Library link to program files:https://doi.org/10.17632/7s2cv9kjfs.1Developer's repository link:https://github.com/omelchert/py-fmasCode Ocean capsule:https://codeocean.com/capsule/8221780Licensing provisions: MITProgramming language: Python3Supplementary material: Reference manual, extended user guide, and usage examples are hosted on gitHub pages under https://omelchert.github.io/py-fmas.Nature of problem: Solves for the z-propagation dynamics of spectrally broad ultrashort optical pulses in single mode nonlinear waveguides in terms of propagation models for the analytic signal of the optical field [1–3]. The implemented models include, e.g., third-harmonic generation and the Raman effect.Solution method: The initial real-valued optical field is defined on a periodic one-dimensional temporal grid and converted to the complex-valued analytic signal. z-stepping is performed via spectral methods. The software implements a selection of algorithms with fixed or adaptive stepsize, commonly used in nonlinear optics for solving nonlinear Schrödinger type equations.Additional comments including restrictions and unusual features: The range of applicability of the provided software is equivalent to that of the forward Maxwell equation [4]. For reasonably chosen initial conditions, it can be used beyond the unidirectional approximation as a bidirectional model for a complex field, allowing to describe forward and backward waves coupled through nonlinear interactions [1]. The software implements various models for the Raman response and allows to calculate spectrograms, detailing the time-frequency composition of the analytic signal. Additionally, a convenience class for analyzing propagation constants is provided.

Image processing algorithms application for the modeling biogeochemical cycles of the Azov Sea

In the article biogeochemical processes of the Azov Sea were researched. Mathematical non-stationary 3D model is proposed which describes the development dynamics of the two most common species of phytoplankton populations in the summer, the growth of which is limited by a single biogenic element, is proposed the linearization of continuous mathematical model on a uniform temporal grid is made. For a continuous model, a discrete analogue is constructed and an optimal method for grid equations solving is selected. To determine the boundary of the considered computational domain of a complex shape an image processing algorithm has been developed, implemented as a software module on Python, which makes it possible to obtain a dynamically changing contour of the Azov Sea from satellite images.

TiGrid: A raster‐based method for time‐interval information representation with geotagged videos as an example

AbstractSpatiotemporal representation is one of the critical areas receiving ongoing interest in GIScience. The ever‐growing spatiotemporal big data, however, pose considerable challenges to temporal representation as existing methods predominantly use one‐dimensional chronological timeline representations. Aiming to represent the lifespans of spatiotemporal objects, a temporal interval grid (TiGrid) is proposed based on a raster structure in two‐dimensional space. The TiGrid is the lower‐triangular part of a square matrix of order N, in which the column and row represent the start time and end time, respectively. Using the graphical transformation method, the lower‐triangular matrix of the TiGrid is transformed to a rectangle matrix. The cells in a TiGrid can be converted into absolute world time using metadata such as the time resolution and start time. The TiGrid generating algorithm, the temporal topological relations, and the unary or binary operations in a TiGrid are defined. Furthermore, spatiotemporal constraint and snapshot retrieval methods are introduced based on the spatial semantics in the TiGrid. A prototype performed evaluation with the case of geotagged video retrieval. The experimental results show that a TiGrid with spatial indexing can efficiently retrieve mass objects with time‐interval information. The TiGrid has the potential to be a new way for spatiotemporal modeling and geovisualization.

Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Spatial–temporal grid clustering method based on frequent stay point recognition

Spatial–temporal grid clustering method based on frequent stay point recognition

Increased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems

We discuss how “parallel-in-space & simultaneous-in-time” Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypical application, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finite difference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearized problems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems.

D'Alembert digitized: A wave pulse method for visualizing electromagnetic waves in matter and for deriving the finite difference time domain method for numerically solving Maxwell's equations

An alternative way of visualizing electromagnetic waves in matter and of deriving the Finite Difference Time Domain (FDTD) method for simulating Maxwell's equations for one-dimensional systems is presented. The method uses d'Alembert's splitting of waves into forward and backward pulses of arbitrary shape and allows for grid spacing and material properties that vary with the position. Constant velocity of waves in dispersionless dielectric materials, partial reflection and transmission at boundaries between materials with different indices of refraction, and partial reflection, transmission, and attenuation through conducting materials are derived without recourse to exponential functions, trigonometric functions, or complex numbers. Placing d'Alembert's method on a grid is shown to be equivalent to the FDTD method and allows for simple and visual proof that the FDTD method is exact for dielectrics when the ratio of the spatial and temporal grid spacing is the wave speed, a straightforward way to incorporate reflectionless boundary conditions and a derivation that the FDTD method retains second-order accuracy when the grid spacing varies with the position and the material parameters make sudden jumps across layer boundaries.

High‐order compact finite difference method for the multi‐term time fractional mixed diffusion and diffusion‐wave equation

In this paper, the multi‐term time fractional mixed diffusion and diffusion‐wave equation is investigated. Firstly, a compact finite difference scheme with fourth‐order spatial accuracy and high‐order temporal accuracy is derived. Then, the unconditional stability and convergence in the maximum norm of the derived high‐order compact finite difference method have been discussed rigorously by means of the energy method. Numerical experiments are presented to test the convergence order in the temporal and spatial direction, respectively. Additionally, to precisely demonstrate the computational efficiency of the derived high‐order compact finite difference method, the maximum norm error and the CPU time are measured in contrast with the second‐order finite difference scheme for the same temporal grid size. Finally, a practical example is presented to show the applicability of the time fractional mixed diffusion and diffusion‐wave model and the efficiency of the derived high‐order compact finite difference method.

A three-level linearized high-order accuracy difference scheme for the extended Fisher–Kolmogorov equation

A three-level linearized difference scheme for the extended Fisher–Kolmogorov equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergence order in maximum norm is $$O(h^4+k^2)$$ , where k and h are the temporal and spatial grid sizes, respectively. The accurateness and effectiveness of the method are tested by taking various examples. The numerical results of the method are compared with the exact solutions and also compared with earlier published results. It is found that the proposed method produces more accurate results than the others available in the literature.

An Efficient Linearly Convergent Regularized Proximal Point Algorithm for Fused Multiple Graphical Lasso Problems

Nowadays, analysing data from different classes or over a temporal grid has attracted a great deal of interest. As a result, various multiple graphical models for learning a collection of graphical models simultaneously have been derived by introducing sparsity in graphs and similarity across multiple graphs. This paper focuses on the fused multiple graphical Lasso model which encourages not only shared pattern of sparsity, but also shared values of edges across different graphs. For solving this model, we develop an efficient regularized proximal point algorithm, where the subproblem in each iteration of the algorithm is solved by a superlinearly convergent semismooth Newton method. To implement the semismooth Newton method, we derive an explicit expression for the generalized Jacobian of the proximal mapping of the fused multiple graphical Lasso regularizer. Unlike those widely used first order methods, our approach has heavily exploited the underlying second order information through the semismooth Newton method. This can not only accelerate the convergence of the algorithm, but also improve its robustness. The efficiency and robustness of our proposed algorithm are demonstrated by comparing with some state-of-the-art methods on both synthetic and real data sets. Supplementary materials for this article are available online.

Methods for frequency and correlation analyses of neurograms

Many physiological functions are based on motor rhythmic activities, among them breathing is a vital issue. The method presented here, or ‘temporal grid extraction’, aims at characterizing the temporal organization of such an activity. Beyond the measurement of the fundamental frequency, defining the successive cycles, some signal processing tools are helpful in order to look for the presence of higher frequency components that potentially structure these cycles. The method is applied to neurograms recorded from frog brainstem preparations, where two cycle types, buccal and lung cycles, may alternate. It relies on:•Continues Wavelet Transform (CWT) for time-frequency maps and frequency profiles•Crosscorrelation analysis for amplitude maps and amplitude profiles•Cycle-by-cycle autocorrelation analysis for autocorrelation maps and autocorrelation profilesUsing this method, the maps and profiles have revealed that a common high frequency clock drives both buccal and lung cycles.

Robust Portfolio Choice With Sticky Wages

We present a robust version of the life-cycle optimal portfolio choice problem in the presence of labor income, as introduced in Biffis, Gozzi and Prosdocimi (Optimal portfolio choice with path dependent labor income: the infinite horizon case, SIAM Journal on Control and Optimization, 58(4), 1906-1938.) and Dybvig and Liu (Lifetime consumption and investment: retirement and constrained borrowing, Journal of Economic Theory, 145, pp. 885-907). In particular, in Biffis, Gozzi and Prosdocimi the influence of past wages on the future ones is modelled linearly in the evolution equation of labor income, through a given weight function. The optimization relies on the resolution of an infinite dimensional HJB equation. We improve the state of art in three ways. First, we allow the weight to be a Radon measure. This accommodates for more realistic weighting of the sticky wages, like e.g. on a discrete temporal grid according to some periodic income. Second, there is a general correlation structure between labor income and stocks market. This naturally affects the optimal hedging demand, which may increase or decrease according to the correlation sign. Third, we allow the weight to change with time, possibly lacking perfect identification. The uncertainty is specified by a given set of Radon measures $K$, in which the weight process takes values. This renders the inevitable uncertainty on how the past affects the future, and includes the standard case of error bounds on a specific estimate for the weight. Under uncertainty averse preferences, the decision maker takes a maxmin approach to the problem. Our analysis confirms the intuition: in the infinite dimensional setting, the optimal policy remains the best investment strategy under the worst case weight.

Isochrony, vocal learning, and the acquisition of rhythm and melody.

A cross-species perspective can extend and provide testable predictions for Savage et al.'s framework. Rhythm and melody, I argue, could bootstrap each other in the evolution of musicality. Isochrony may function as a temporal grid to support rehearsing and learning modulated, pitched vocalizations. Once this melodic plasticity is acquired, focus can shift back to refining rhythm processing and beat induction.