Backward Recursion Research Articles

Arbitrarily Parallel Turbo Decoding for Ultra-Reliable Low Latency Communication in 3GPP LTE

In order to meet the latency requirements of the ultra-reliable low latency communication (URLLC) mode of the third-generation partnership project’s long term evolution (LTE) mobile communication standard, this paper proposes a novel turbo decoding algorithm that supports an arbitrarily high degree of parallel processing, facilitating significantly higher processing throughputs and substantially lower processing latencies than the state-of-the-art (SOTA) LTE turbo decoder. As in conventional turbo decoding algorithms, the proposed Arbitrarily Parallel Turbo Decoder (APTD) decomposes each frame of information bits into a sequence of windows, where the bits within different windows are processed simultaneously using forward and backward recursions in a serial manner. However, in contrast to conventional turbo decoding algorithms, the APTD does not require different windows to be composed of an identical number of bits, which allows the use of an arbitrary number of windows and hence an arbitrary degree of parallelism, when decoding information bits of an arbitrary frame length. Furthermore, conventional turbo decoding algorithms alternate between simultaneously processing the windows in the upper decoder and those in the lower decoder. By contrast, the APTD processes the odd-indexed windows in the upper decoder at the same time as the even-indexed windows in the lower decoder and alternates between this and the reversed arrangement, hence further improving the decoding throughput and latency. Furthermore, the APTD achieves a reduced hardware resource requirement by calculating the extrinsic information based only on the outputs of the forward recursions, rather than based on both the forward and backward recursions of conventional turbo decoding algorithms. We demonstrate that the proposed APTD achieves superior latency, throughput, and computational efficiency than the SOTA LTE turbo decoder at all frame lengths, but particularly at the short frame lengths that are typically used in URLLC approaches. For example, at a frame length of $N={504}$ bits, the proposed APTD achieves an FER of 10−5 at the same $E_{b}/N_{0}$ as $I=8$ iterations of a conventional turbo decoder but with a computational efficiency that is 6 times higher than that of the SOTA turbo decoder, while achieving a latency and throughput that are 0.7 and 1.4 times those of the SOTA decoder, respectively.

Stochastic Gradient Estimation for Artificial Neural Networks

We investigate a new approach to compute the gradients of artificial neural networks (ANNs), based on the so-called push-out likelihood ratio method. Unlike the widely used backpropagation (BP) method that requires continuity of the loss function and the activation function, our approach bypasses this requirement by injecting artificial noises into the signals passed along the neurons. We show how this approach has a similar computational complexity as BP, and moreover is more advantageous in terms of removing the backward recursion and eliciting transparent formulas. We also formalize the connection between BP, a pivotal technique for training ANNs, and infinitesimal perturbation analysis, a classic path-wise derivative estimation approach, so that both our new proposed methods and BP can be better understood in the context of stochastic gradient estimation. Our approach allows efficient training for ANNs with more flexibility on the loss and activation functions, and shows empirical improvements on the robustness of ANNs under adversarial attacks and corruptions of natural noises.

A Systematic Process for Evaluating Structured Perfect Bayesian Equilibria in Dynamic Games With Asymmetric Information

We consider both finite-horizon and infinite-horizon versions of a dynamic game with $N$ selfish players who observe their types privately and take actions that are publicly observed. Players’ types evolve as conditionally independent Markov processes, conditioned on their current actions. Their actions and types jointly determine their instantaneous rewards. In dynamic games with asymmetric information, a widely used concept of equilibrium is perfect Bayesian equilibrium (PBE) which consists of a strategy and belief pair that simultaneously satisfy sequential rationality and belief consistency. In general, there does not exist a universal algorithm that decouples the interdependence of strategies and beliefs over time in calculating PBE. In this paper, for the finite-horizon game with independent types, we develop a two-step backward–forward recursive algorithm that sequentially decomposes the problem (w.r.t. time) to obtain a subset of PBEs, which we refer to as structured Bayesian perfect equilibria (SPBE). In such equilibria, a player's strategy depends on his/her history only through a common public belief and its current private type. The backward recursive part of this algorithm defines an equilibrium generating function. Each period in the backward recursion involves solving a fixed-point equation on the space of probability simplexes for every possible belief on types. Using this function, equilibrium strategies and beliefs are generated through a forward recursion. We then extend this methodology to the infinite-horizon model, where we propose a time-invariant single-shot fixed-point equation, which in conjunction with a forward recursive step, generates the SPBE. Sufficient conditions for the existence of SPBE are provided. With our proposed method, we find equilibria that exhibit signaling behavior. This is illustrated with the help of a concrete public goods example.

Event-Triggered Strategy Design for Discrete-Time Nonlinear Quadratic Games With Disturbance Compensations: The Noncooperative Case

In this paper, the event-triggered strategy design problem is addressed for a class of discrete-time nonlinear quadratic noncooperative games subject to matched disturbances. The event-triggered scheme is proposed based on the relative error of the input signals with aim to determine whether such signals should be transmitted to the actuator or not. The disturbance-observer-based game strategy is put forward to compensate the matched disturbance and also optimize the individual cost function for each player. The main purpose of the addressed problem is to design the time-varying strategy parameters such that the upper bound of the individual cost function of each player is minimized unilaterally over a finite horizon $[{0,N}]$ . Sufficient conditions are first established for the existence and uniqueness of the game strategies through backward Riccati-like recursions and then the desired strategy parameters are computed iteratively by utilizing the Moore–Penrose pseudo inverse. Finally, a simulation example is provided to verify the effectiveness of the proposed design method.

Company Value with Ruin Constraint in Lundberg Models

In this note we study the problem of company values with a ruin constraint in classical continuous time Lundberg models. For this, we adapt the methods and results for discrete de Finetti models to time and state continuous Lundberg models. The policy improvement method works also in continuous models, but it is slow and needs discretization. Better results can be obtained faster using the barrier method for discrete models which can be adjusted for Lundberg models. In this method, dividend strategies are considered which are based on barrier sequences. In our continuous state model, optimal barriers can be computed with the Lagrange method leading to a backward recursion scheme. The resulting dividend strategies will not always be optimal: in the case without ruin constraint, there are examples in which band strategies are superior. We also develop equations for optimal control of dynamic reinsurance to maximize the company value under a ruin constraint. These identify the optimal reinsurance strategy in no action regions and allow for an interactive computation of the value function. We apply the methods in a numerical example with exponential claims.

Near-Optimal Resilient Control Strategy Design for State-Saturated Networked Systems Under Stochastic Communication Protocol.

In this paper, the near-optimal resilient control strategy design problem is investigated for a class of discrete time-varying system in simultaneous presence of stochastic communication protocols (SCPs), gain perturbations, state saturations, and additive nonlinearities. In the sensor-to-controller network, only one sensor is permitted to get access to the communication media so as to avoid possible data collisions. Described by a Markov chain, the SCP is employed to determine which sensor should obtain the access to the network at a certain time. Furthermore, two kinds of well-recognized complexities (i.e., state saturations and additive nonlinearities) are considered in the system model and the phenomenon of controller gain perturbation is also taken into special consideration. Accordingly, the resilient control strategy is designed by: 1) deriving a certain upper bound on the associate cost function of underlying systems and 2) minimizing such an upper bound through the utilization of the completing-the-square technique and the Moore-Penrose pseudo inverse. The resilient control strategy is obtained in an iterative manner by solving a set of coupled backward Riccati-like recursions. Furthermore, based on the proposed control strategies, the infinite horizon case is considered and the corresponding upper bound of the cost function is explicitly provided. Finally, numerical simulations are carried out on power systems in order to verify the validity of the proposed resilient control algorithms.

Reinforcement learning, Sequential Monte Carlo and the EM algorithm

Using the expression for the unnormalized nonlinear filter for a hidden Markov model, we develop a dynamic-programming-like backward recursion for the filter. This is combined with some ideas from reinforcement learning and a conditional version of importance sampling in order to develop a scheme based on stochastic approximation for estimating the desired conditional expectation. This is then extended to a smoothing problem. Applying these ideas to the EM algorithm, a reinforcement learning scheme is developed for estimating the partially observed log-likelihood function. A stochastic approximation scheme maximizes this function over the unknown parameter. The two procedures are performed on two different time scales, emulating the alternating ‘expectation’ and ‘maximization’ operations of the EM algorithm. We also extend this to a continuous state space problem. Numerical results are presented in support of our schemes.

A real options approach for joint overhaul and replacement strategies with mean reverting prices

Due to its significant impact on economic performance, an effective equipment overhaul and replacement strategy is a key aspect of physical asset management in capital-intensive industries, such as the mining industry. Classical approaches suggest periodic interventions based on the physical condition of the equipment, considering factors such as availability and operational costs. These fixed models generally ignore two important aspects: first, the flexibility of the decision to overhaul or replace, which may be re-evaluated within a given period, and second, the uncertainty around economic factors that may affect future maintenance decisions, such as the product price. This work improves on classical models by considering the effect of integrated price uncertainty in the definition of joint overhaul and replacement strategy, using a real options approach and a mean reversion binomial model to represent the uncertainty in price. More specifically, we develop a real options model and use a backwards recursion algorithm to determine an optimal intervention policy that maximizes expected profits. We then present a numerical study of the mining industry to validate the effectiveness of the proposed methodology. Results show that the option-based decision model economically outperforms the classical periodic strategy approach from with net present value increments ranging from 36.8 to 8.6%, according to the number of periods in the maintenance cycle, offering evidence that a new approach to equipment overhaul and replacement strategy is needed.

The generalized cross validation filter

Generalized cross validation (GCV) is one of the most important approaches used to estimate parameters in the context of inverse problems and regularization techniques. A notable example is the determination of the smoothness parameter in splines. When the data are generated by a state space model, like in the spline case, efficient algorithms are available to evaluate the GCV score with complexity that scales linearly in the data set size. However, these methods are not amenable to on-line applications since they rely on forward and backward recursions. Hence, if the objective has been evaluated at time t−1 and new data arrive at time t, then O(t) operations are needed to update the GCV score. In this paper we instead show that the update cost isO(1), thus paving the way to the on-line use of GCV. This result is obtained by deriving the novel GCV filter which extends the classical Kalman filter equations to efficiently propagate the GCV score over time. We also illustrate applications of the new filter in the context of state estimation and on-line regularized linear system identification.

A new Fortran 90 program to compute regular and irregular associated Legendre functions (new version announcement)

This is a revised and updated version of a modern Fortran 90 code to compute the regular Plm(x) and irregular Qlm(x) associated Legendre functions for all x∈(−1,+1) (on the cut) and |x|>1 and integer degree (l) and order (m). The necessity to revise the code comes as a consequence of some comments of Prof. James Bremer of the UC//Davis Mathematics Department, who discovered that there were errors in the code for large integer degree and order for the normalized regular Legendre functions on the cut. New version program summaryProgram Title: Associated Legendre FunctionsProgram Files doi:http://dx.doi.org/10.17632/29j2mvwgwt.1Licensing provisions: GPLv2Programming language: Fortran 90Does the new version supersede the previous version?: YesJournal reference of previous version: Comput. Phys. Commun. 181 (2010) 2091Reason for new version: The necessity to revise the code comes as a consequence of some comments of Prof. James Bremer of the UC//Davis Mathematics Department, who discovered that there were errors in the code for large integer degree and order for the normalized regular Legendre functions on the cut. This occurred because renormalization of the un-normalized functions involved the manipulation of factorials of large integer value. The problem was solved by a direct computation of the normalized functions from a revised recursion formula. This completely avoids the manipulation of the factorials that were present in the first version. In re-examining the code we also noticed that there were cases for large degree, order and argument that led to overflows causing the results to display infinities or NaN when printed out, without crashing. There does not appear to be any good approach, other than resorting to extended precision arithmetic, to cure this pathology. Since it was not our original intention to provide such a code, all we can do is to point out that this can happen and that the user should take care not to blindly accept the results of the computation for large degree and order. We have not found a general approach to predict in advance when this will occur in this three dimensional parameter space.Summary of revisions: None of the basic algorithms has been changed. We have eliminated the computation of factorials for large integer values and now calculate the normalized regular Legendre functions on the cut (-1,+1) directly from the recursion relation for those functions rather than computing the unnormalized functions and then normalizing after the computation. This required some modifications of the original Fortran subroutines as well as the addition of a new module.Nature of problem: Compute the regular and irregular associated Legendre functions for integer values of the degree and order and for all real arguments. The computation of the interaction of two electrons, 1∕|r1−r2|, in prolate spheroidal coordinates is used as one example where these functions are required for all values of the argument and we are able to easily compare the series expansion in associated Legendre functions and the exact value.Solution method: The code evaluates the regular and irregular associated Legendre functions using forward recursion when |x|<1 starting the recursion with the analytically known values of the first two members of the sequence. For values of the argument |x|<1, the upward recursion over the degree for the regular functions is numerically stable. For the irregular functions, backward recursion must be applied and a suitable method of starting the recursion is required. The program has two options; a modified version of Miller’s algorithm and the use of the wronskian relation between the regular and irregular functions, which was the method considered in [1]. Both approaches require the computation of a continued fraction to begin the recursion. The wronskian method (which can also be described as a modified Miller’s method) is a convenient method of computations when both the regular and irregular functions are needed.

Convergence and Density Evolution of a MIMO Detector Based on a Forward–Backward Recursion Over a Ring

Convergence and density evolution of a low-complexity iterative multiple-input multiple-output detection based on belief propagation over a ring-type pairwise graph are presented for binary data. The detection algorithm to be considered is effectively a forward–backward recursion and was originally proposed by Yoon and Chae in a work published in 2014, in which link-level performance, computational complexity, and convergence for Gaussian input were analyzed in detail. This paper presents the convergence proof and the density evolution framework for binary input to give an asymptotic performance in terms of average signal-to-interference-plus-noise ratio and bit error rate (BER) without channel coding. The BER curve obtained via density evolution shows a good match with the simulation results for uncoded BER in the paper by Yoon and Chae verifying the effectiveness of the analysis provided and the performance of the detection algorithm.

Valuation of variable annuities with Guaranteed Minimum Withdrawal Benefit under stochastic interest rate

This paper develops an efficient direct integration method for pricing of the variable annuity (VA) with guarantees in the case of stochastic interest rate. In particular, we focus on pricing VA with Guaranteed Minimum Withdrawal Benefit (GMWB) that promises to return the entire initial investment through withdrawals and the remaining account balance at maturity. Under the optimal (dynamic) withdrawal strategy of a policyholder, GMWB pricing becomes an optimal stochastic control problem that can be solved using backward recursion Bellman equation. Optimal decision becomes a function of not only the underlying asset but also interest rate. Presently our method is applied to the Vasicek interest rate model, but it is applicable to any model when transition density of the underlying asset and interest rate is known in closed-form or can be evaluated efficiently. Using bond price as a numéraire the required expectations in the backward recursion are reduced to two-dimensional integrals calculated through a high order Gauss–Hermite quadrature applied on a two-dimensional cubic spline interpolation. The quadrature is applied after a rotational transformation to the variables corresponding to the principal axes of the bivariate transition density, which empirically was observed to be more accurate than the use of Cholesky transformation. Numerical comparison demonstrates that the new algorithm is significantly faster than the partial differential equation or Monte Carlo methods. For pricing of GMWB with dynamic withdrawal strategy, we found that for positive correlation between the underlying asset and interest rate, the GMWB price under the stochastic interest rate is significantly higher compared to the case of deterministic interest rate, while for negative correlation the difference is less but still significant. In the case of GMWB with predefined (static) withdrawal strategy, for negative correlation, the difference in prices between stochastic and deterministic interest rate cases is not material while for positive correlation the difference is still significant. The algorithm can be easily adapted to solve similar stochastic control problems with two stochastic variables possibly affected by control. Application to numerical pricing of Asian, barrier and other financial derivatives with a single risky asset under stochastic interest rate is also straightforward.

Efficient Dynamic Programming Solution to a Platoon Coordination Merge Problem With Stochastic Travel Times

The problem of maximizing the probability of two trucks being coordinated to merge into a platoon on a highway is considered. Truck platooning is a promising technology that allows heavy vehicles to save fuel by driving with small automatically controlled inter-vehicle distances. In order to leverage the full potential of platooning, platoons can be formed dynamically en route by small adjustments to their speeds. However, in heavily used parts of the road network, travel times are subject to random disturbances originating from traffic, weather and other sources. We formulate this problem as a stochastic dynamic programming problem over a finite horizon, for which solutions can be computed using a backwards recursion. By exploiting the characteristics of the problem, we derive bounds on the set of states that have to be explored at every stage, which in turn reduces the complexity of computing the solution. Simulations suggest that the approach is applicable to realistic problem instances.

Research on the indexed stock options’ quadratic hedging under jump-diffusion settings

This paper studied the hedging of European indexed stock options, whose exercise price is a variable varying with the market index. Firstly, using the expectation of the squared error between option’s terminal value and hedging portfolio to measure market risk , and with the constraint of self-financing, a squared hedge decision model was built; Then, by using the Dynamic Programming Principle and backward recursion method, the optimal hedging positions’ analytical expressions were acquired; Lastly, all hedging results with different option maturities and different position rebalancing frequencies show that there is dependent relationship between hedging positions and option’s maturity time limit; hedging positions also rely on the relative variation of the underlying asset price.

Semi-global solutions to DSGE models: perturbation around a deterministic path

AbstractThis study proposes an approach based on a perturbation technique to construct global solutions to dynamic stochastic general equilibrium models (DSGE). The main idea is to expand a solution in a series of powers of a small parameter scaling the uncertainty in the economy around a solution to the deterministic model, i.e. the model where the volatility of the shocks vanishes. If a deterministic path is global in state variables, then so are the constructed solutions to the stochastic model, whereas these solutions are local in the scaling parameter. Under the assumption that a deterministic path is already known the higher order terms in the expansion are obtained recursively by solving linear rational expectations models with time-varying parameters. The present work also proposes a method rested on backward recursion for solving general systems of linear rational expectations models with time-varying parameters and determines the conditions under which the solutions of the method exist.

Inter-domain optical path provisioning with blocking avoidance by intra-domain immediate reroute-based semi-proactive hierarchical PCE

Optical fiber networks have become the foundations of communication system to provide enormous transmission capacity with less cost. Connection blocking is an inherent attribute and influences optical networks' performance, especially in multi-domain network scenarios. We analyze the reason and effect of blockings including routing blockings and signaling blockings. We regard the cause of signaling blockings as the information asynchronization in resource reservation process. We define the concept of Information Asynchronization Period to describe the effect of signaling blockings. To reduce signaling blockings in end-to-end optical path provisioning for multi-domain scenarios, we propose a novel network routing and control scheme, named Intra-Domain Immediate Reroute based Semi-Proactive Hierarchical Path Computation Element (IDIRSP H-PCE). The proposed routing and control scheme mainly consists of two parts, Semi-Proactive routing and Intra-Domain Immediate Reroute mechanism. Dynamic network simulations verify our proposed scheme. We compare the network performance with Reactive Backward Recursive PCE-based Computation (BRPC) based PCE, Reactive H-PCE and Proactive H-PCE. Simulation results indicate that IDIRSP H-PCE can provide connection with a very low blocking probability in light load case, which is close to Proactive H-PCE, and obviously better than BRPC based Reactive PCE and Reactive H-PCE. For heavy load case, IDIRSP H-PCE has a remarkably lower blocking probability than other three methods. Moreover, we test our proposed routing and control scheme when facing link faults. Simulation results indicate that IDIRSP H-PCE can greatly improve the traffic access rate and optimize the network performance.

Optimal extraction policy when the environmental and social costs of the opencast coal mining activity are internalized: Mining District of the Department of El Cesar (Colombia) case study

Several articles have confirmed the social and environmental consequences of opencast coal mining. The main purpose of this study is to simulate the optimal extraction policy of coal mining with and without the internalization of the environmental and social monetary costs that occur in the Mining District (located in the central part of the Department of El Cesar) using discrete dynamic programming (backward recursion, discrete state Markov decision model and Bellman equation). Results indicate that the private optimal of the overproduction policy for the terminal phase of the resource extraction program can be reduced once the negative externalities produced by mining practices are internalized into the cost function of the mining investment companies in Colombia. This means that if there is an increase in the total cost of extraction to offset the environmental and social impacts generated, the negative externalities would be less than or equal to the current level. Likewise, profits would continue being positive for the mining firms at the Mining District.

Recursive Implementation of the Gaussian Filter Using Truncated Cosine Functions

This paper proposes a new recursive implementation of the Gaussian filter. Previous recursive implementations of the Gaussian filter suffer from one or several drawbacks. Some methods require employment of both causal and anti-causal filters. Implementing such filters requires both forward and backward recursions, and thus, significant amounts of memory, in the case of long sequences, to store and combine both recursion results. Moreover, most existing techniques do not ensure that the sum of impulse response weights equals unity, or that the standard deviation of the filter is exactly the desired one. The proposed technique is based on a recursive implementation of cosine-based finite impulse response filters and only requires forward recursions. The sum of filter weights and the standard deviation of the filter are constrained to the desired values during the design process. Experimental results confirm that the proposed filter provides a more suitable tradeoff between computational efficiency and filtering accuracy compared to existing recursive implementations. The performance and advantages of the proposed filter are demonstrated on signal and image processing examples.

A Counter example to the Forward Recursion in Fuzzy Critical Path Analysis Under Discrete Fuzzy Sets Full Text

Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy version of the backward recursion has been shown to produce results that incorrectly amplify the level of uncertainty. However, the fuzzy version of the forward recursion has been widely proposed as an approach for determining the fuzzy set of critical path lengths. In this paper, the direct application of the extension principle leads to a proposition that must be satisfied in fuzzy critical path analysis. Using a counterexample it is demonstrated that the fuzzy forward recursion when discrete fuzzy sets are used to represent activity durations produces results that are not consistent with the theory presented. The problem is shown to be the application of the fuzzy maximum. Several methods presented in the literature are described and shown to provide results that are consistent with the extension principle.

Multi-period mean–variance portfolio optimization based on Monte-Carlo simulation

We propose a simulation-based approach for solving the constrained dynamic mean–variance portfolio management problem. For this dynamic optimization problem, we first consider a sub-optimal strategy, called the multi-stage strategy, which can be utilized in a forward fashion. Then, based on this fast yet sub-optimal strategy, we propose a backward recursive programming approach to improve it. We design the backward recursion algorithm such that the result is guaranteed to converge to a solution, which is at least as good as the one generated by the multi-stage strategy. In our numerical tests, highly satisfactory asset allocations are obtained for dynamic portfolio management problems with realistic constraints on the control variables.