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Closed-Form Solutions to Differential Equations via Differential Evolution

We focus on solving ordinary differential equations using the evolutionary algorithm known as differential evolution (DE). The main purpose is to obtain a closed-form solution to differential equations. To solve the problem at hand, three steps are proposed. First, the problem is stated as an optimization problem where the independent variables are elementary functions. Second, as the domain of DE is real numbers, we propose a grammar that assigns numbers to functions. Third, to avoid truncation and subtractive cancellation errors, to increase the efficiency of the calculation of derivatives, the dual numbers are used to obtain derivatives of functions. Some examples validating the effectiveness and efficiency of our method are presented.

1. Introduction

Most of the problems in engineering and physics can be modeled as ordinary differential equations (ODEs). For this reason there are many studies addressing their solution. Regarding the deterministic arena, the most used methods are the Runge-Kutta methods [1–4], predictor-corrector methods [5–7], and radial basis functions methods [8–10]. Recently, some studies dedicated to solve differential equations using nondeterministic methods have been published. In [11], genetic algorithms are used to solve some differential equations appearing in economic sciences. In [12] a variational approach has been used in order to solve elliptic partial differential equations, and a genetic algorithm is used as the optimization method. In all the previously referenced articles—deterministic or not—the solution is given in a numerical approximated form. There are very few studies reporting closed-form solutions to differential equations. For example, using the evolutionary method known as grammatical evolution, [13] reports a method that produces closed-form solutions. Another approach that produces closed-form solutions to differential equations is [14], and the method used there is a hybrid method combining grammatical evolution and neural networks.

In this paper we propose a method based on the DE algorithm that obtains solutions to second-order ODEs as closed-form expressions. When the exact solution is not reached, the algorithm we propose converges to a solution that minimizes the objective functional of an optimization problem considering both, the differential equation and the boundary conditions. As DE was proposed to minimize real valued functions (i.e., not functionals), we propose a one-to-one grammar that assigns integer numbers to elementary functions; in this way, we are able to use DE to minimize the objective functional, which measures if a candidate function (a candidate function is the result of the evolution process obeying the DE algorithm) satisfies or not the ODE we want to solve.

In order to compute the first and second derivatives of the candidate function, we use the dual number approach. In this way, the derivatives are directly obtained without the use of a limit process, thus avoiding truncation and subtractive cancellation errors. All the programming functions required to solve an ODE are coded in Fortran language and they are included in a folder, which is provided as additional material to this paper, whose download link is as follows:

The rest of the paper is organized as follows. Section 2 states the optimization problem and describes the classical DE algorithm. Section 3 presents the proposal of the one-to-one grammar which allows using DE for minimization of functionals. Section 4 presents the dual number approach to obtain the first and second derivatives of the candidate functions. Section 5 works out several examples and applications of our method. Conclusions are presented in Section 6. Finally, two appendixes close the paper. Appendix A presents the way in which the candidate function is generated and evaluated. Appendix B presents graphs of the behavior of the DE algorithm for each worked-out example.

2. Statement of the Problem

Let us consider a second-order ordinary differential equation (1) defined on the real interval , with boundary conditions (2) and (3), where and . Note that it is not required that the functions , , and be differentiable:The problem that we address in this paper is to find a closed-form expression for satisfying (1), (2), and (3). Therefore, we rewrite the problem as that of minimizing the functional (4) under , where and are weighting factors (chosen by the user):If there exists a function for which , then will be the solution to (1), satisfying (2) and (3). As the approach we follow to minimize (4) is evolutive, we use the differential evolution algorithm which is a simple yet powerful evolutionary algorithm for global optimization introduced by Storn and Price [15], which is presented below.

2.1. Differential Evolution

The DE algorithm has gradually become more popular and has been used in many practical cases. It only requires information about the objective function itself, which does not need to be a differentiable function, and the state space of possible solutions can be disjoint and can encompass infeasible regions [16]. Below, the original version of the method—known as DE/rand/1/bin—is outlined [17].(1)The population is described by where , , and represent the dimensionality of , the number of individuals, and the number of generations, respectively.(2)Initialization of population is as follows: Vectors and are the parameter limits and is a random number in generated for each parameter.(3)Mutation is as follows: is called the base vector which is perturbed by the difference of two other vectors., . is a scale factor greater than zero.(4)Crossover is as follows.A dual recombination of vectors is used to generate the trial vector: The crossover probability, , is a user-defined value, .(5)Selection is as follows.The selection is made according to In our study we use the DE/rand/1/bin method with the dither variant, which means that the parameter is taken to be a random number—in our case .

3. Construction of Functions

As we can see, the DE method was designed to seek the optimum individual on a real continuum domain. Since we are interested in solving differential equations (i.e., minimizing a functional), our individuals are functions. Therefore, we associate a function with a real vector; once this is done, the DE method can be applied as usual.

In order to assign a function to a real vector we propose the grammar shown in Table 1. The relation between a set of numbers and a function can be done by using a parse tree. This parse tree should be read from top to down and from left to right. Table 2 shows an example of a function construction and Figure 1 shows the corresponding parse tree. We can get an easy understanding of the function construction by conceptualizing the operators , as functions of two arguments. For example, the expression can be seen as where the function is defined as .

String Associated number

Integer from one to ten 1 : 10
pow() 15

Chromosome String

12, 17:
12, 17, 11:
12, 17, 11, 18:
12, 17, 11, 18, 11:

The set of integers related to a mathematical function can be manipulated by the DE method but the mutation operator will produce a set of real numbers that will not necessarily be a set of integers. This is addressed by taking the integer part of the numbers or by using the floor (ceiling) function.

4. Evaluation of the Constructed Function and Its Derivatives

For the evaluation of the constructed function we have written a Fortran parse function that receives the integer vector generated by the proposed grammar and produces a candidate function and its derivatives (first and second) evaluated at some specified point. The construction of this programming function is explained in Appendix A.

Traditional methods for calculating numerical derivatives (finite-difference) are subject to both truncation and subtractive cancellation errors. These problems are avoided by using dual functions. The approach to obtain first order derivatives by using dual functions is well known [18–20]. However, in order to make the paper self-contained this section presents the essential ideas as follows [20].

A dual number is a number of the form where , the field of the real numbers, and . From the Taylor theorem if a function is analytic, then


Closed-form solution for the Kalman filter gains of time-varying systems

Abstract: A method for derivation of closed-form solutions for the differential Riccati matrix equation for specific time-varying systems is presented. It allows more insight into the nature of the solution. It reduces the on-line computation requirements, since it does not require on-line solution of a differential equation. Sufficient conditions for the existence of the closed-form solution are given. The method is applied to a target tracking problem.

Published in: IEEE Transactions on Aerospace and Electronic Systems ( Volume: 34 , Issue: 2 , Apr 1998 )

Article #:

Page(s): 635 - 638

Date of Publication: Apr 1998

ISSN Information:

Print ISSN: 0018-9251

Electronic ISSN: 1557-9603

CD: 2371-9877

Sours: /document/
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Closed-form solution for minimizing power consumption in coordinated transmissions

EURASIP Journal on Wireless Communications and Networkingvolume 2012, Article number: 122 (2012) Cite this article

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The growth in the demand of energy, and its consequent contribution to the greenhouse effect, gives rise to new challenges in the design of future wireless networks. Keeping in mind these requirements, in this article we study the power allocation problem in the downlink of an orthogonal frequency division multiple access (OFDMA) system, where two (or more) coordinated transmission points (CTPs) should find the best way to allocate their transmit power through the multiple orthogonal sub-channels of the system. The ultimate goal of the power allocation scheme is to minimize the joint power consumption of the system, but verifying at the same time the target throughput and the individual power constraint per CTP. The power allocation problem is formulated as a constrained optimization problem, and a group of closed-form power allocation solutions are derived. Based on the derived solutions (that take the form of the traditional water-filling but demanding cooperation among CTPs), a novel power allocation algorithm with joint minimization power consumption (JMPC-PA) is proposed. Numerical results are presented to verify the optimality of the results that were obtained by the JMPC-PA scheme. It is important to note that, due to the flexibility that exist in the definition of CTPs in this article, the derived power allocation scheme is valid for any kind of network that incorporates the coordinated multipoint transmission feature in its design.

1. Introduction

Orthogonal frequency division multiplexing access (OFDMA) is a promising technique for high data rate transmission in wideband wireless systems [1, 2]. In OFDMA systems, the total bandwidth is divided into many orthogonal narrowband subcarriers, with different subcarriers allocated to different users to enable flexible multi-user access. Transmission power allocation represents an effective way to increase the throughput of wireless communication systems [3]. According to different optimization objectives and constraints, adaptive power allocation schemes for OFDMA systems can be roughly divided into two categories: rate adaptive (RA) schemes and margin adaptive (MA) schemes. Rate adaptive schemes seek the maximization of the system throughput under a total and/or individual transmit power constraints [4–6], while MA schemes try to minimize the overall transmission power with constraints on individual and/or system data rates [7–9]. It is shown that the optimal power allocation policy of the above schemes often leads to the form of a WF solution. However, traditional WF solutions are simple to evaluate because all of them consider a single power constraint (known as waterlevel). As a consequence, it is quite straightforward to compute the solution numerically. However, some other optimization problems (such as multi-user or multiple transmission points) would result in complicated non-convex problems. For such cases, it may be difficult to obtain closed-form solutions or practical algorithms.

Recently, the concept of cooperative communications has been proposed for wireless networks, such as macrocellular networks and wireless ad-hoc networks [10–13]. The basic idea behind cooperative communications is to allow the nodes in the wireless networks to transmit their information signals coordinately, to improve the quality of the communication (via spatial diversity) or to increase the achievable data rates (through spatial multiplexing). In both cases, the ultimate goal is to boost the system performance and achieve a better usage of the system resources. One typical technology in this context is the so-called coordinated multi-point transmission/reception (CoMP), which has been considered as an effective tool to improve the coverage of high data rates and the cell-edge throughput for 3GPP long term evolution (LTE) advanced [14].

Motivated by the coordinated feature of CoMP, a new cooperative power allocation scheme was presented in [15], where two coordinated transmission points (CTPs) allocated jointly their constrained transmit power to the multiple orthogonal subchannels based on their channel state information (CSI) that they exchanged. In order to maximize the system throughput, a closed-form solution known as joint-waterfilling (Jo-WF) solution was obtained by Luo et al. [15], when solving the constrained non-convex optimization problem that arose in that situation. The solution turned out to take the form of tradition WF, and also had a cooperative feature. In green radio framework, however, the myopic maximization of system throughput does not make much sense. On the contrary, it is more important to minimize the total cost of transmission (i.e., the transmission power), while setting the constraints in the individual transmit power of each CTP power to attain a given target throughput.

In this article, we extend our previous study and analyze the dual power allocation problem in an OFDMA system. We present an effective way to coordinate the transmission between adjacent CTPs, seeking the minimization of the total transmit power while meeting the throughput requirements. To simplify the analysis we assume that the subchannels that are available to serve a given user are selected beforehand, e.g., using an arbitrary scheduling algorithm. This simplification allows us to put the emphasis in the design of the power allocation scheme, focusing our attention in the formulation of our objective task into a non-convex optimization problem. A group of closed-form solutions are obtained. Based on the derived solutions, a novel power allocation algorithm with joint minimization power consumption (JMPC-PA) is proposed. Numerical simulation results allow to show that, when compared to equal power allocation (EPA) scheme, the proposed JMPC-PA scheme provides a significant gain in terms of total power consumption.

Motivated by the regular theoretical derivation that was obtained in the two-CTP case, we also extend the derived results into an arbitrary K-CTP case, an scenario that can be considered more significant for practical networks. Due to the definition of CTP in this study is very flexible (i.e., it could represent both, a base station or a relay node), it is important to highlight that the JMPC-PA scheme can be applied to any kind of cooperative networks, such as next-generation macrocellular networks, heterogeneous networks, and ad-hoc networks.

The rest of the article is organized as follows. Section 2 presents the coordinated transmission model. Section 3 introduces the closed-form power allocation solutions for the 2-CTP case, while Section 4 extends the analysis to the more generic K-CTP (i.e., with an arbitrary large number of coordinated sources). Simulation results are presented in Section 5. Finally, the concluding remarks of the paper are summarized in Section 6.

2. Coordinated transmission model

We consider a downlink multi-user OFDM system, in which two CTPs coordinately transmit their power to the coordinative zone users as depicted in Figure 1. In Figure 1, the CTPs are denoted by the solid squares and the users are denoted by the circles. Each CTP has all the information intended to the users in the coordinative zone via a wireline connection. In order to simplify the mathematical derivation, we assume that the two CTPs have the same power constraint P0, and that they share the same overall bandwidth B, which is divided into N orthogonal narrow-band subcarriers. Each user feeds back the CSI to its corresponding CTP via a feedback channel, and the instantaneous CSI can be exchanged reliably and fast between the two CTPs.

System model with two coordinated transmission points (CTP).

Full size image

In order to focus solely on power allocation, we do not explicitly consider subcarrier scheduling here. However, it is noted that the power allocation results presented in this article are valid for any scheduling strategy.a Besides, in this article we only focus on the total system capacity, leaving aside fairness issues concerning the way in which the common resources are shared among users. We also emphasize that our analysis is valid for any kind of coordinated transmission network, as long as the sum of orthogonal subcarrier capacities is a relevant performance metric.

In a multi-carrier system, the sum of the individual capacities per carrier represents the total system capacity. We suppose that a certain group of N-subcarriers has been selected for coordinated transmission by an arbitrary scheduling algorithm. Let us assume that the subcarriers are narrow enough to undergo flat fading, and that the channel gains are constant within a given transmission time interval. Thus, the achievable (sum rate) throughout given by the Shannon formula attains the form


where represents the power of additive white Gaussian noise at the j th subcarrier, Pijdenotes the allocated transmission power from CTP i to the j th subcarrier, and Hi,jrepresents the corresponding channel gain between the CTP i and the j th subcarrier.

The aim of this article is to minimize the joint transmit power for both CTPs, while satisfying at the same time the system throughput requirement RSas well as the individual transmit power constraint per CTP. In mathematical terms, this is equivalent to solve the following optimization problem:


3. Closed-form power allocation solutions

We search for the optimal coordinated power allocation by approaching the following optimization problem:


where is the feasible set. Since ΩNis a closed and bounded set, and R : ΩN→ ℝ is continuous, function (1) has a solution [[16], Theorem 0.3].

For the sake of mathematical derivation, we denote and xi,j= Pi,j/P0. It is indicated that γi,jis the SNR associated with CTP i over the j th subchannel when assuming the entire power P0 is allocated to the j th subchannel, and xi,jrepresents the power allocation ratio. Since the logarithm is monotonically increasing function, the objective (2) combined with the constraints can be described as


where Z ≤ 2 represents the total power consumption ratio.

In order to obtain a closed-form expression for the required power allocation, we divide the constrained problem into two different cases:

  1. a)

    The power constraint is assumed to be large enough, so that solving the primal objective is equivalent to solving an unconstrained problem. In this case, the power constraint at each CTP (i.e., for i = 1, 2) is always satisfied.

  2. b)

    The power constraint is not large enough, so that one of the CTPs may exceed the power constraint if the distribution of the users is not even. In this situation, the solution in (a) leads to or . In this situation the exceeded CTP should provide its full power for transmission, while the other CTP should increase its transmit power until it meets the throughput requirement for the system. In mathematic terms, the constraint in the primal objective should be changes to and , or to and . Actually, this case is more meaningful in practical systems.

3.1. Optimal solution for the unconstrained case

It is noted that for arbitrary γi,j, the likelihood of having γ1,j= γ2,jfor j∈ {1, 2} in an actual system is almost zero. Without loss of generality, we divide the N subchannels into two parts: the first part contains M subchannels that satisfy γ1,m> γ2,mfor m∈ {1, 2,..., M}, while the second part contains K subchannels that satisfy γ1,k< γ2,kfor k∈ {1,2,..., K}. According to this model, N = M + K.

We first present the following lemma:

Lemma 1: When ignoring the power constraint , the objective problem (4) can be degenerated into the unconstrained function


Proof: To prove this lemma we use reductio ad absurdum. Assume that vectors and are optimal solutions for m∈ {1,..., M} and k∈ {1,..., K}, which satisfy the throughput requirement and achieve the minimum Z*. However, we could find another set of vectors and which also satisfy the throughput requirement, and achieve a smaller Z**, due to γ1,m> γ2,m, γ1,k< γ2,kand


Therefore, the objective (4) could be equivalently transformed into (5). The degeneration implies that a group of M subchannels (that verify γ1,m> γ2,m) should receive power allocation from CTP 1, while the other K = N - M subchannels (that verify γ1,k< γ2,k) should receive power allocation from CTP 2. In other words, to achieve the goal of minimizing the total transmit power in the system, each CTP should select the better subchannels (according to the channel quality) to allocate power on them.

The degraded objective function (5) is strictly convex. Then, by Lagrange dual function we have that


First, we derive λ as


Then, the closed-form solution for the power allocation ratio at each CTP is obtained as follows:


Finally, the minimum total power consumption ratio is given by the following expression:


Note: As mentioned before, formula (9) only represents the optimal solution for the primal objective when the power constraint for i = 1,2 holds. However, when the power constraint is not large enough and the user density is not evenly distributed, one of the CTPs may exceed its individual power constraint. In this situation, the solution in (9) leads to or . Therefore, the exceeded CTP should provide its full power for transmission, and the power allocation for the other user should be recalculated to meet the throughput requirement for the system. The analyisis for this case is specified in the following section.

3.2. Optimal solution for the constrained case

Without loss of generality, we assume that CTP 1 is the overloaded transmission point. Thus, it is straightforward to know that CTP 1 should provide its full power for transmission (i.e., ). Then the objective (4) can be redefined as


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Form solution closed

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Closed form from a recursive definition

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