Differential evolution algorithm of solving an inverse problem for the spatial Solow mathematical model
Sergey Kabanikhin, Olga Krivorotko, Maktagali Bektemessov, Zholaman Bektemessov, Shuhua Zhang
aa r X i v : . [ m a t h . O C ] A p r J. Inverse Ill-Posed Probl. ? ( ???? ), 1 – 19DOI 10.1515 /jip- ???? - ??? © de Gruyter ???? Differential evolution algorithm of solving an inverseproblem for the spatial Solow mathematical model
Sergey Kabanikhin, Olga Krivorotko, Maktagali Bektemessov,Zholaman Bektemessov and Shuhua Zhang
Abstract.
The differential evolution algorithm is applied to solve the optimization prob-lem to reconstruct the production function (inverse problem) for the spatial Solow mathe-matical model using additional measurements of the gross domestic product for the fixedpoints. Since the inverse problem is ill-posed the regularized differential evolution is ap-plied. For getting the optimized solution of the inverse problem the differential evolutionalgorithm is paralleled to 32 kernels. Numerical results for different technological levelsand errors in measured data are presented and discussed.
Keywords.
Solow model, spartial Solow model, economy, inverse problem,reconstruction of parameters, PDE, parameter identification, optimization, differentialevolution, regularization, identifiability.
As the Solow growth model [24] was build using production function (naturallyCobb-Douglas production function), law of motion for the stock of capital andsaving/investment function, model can be easily extended to include a householdsproblem (the Ramsey-Cass-Koopmans model). Usually the interest of the Solowmodel is that it perpetual growth, that can be obtained using balanced growth pathand technological progress over time. Output per worker can grow only as longas capital per worker grows and the key to constant growth is the existence ofnon-diminishing marginal product of capital. Another way of perpetual econom-ical growth is letting technological progress change in model, it means allowingtechnological parameter to grow exogenously over time.
This work is supported by the Russian Science Foundation (grant No. 18-71-10044), i.e. numericalinvestigation of spatial Solow mathematical model, and by the grant of the Ministry of Educationand Science of the Republic of Kazakhstan (project No. AP05134121 ”Numerical methods of iden-tifiability of inverse and ill-posed problems of natural sciences”), i.e. inverse problem statement(Section 2).
S. Kabanikhin, O. Krivorotko, M. Bektemessov and others
As it has been written in [19], the Solow growth model was considered with as-sumptions as concave homogeneous production function (instead of Cobb-Douglasproduction function), exponentially growing labor and constant saving function.In addition, they considered the per capita consumption maximization problemsubject to economic equilibria. Authors considered two cases when productionfunction is logistic and the labor grow exponentially and when both of them arelogistic, to reduce them to one variable parametric maximization problem. Aftermaking sure that production function is nonconvex and satisfies the Lipschitz con-ditions, authors solved nonconvex optimization problem by global optimizationtechniques, considered on a sufficiently large interval. The solution can be foundby the method of piecewise linear function [20].In the article [23] by Smirnov and Wang, the work of Ryuzo Sato [22], de-voted to the development of economic growth models within the framework ofthe Lie group theory, was extended to a new growth model based on the assump-tion of logistic growth by using the Solow economic growth model as a startingpoint. Authors claimed that the Cobb-Douglas function can no longer adequatelydescribe the growth of the economy over a long-run, it was aimed to develop anew mathematical paradigm that can be used to study the current state of econ-omy and to replace neoclassical growth model in the sense of Sato representingexponential growth with a logistic growth. Also they used the new “S-shaped”production function, the consequence of logistic growth in factors, to solve maxi-mization problem of profit under condition of perfect competition, using the samearguments of subject to relevant changes by assuming that the revenue of the firmfrom sales is determined.The logistic growth in other words can be described by spatial Solow modeland that was used in [5], where they did identification of production function using(noisy) data that is an ill-posed inverse problem, using non-parametric approachand applied Tikhonov regularization to stabilize the computations. As there is noclear choice which production function will fit the situation best, it was proposedto identify production function from data about the capital distribution of somespatial economy, further they obtained the following optimization problem thathas to be minimized. To solve the minimization problem authors applied the gra-dient descent algorithm. As the objective function was Freshet differentiable, theyused the directional derivatives of the Langrangian and to find the minimum ofthe functional the simple steepest descent method and a backtracking line searchmethod [7] were used. So they reconstructed the production function to a spatialSolow model with the different noise levels and different technology terms, whenit is constant and space-dependent.We use the spatial Solow mathematical model as in [5] and investigate the in-verse problem for its using stochastic approach for global optimization. ifferential evolution algorithm for spatial Solow model
In this Section the derivation of neoclassical Solow mathematical model for ordi-nary differential equation is demonstrated at subsection 2.1. Based on that deriva-tion the statement of the spatial Solow model is considered in subsection 2.2 andthe inverse problem statement for spatial Solow model is formulated at subsec-tion 2.3.
Neoclassical economical Solow model describes evolution of gross output – Y ( t ) ,using next (due to such) indicators as: used labor resources – L ( t ) , saving capital– K ( t ) and technological progress – A ( t ) . And since the output parameter of themodel should be a stable indicator of a productive economy, then the gross domes-tic product (GDP) is taken, which is a macroeconomic index reflecting the marketvalue of all final goods and services produced during the year in the state [15]. Amathematical notation connecting these variables is Y ( t ) = A ( t ) Q ( K ( t ) , L ( t )) ,where Q represents production function. It is assumed that the production functionis homogeneous, which means Q ( αK ( t ) , αL ( t )) = αQ ( K ( t ) , L ( t )) . Also it canbe noted that the production function satisfies the following condition Q ( , L ( t )) = = Q ( K ( t ) , ) . It is worth saying that the description of the development of any economy onlydue to the absolute value of any gross output is useless, it is hard to say whetherthe economy is doing well or not. Simon Kuznets, one of the architects of the USnational accounting system, the man who first introduced the concept of GDP in1934, warned against identifying GDP growth with increasing economic or socialwelfare.What we are interested in is the rate of economic growth [13, 14]. There-fore, we consider the rate of change in capital, which looks like dKdt = Y ( t ) − C ( t ) − δK ( t ) . (2.1) S. Kabanikhin, O. Krivorotko, M. Bektemessov and others
It means the change in fixed capital stock negatively depends on the volume ofconsumption C ( t ) and on the amount of depreciation that is supposed to occur withthe rate δ . Moreover, we assume that the difference in production and consumptionpersists for each period of time, namely Y ( t ) − C ( t ) = sY ( t ) . (2.2)Then using that and inserting (2.2) into (2.1), we have the following dKdt = sA ( t ) Q ( K ( t ) , L ( t )) − δK ( t ) . (2.3)Next, we introduce a new variable, namely k ( t ) = K ( t ) L ( t ) , the capital per capita.Then it turns out, using the homogeneous of function Q we can write the following q ( k ( t )) = L ( t ) Q ( K ( t ) , L ( t )) = Q (cid:18) K ( t ) L ( t ) , (cid:19) and calculate dk ( t ) dt = ddt (cid:18) K ( t ) L ( t ) (cid:19) = dK ( t ) dt L ( t ) − n K ( t ) L ( t ) , where n = dL ( t ) dt L ( t ) denotes a constant growth rate of labor costs (labor intensity).With these designations and abbreviations, we can rewrite (2.3) as follows dk ( t ) dt = sA ( t ) q ( k ( t )) − ( δ + n ) k, (2.4)which is the basic equation for spatial structured Solow model [12, 16, 24].It is worthy to clarify that we are interested in the change in capital for a workunit (that is, an employee) - the capital-labor ratio, or more precisely, the situationwhere the capital per work unit reaches its steady state. To do this, consider astationary solution of equation (2.4)0 = sA ( t ) q ( k ( t )) − ( δ + n ) k. If we assume that A ( t ) =
1, then it means that there is no technological progressat all. Then there are only three variables describing the capital-labor ratio: savingrate - s , depreciation rate - δ and the rate of population growth or unit of labor used- n . Consequently, capital intensity will increase (grow) if sq ( k ( t )) > ( δ + n ) k ifferential evolution algorithm for spatial Solow model k E , i.e. there are enough savings to cover thecosts associated with population growth and the amount of capital lost due to de-preciation. Moreover, the economic growth rate in steady state equals to rate ofpopulation growth (i.e. n ). Further, we assume that parameters such as populationgrowth rate and depreciation coefficient are always constant, then the only vari-able affecting the model is the savings rate - s . It is also assumed that when savingchanges from s to s at ( s > s ), the function shows a sharp rise, and then thesteady state increases from k to k . It is good for economy for a short period oftime, because economic growth occurs faster, but in the long run the economy willtend to a new steady state and then the economic growth rate will again be equalto n . So n is not only constant, but also equals zero, since the population doesnot change at all. The rate of savings over a large time interval, in turn, does nothave any effect either on the rate of economic growth. The only option to obtaineconomic growth is a technological progress [4]. Thus, if the parameter as n doesnot have any effect on model - n =
0, noting that we set s = dk ( t ) dt = A ( t ) q ( k ( t )) − δk. (2.5) Consider the scaled initial-boundary value problem for the mathematical modeldescribed dynamic of the capital stock held by the representative household lo-cated at x at date t [2, 5]. Then the mathematical model (2.5) with adding initialand boundary conditions is rewritten as follows: ∂k∂t − d △ k ( x, t ) = g ( k, x, t ) , x ∈ Ω, t ∈ [ , T ] ,k ( x, ) = k ( x ) > , x ∈ Ω, ∇ k · n = , on ∂Ω × [ , T ] . (2.6)Here d = δL is a scaled coefficient, δ is the depreciation rate, g ( k, x, t ) = A ( x,t ) δ q ( k ) − k , A ( x, t ) denotes the technological level at x and time t . The stan-dard neoclassical production function is assumed to be non-negative, increasingand concave, and verifies the Inada conditions, that is,lim k → q ′ ( k ) = + ∞ , lim k →∞ q ′ ( k ) = , q ( ) = . We will depart from the assumptions with respect to concavity in particular aroundzero as well as the first Inada condition and allow for general convex-concave
S. Kabanikhin, O. Krivorotko, M. Bektemessov and others production functions, an example being [5] q ( k ) = α k p + α k p , α , α ≥ , p > . (2.7)Such examples of q are of particular interest, because they are related to the poten-tial existence of poverty traps. Define the set of admissible production functions Q adm = (cid:8) q ∈ H ( , K ) | q ( ) = , ≤ q ′ ( k ) ≤ q ′ max for k ∈ ( , K ) ,q ′ ( k ) = } , where q ′ max being a fixed constant, which can be understood as the maximal growththat an economy is capable of.The technological level A ( x, t ) is determined via a diffusion equation of theform ∂A∂t − △ A = Ag A , x ∈ Ω, t ∈ [ , T ] ,A ( x, ) = A ( x ) , x ∈ Ω, ∂A∂x = , on ∂Ω × [ , T ] , (2.8)with g A being either constant, a function depending only on space or a functiondepending on space as well as on time.The Neumann boundary condition in problem (2.6) represents no capital flowthrough the boundary and thereby a closed economy.In paper [5] authors proved a well-posedness of direct problem (2.6) at space L ([ , T ] , H ( Ω )) ∩ H ([ , T ] , H − ( Ω )) if k ∈ L ∞ ( Ω ) , q ∈ Q adm and A ∈ C ( Ω × [ , T ]) . A more detailed analysis of this model can be found in [3]. The choice of the production function is crucial for an economic model, as itsshape will greatly influence the capital distribution. In general, data about theeconomic situation, such as the gross domestic product (GDP), of different regionsand different countries are readily available. Suppose, that we have additionalinformation about GDP of some spatial economy at fixed space and time points: k ( x m , t j ) = f mj + ε mj , x m ∈ Ω, t j ∈ [ , T ] , m = , . . . , M, j = , . . . , N. (2.9)Here ε mj are Gaussian noise in measurements.The inverse problem (2.6), (2.9) consists in identification of production func-tion (2.7) (or identification of parameters α , α , p ) of initial-boundary value prob-lem (2.6) using additional measurements (2.9). It means that we have the nonlinear ifferential evolution algorithm for spatial Solow model A : q ∈ Q adm f ε ∈ E MN mapping the productionfunction q to the respective capital distribution f ε = { f mj + ε mj } m = ,...,M,j = ,...,N , i.e. A ( q ) = f ε . Here E is an Euclidean space of measurements.The inverse problem (2.6), (2.9) is ill-posed [9], i.e. the solution q ( k ) is non-unique and can be unstable [5]. That is we apply the regularization techniquedescribed in Section 3. Reduce our inverse problem (2.6), (2.9) to an optimization problem that consistsin minimization of the misfit function J ( q ) = k A ( q ) − f ε k L χ ( Ω × [ ,T ]) : = T Z Z Ω χ ( x, t )( A ( q ) − f ε ) dxdt. (3.1)Here χ ( x, t ) is a characteristic function of incomplete measurements (2.9). In ourcase the misfit function (3.1) has the form: J ( q ) = N M N X j = M X m = ( k ( x m , t j ; q ) − f εmj ) . Optimization problem can be solved by various methods such as gradient ap-proaches, stochastic methods, etc [8]. The misfit function (3.1) has a lot of lo-cal minimums due to ill-posedness of inverse problem (2.6), (2.9). In paper [5]authors applied the Tikhonov regularization approach based on gradient methodwith Tikhonov regularization term. The main weaknesses of this approach are thedifficulty of choosing the regularization parameter and the dependence of the con-vergence of the gradient method on the choice of the initial approximation (localconvergence). We choose the stochastic algorithm of global optimization basedon solving more simple evolutionary problems from biology named differentialevolution algorithm [26].
Differential evolution algorithm (DE), a class of evolutionary algorithms, was in-troduced by Storn and Price at 1995 [17, 25–27] for solving a polynomial fittingproblem. The algorithm is generally called as a very simple but very powerfulpopulation-based meta-heuristic algorithm [18]. The algorithm is generally char-acterized by the features of simplicity, effectiveness and robustness. Also, it is
S. Kabanikhin, O. Krivorotko, M. Bektemessov and others easy-to-use, and it requires few controlling parameters, and it has fast convergencecharacteristic [27]. Due to these advantages, it presents a wide range of imple-mentation examples in different areas such as acoustics, biology, material science,mechanic, medical imaging, optic, mathematics, physics, seismology, economicsetc. More details and examples about the implementation of DE to solve variousproblems are given in [18]. Even though previous comprehensive studies overreal-world problems have shown that DE performs better in terms of convergencerate and robustness [6, 21] than the other evolutionary algorithms such as geneticalgorithm, particle swarm optimization [10], simulated annealing [11], etc.An algorithm of differential evolution is follows:1.
Initialization . Create an initial population of target vectors of parameters q i,G = (cid:16) q i,G , q i,G , q i,G (cid:17) , i = , . . . , N p , where N p is the population size, G denotes current generation. Here q i,G = α i,G , q i,G = α i,G , q i,G = p i,G . The algorithm is initialized by a randomly created population within apredefined search space considering the upper (index u ) and lower (index l )bounds of each parameter q ji,G ∈ [ q jl , q ju ] , j = , , Choose stopping criteria . Set the stopping parameter ε stop for misfit functionand maximum number of iterations G max . If J ( q i,G ) < ε stop for some i = , . . . , N p or G = G max then stop iterations and choose i with minimumvalue of misfit function J ( q i,G ) . Otherwise go to step 3.3. Mutation . At each iteration, the algorithm generates a new generation ofvectors, randomly combining vectors form the previous generation. For eachnew generation ( G +
1) of a vector from a given target vector q i from theold generation ( G ) algorithm randomly selects three vectors q r ,G , q r ,G and q r ,G such that i, r , r , r are distinct and creates a donor vector v i,G + = q r ,G + F ( q r ,G − q r ,G ) , F ∈ [ , ] is a differential weight.4. Crossover (recombination) . Create the trial vector u i,G from the elements ofthe target vector q i,G and donor vector v i,G + with probability Cr ∈ [ , ] using formula: u ji,G + = ( v ji,G + , if rand i,j ≤ Cr or j = j rand ,q ji,G , otherwise , j = , , . Here rand i,j represents a uniformly distributed random variable in the rangeof [ , ) , j rand is a randomly chosen integer in the range [ , ] to provide thatthe trial vector does not duplicate the target vector. ifferential evolution algorithm for spatial Solow model Selection . The vector obtained after crossover is the test vector. If it is betterthan the base vector, then in the new generation the base vector is replaced bytrial one, otherwise the base vector is stored in the new generation. Choosethe next generation as follows: q i,G + = ( u i,G + , J ( u i,G + ) ≤ J ( q i,G ) ,q i,G , otherwise . , i = , . . . , N p and go to step 2 till G + < G max . We will show some numerical results of the inverse problem (2.6), (2.9) usingDE algorithm described is Section 3.1. We start by giving some details aboutthe simulated dataset used for the calculations and then show the identificationresults for a constant technological level A (Section 4.2) and for a space-dependenttechnology term A ( x ) (Section 4.3). Consider the modelling scaled domain [ , L ] × [ , T ] with L =
50 and T = L can numerated regions with different GDP and T described time in years).After nondimensionalization we get new computational domain [ , ] × [ , δT ] where mathematical problem (2.6) is formulated. We put δ = .
05. We set anequidistant grid with N x =
26 nodal points in space and N t =
251 nodal pointsin time, which leads to a spatial-step size h x = /N x = .
04 and a time-step size h t = δT /N t = .
03. The classic second-order difference approximation has beenused to discretize the diffusion. The time derivative is approximated by backwarddifference of the first-order.We put an initial condition k ( x ) as a piece-wise function on the interval [ , ] : k ( x ) = , x ∈ [ , . ) , ( x − . ) , x ∈ [ . , . ] , , x ∈ ( . , ] . We obtain the synthetic data f mj from (2.9) for different M and N by solving thedirect problem (2.6) with the production function q ex ( k ) = . k + . k S. Kabanikhin, O. Krivorotko, M. Bektemessov and others presented at figure 1 (left) and two types of technological terms A ( x, t ) (see be-low). Measurements are uniformly distributed in space on [ , ] and time on [ δT / , δT ] (see example for M = N = q ex ( k ) k x t -2 0 2 4 6 8 10 12 14 16 18 20 Figure 1. The exact production function q ex ( k ) (left) and map of direct problem so-lution k ( x, t ; q ex ) with points m = , . . . , M , j = , . . . , N of measurements (2.9)for M = N = the Gaussian noise to inverse problem data (2.9) as follows f εmj = f mj + εf mj ξ mj , m = , . . . , M, j = , . . . , N. Here ξ mj ∼ N , is a normally distributed modeled random variables with zeromean and unit dispersion, ε is an error level.For DE algorithm we put population size N p =
100 and we choose param-eters F = . Cr = . G max = ε stop = − . For getting the optimized solution of the inverse problem we launchthe DE algorithm 1000 times for all decribed numerical calculations using thecluster NKS-30T in the Siberian Supercomputer Center in the Institute of Compu-tational Mathematics and Mathematical Geophysics of the SB RAS and then takethe arithmetic average. A ( x, t ) = We solve optimization problem (3.1) with constant technological term A ( x, t ) = ε = . q ε ( k ) for four variants of M and N . Figure 2 (left) demonstratesthe difference δ ( k ) = q ex ( k ) − q ε ( k ) of exact and calculated solutions of inverseproblem with four variants of measurements. Table 1 shows that the M = N = ifferential evolution algorithm for spatial Solow model
116 is sufficient for reconstruction of production function with necessary accuracyin relative error ρ ε = k k ( · , · ; q ex ) − k ( · , · ; q ε )) k L / k k ( · , · ; q ex ) k L . The smallernumber of measurement points the greater the difference δ ( k ) (see figure 2 left). -0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0 5 10 15 20 δ ( k ) k M=3, N=2M=4, N=4M=5, N=6M=13, N=10 -0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0 5 10 15 20 δ ( k ) k ε = 0 ε = 0.05 ε = 0.1 Figure 2. The difference δ ( k ) of exact and approximate solutions for different pointsof measurements M and N for fixed error level in data (2.9) ε = . δ ( k ) of exact and approximate solutions for different noise levels ε = , . , . M = N = J ( q ε ) for different numberof measurements (2.9) with error level ε = . A ( x, t ) = Values of M and N max | δ ( k ) | ρ ε J ( q ε ) M = N = M = N = M = N = M = N =
10 0.01 0.006 0.199
For ε =
0, 0 .
05 and 0 . M = N =
6, we get the inverseproblem solution q ε ( k ) (the differences δ ( k ) are plotted on figure 2 right).Table 2shows the reconstructed parameters α , α and p in function (2.7) for differenterror level in measured data. If we have noise free data of inverse problem thenthe difference δ ( k ) is close to zero. It means that reconstruction of parameters α , α , p is close to the tested ones (the maximum of the absolute difference δ ( k ) is equal to 0.005 as given in table 2). Note, that maximum absolute error of inverseproblem solutions for M = N = | δ ( k ) | ≤ .
02 formaximum error level in inverse problem data (2.9) ε = . k ( x, t ; q ε ) of spatial Solow mathematical model for reconstructed2 S. Kabanikhin, O. Krivorotko, M. Bektemessov and othersTable 2. Reconstructed parameters in function q ε ( k ) for different error levels ε = , . , . M = N = A ( x, t ) = Parameters Exact values Error level in inverse problem data ε = ε = . ε = . α α p | δ ( k ) | x t = 75 0 5 10 15 20 0 10 20 30 40 50 x t = 100.2 0 5 10 15 20 0 10 20 30 40 50 x t = 138 Figure 3. The solution k ( x, t j ; q ε ) of the direct problem (2.6) for reconstructed q ε ( k ) with error level in measurements ε = . M = N = t = t = . t = q ε ( k ) and measured data (2.9) for ε = . k ( x, t ; q ε ) (red line for fixed time point)with measured synthetic noisy data with noise level ε = . Sensitivity analysis of spatial Solow mathematical model.
Investigate the influence of parameters α , α and p to the mathematical model (2.6)namely to the right-hand side g ( k, x, t ) = γ α k p + α k p − k, γ = A ( x, t ) δ . For this function g , consider its gradient by parameters: ∂g∂α = γk p + α k p , ∂g∂α = − γα k p ( + α k p ) , ∂g∂p = − γα ln ( k ) k p ( + α k p ) . For different values of function k ( x, t ) we construct the gradient field of function ifferential evolution algorithm for spatial Solow model ∂ g/ ∂ p ∂ g/ ∂α ∂ g/ ∂α ∂ g/ ∂ p a) ∂ g/ ∂ p ∂ g/ ∂α ∂ g/ ∂α ∂ g/ ∂ p b) ∂ g/ ∂ p ∂ g/ ∂α ∂ g/ ∂α ∂ g/ ∂ p c) Figure 4. The gradient field ( ∂g∂α , ∂g∂α , ∂g∂p ) for fixed values of k ( x, t ) : a) k ( x, t ) = .
5, b) k ( x, t ) = . k ( x, t ) = .
25. Red color arrows show the projectionof the gradient field on ( α , α ) plane. g . Figure 4 shows that the maximum rate of gradient variability for small valuesof capital stock k ( x, t ) corresponds to parameters α and α . For bigger values4 S. Kabanikhin, O. Krivorotko, M. Bektemessov and others of k ( x, t ) (figure 4 right) and for small values of parameters α and α gradientgrows to the direction of parameter p , but when the values of parameters α and α became bigger the gradient growth turns to parameters α and α again. A ( x ) We consider a space-dependent technological level A ( x ) demonstrated at figure 5(left). Then the solution of the direct problem (2.6) for the exact function q ex demonstrates on figure 5 (right). A ( x ) x
0 40 80 120 0 10 20 30 40 50 5 10 15 20 t x
Figure 5. The space-dependent technological term A ( x ) (left) and the solution ofthe direct problem (2.6) for space-dependent technological level A ( x ) and q ex ( k ) (right). Using the same simulated dataset (see Section 4.1) the inverse problem (2.6), (2.9)is solved for number of measurements M = N = ε = , . , .
1. The results are collected to table 3 and demonstrated at figure 6.Note, that the results of inverse problem solution are the same as for constant tech-nological term A (see Section 4.2), i.e. accuracy in relative error ρ ε is less than10 − , maximum of absolute difference of exact and approximated solutions of in-verse problem max | δ ( k ) | is the same order of 10 − . The difference of exact andapproximated solutions of inverse problem δ ( k ) for ε = , . , . α , α and p (see table 3) is not critical to the behaviorof the function q ( k ) (see figure 6 from the left that demonstrated the exact andreconstructed solutions of inverse problem for error level in data (2.9) ε = . M = N = k ( x, t ; q ε ) of spatial Solow mathematical model for reconstructed q ε ( k ) and measured data (2.9) for ε = . ifferential evolution algorithm for spatial Solow model Table 3. Reconstructed parameters in function q ε ( k ) for different error levels ε = , . , . M = N = A ( x ) . Parameters Exact values Error level in inverse problem data ε = ε = . ε = . α α p | δ ( k ) | ρ ε J ( q ε ) · − . · − q ( k ) k Exact solutionApproximate solution -0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0 5 10 15 20 δ ( k ) k ε = 0 ε = 0.05 ε = 0.1 Figure 6. The exact q ex ( k ) and approximate q ε ( k ) solutions of inverse problemfor spatial Solow model for error level in data (2.9) ε = . δ ( k ) of exact and approximate solutions of inverse problem for different noise levels ε = , . , . M = N = A ( x ) . of space-dependent technological level A ( x ) . Note that capital stocks k ( x, t j ; q ε ) , j = , , f ε (black triangles)as expected. Today, economists use Solow’s sources-of-growth accounting to estimate the sep-arate effects on economic growth of technological change, capital, and labor. Oneimportant use of the Solow growth model is to estimate the share of observed6
S. Kabanikhin, O. Krivorotko, M. Bektemessov and others xt = 75 0 5 10 15 20 0 10 20 30 40 50 xt = 100.2 0 5 10 15 20 0 10 20 30 40 50 xt = 138 Figure 7. The solution k ( x, t j ; q ε ) of the direct problem (2.6) for reconstructed q ε ( k ) with error level in measurements ε = . M = N = t = t = . t = growth that has resulted from growth in Total Factor Productivity (TFP), ratherthan from the application of increased inputs - labor, capital, and human capi-tal (increased productive skills resulting from education and training.) Using theSolow model to approximate the output that would result in the absence of anychange in TFP, you can then subtract this value from the output actually produced,and attribute the difference to TFP growth. The growth Solow model is the startingpoint of all analyses in modern economic growth theories, thus understanding ofthe model is essential to understanding the theories of the Solow growth.The differential evolution algorithm is applied to the optimization problem ofthe production function q ( k ) reconstruction for the spatial Solow model using ad-ditional measurements of GDP type for fixed space and time. Despite the fact thatthe considered inverse problem is ill-posed, numerical calculations show a goodresult with the accuracy of recovery of the production function is more than 95%(in case of error level in measured data 10%). We compare the results with cal-culations from paper [5] where the authors applied Tikhonov regularization andgradient method for solving the regularized optimization problem. In the case offull measured data (that means f ε ( x, t ) = k ( x, t )+ ε ( x, t ) ) and error level 10% theaccuracy of reconstruction of production function was 80% for both cases of tech-nological levels. The reason consists in sensitivity of local regularization methodsto an initial approximation while the reconstruction results for DE approach do notdepend on initial population. Acknowledgments.
Authors thank to Professor Daniyar Nurseitov for problemstatement and fruitfull discussions for concerning Solow mathematical model andthank to Dr. Igor Chernykh for help in parallel realization of DE algorithm. ifferential evolution algorithm for spatial Solow model Bibliography [1] Ç. Balkaya, An implementation of differential evolution algorithm for inversion ofgeoelectrical data,
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Sergey Kabanikhin, Institute of Computational Mathematics and MathematicalGeophysics SB RAS, Prosp. Akad. Lavrentyeva 6, Novosibirsk State University,Pirogova str. 2, 630090 Novosibirsk, Russia.E-mail: [email protected]
Olga Krivorotko, Institute of Computational Mathematics and Mathematical GeophysicsSB RAS, Prosp. Akad. Lavrentyeva 6, Novosibirsk State University, Pirogova str. 2,630090 Novosibirsk, Russia.E-mail: [email protected] ifferential evolution algorithm for spatial Solow model Maktagali Bektemessov, Abai Kazakh National Pedagogical University, Dostyk ave. 13,050010 Almaty, Republic of Kazakhstan.E-mail: [email protected]
Zholaman Bektemessov, Al-Farabi Kazakh National University, Kazybek Bi st. 30,050040 Almaty, Republic of Kazakhstan.E-mail: [email protected]