convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

a clustering technique for the identification of piecewise affine systems we propose a new technique for the identification of discrete time hybrid systems in the piecewise affine pwa form this problem can be formulated as the reconstruction of a possibly discontinuous pwa map with a multi dimensional domain in order to achieve our goal we provide an algorithm that exploits the combined use of clustering linear identification and pattern recognition techniques this allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures moreover the clustering step used for classifying the datapoints is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure

recursive prediction error identification using the nonlinear wiener model the nonlinear wiener model consisting of a linear dynamic block in cascade with a static nonlinearity is considered a recursive prediction error identification algorithm based on the wiener model is derived the linear dynamic block is modelled as a siso transfer function operator and the static nonlinearity is approximated with a piecewise linear function a theoretical analysis of the method is carried out and conditions for local convergence to the true parameter vector are given in particular the analysis shows that the input signal should be such that there is signal energy in the whole range of the piecewise linear approximation a numerical example illustrates the performance of the algorithm further practical guidelines on how to apply the algorithm are also included in the paper

numerical solution for a nonlinear one dimensional problem of thermoelasticity a numerical solution for a nonlinear one dimensional boundary value problem of thermoelasticity for the elastic half space is presented the resulting equations are discussed and the numerical method is investigated for stability comparison with other existing numerical schemes is carried out the obtained results clearly indicate the process of shock formation the presented figures show the effects of different nonlinear coupling constants on the distributions of the mechanical displacement and temperature in the medium a special case is briefly discussed

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

the pointwise estimates to solutions for 1 dimensional linear thermoelastic system with second sound in this paper we study the thermoelastic system with second sound in one space variable the mathematical model is a hyperbolic partial differential system with damping term which brings about some parabolic properties the pointwise estimates of the solution are obtained and from which the generalized huygens principle is shown our approach is based on detailed analysis on the green function of the system

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

the stefan problem for a hyperbolic heat equation abstract in this paper we consider the 1 phase 1 dimensional stefan problem corresponding to the hyperbolic heat equation obtained by relaxing the fourier law taking q t q k x where q is the flux the temperature k the conductivity and 0 the relaxation coefficient we prove existence and uniqueness of a smooth solution with smooth free boundary we also study the limiting solution as 0 showing under some conditions that it converges to the solution of the classical stefan problem

a one phase hyperbolic stefan problem in multi dimensional space the hyperbolic heat transfer model is obtained by replacing the classical fourier s law with the relaxation relation q t q k t the sufficient and necessary conditions are derived for the local existence and uniqueness of classical solutions for multi d stefan problem of hyperbolic heat transfer model where phase change is accompanied with delay of latent heat storage

recursive identification method for piecewise arx models a sparse estimation approach this paper deals with the identification of nonlinear systems using piecewise linear models by means of a sparse overparameterization this challenging problem is turned into a convex optimization problem the proposed method uses a likelihood based methodology which adaptively penalizes model complexity and directly leads to a recursive implementation in this sparse estimation approach the tuning of user parameters is avoided and the computational complexity is kept linear in the number of data samples numerical examples with both simulated and experimental data are presented and the results are compared with previously published methods

the existence and stability of conic shock waves we study the existence of the nonsymmetrical conic shock wave produced by a supersonic flow past a distorted conic projectile for the weak conic shock wave we establish the existence and its linear stability using the mathematical model of an isentropic irrotational flow

asymptotic behavior of solutions to a class of nonlinear evolution equations on etudie les equations de la chaleur a controle de temperature a l interieur ou sur la frontiere du domaine pour cela on considere un type de problemes variationnels paraboliques avec obstacles

a bounded error approach to piecewise affine system identification this paper proposes a three stage procedure for parametric identification of piecewise affine autoregressive exogenous pwarx models the first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the partition into a minimum number of feasible subsystems min pfs problem for a suitable set of linear complementary inequalities derived from data second a refinement procedure reduces misclassifications and improves parameter estimates the third stage determines a polyhedral partition of the regressor set via two class or multiclass linear separation techniques as a main feature the algorithm imposes that the identification error is bounded by a quantity spl delta such a bound is a useful tuning parameter to trade off between quality of fit and model complexity the performance of the proposed pwa system identification procedure is demonstrated via numerical examples and on experimental data from an electronic component placement process in a pick and place machine

global solution and large time behavior of the 3d compressible euler equations with damping national natural science foundation of china nsaf 10976026 national natural science foundation of china 11271305

some convergence results for a class of nonlinear phase field evolution equations two heat diffusion problems in the framework of the parabolic phase field model are presented the first problem is related to a single isotropic fluid and the other describes the heat transmission between two different substances in contact some known existence and uniqueness results are briefly recalled then an asymptotic analysis of both situations is carried out as the kinetic equation collapses to a temperature phase relation of stefan type in the first case in the whole material and in the second in only one of the substances in both cases a convergence result for the solutions is proved the second problem shows some more mathematical difficulties that are due to the presence of nontrivial terms on the common boundary in order to control the latter some tools are used from the convergence theory for convex functionals

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

online hyperparameter free sparse estimation method in this paper we derive an online estimator for sparse parameter vectors which unlike the lasso approach does not require the tuning of any hyperparameters the algorithm is based on a covariance matching approach and is equivalent to a weighted version of the square root lasso the computational complexity of the estimator is of the same order as that of the online versions of regularized least squares rls and lasso we provide a numerical comparison with feasible and infeasible implementations of the lasso and rls to illustrate the advantage of the proposed online hyperparameter free estimator

global solutions of the neumann problem in one dimensional nonlinear thermoelasticity for the equations of one dimensional nonlinear thermoelasticity slemrod 17 established in 1981 the global existence of smooth solutions for small initial data in the case of a bounded reference configuration for a homogeneous medium the boundary conditions traction free constant temperature or rigidly clamped insulated he considered yield through the differential equation an additional boundary condition which was crucial for obtaining the necessary energy estimates subsequently jiang 7 discussed the same boundary conditions as in 17 in the case of an unbounded reference configuration the half line kawashima lo zheng and shen 181 and hrusa and tarabek 6 dealt with the one dimensional cauchy problem recently racke and shibata 13 proved a global existence theorem for the dirichlet initial boundary value problem rigidly clamped constant temperature in a bounded interval by using the decay estimates for the linearized equations and the l energy method and the author 8 obtained a similar result for the problem in an unbounded interval the half line by only applying the l energy method more recently using a method similar to that in 13 shibata 16 showed that global small solutions exist for the neumann problem traction free insulated in the case of a bounded reference configuration for an unbounded reference configuration the half line the case of the neumann boundary conditions remained open our purpose in this paper is to prove a global existence theorem for the neumann problem in the half line and in a bounded interval for small data by applying the l2 energy method the proof given here closely follows the pattern of the proof in 8 subject to the necessary modifications we note that a globally defined classical solution should not be expected for large data indeed dafermos and hsiao 4 and hrusa and messaoudi 5 showed that for specialized constitutive equations if the initial data are large then the solution to the cauchy problem will blow up in finite time for three dimensional nonlinear models we refer the reader to the papers 3 9 10 12 151 and the references cited therein this paper is organized as follows in section 1 the equations of one dimensional nonlinear thermoelasticity are presented and the main theorem is stated in section 2 we give a local

l p convergence rates of planar waves for multi dimensional euler equations with damping abstract in this paper the l p convergence rates of planar diffusion waves for multi dimensional euler equations with damping are considered the analysis relies on a newly introduced frequency decomposition and green function based energy method it is a combination of the l p estimate on the low frequency component by using an approximate green function and l 2 estimate on the high frequency component through the energy method by noticing that the low frequency component in the approximate green function has the algebraic decay which governs the large time behavior while the high frequency component has the exponential decay but with singularity their combination leads to a global algebraic decay estimate to use the decay property only of the low frequency component in the approximate green function avoids the singularity in the high frequency component so that it simplifies and improves the previous works on this system this new approach of the combination of the green function and energy method through the frequency decomposition can also be applied to the hyperbolic parabolic systems satisfying the kawashima condition and also the systems whose derivatives of the coefficients have suitable time decay properties

the lp decay estimates for the chemotaxis shallow water system abstract in this paper we consider the large time behavior of solution for the chemotaxis shallow water system in two dimension by the method of frequency decomposition first we construct the green s function of the linearized system and get the corresponding estimates for low frequency part then we introduce an energy method we named double cycle method to get the decay rate for high frequency part due to the dissipative mechanism of this system the high frequency part decays faster than the low frequency part

global smooth solutions and asymptotic stability in one dimensional nonlinear thermoelasticity the global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one dimensional homogeneous reference configuration dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t is described

recursive prediction error identification and scaling of non linear state space models using a restricted black box parameterization a recursive prediction error algorithm for identification of systems described by non linear ordinary differential equation ode models is presented the model is an ode model parameterized with coefficients of a multi variable polynomial that describes one component of the right hand side function of the ode this avoids over parameterization problems the selected model can also handle systems with more complicated right hand side structure by identification of a local input output equivalent system in the coordinate system of the selected state variables a novel technique based on scaling of the sampling period is proposed the technique can improve the conditioning of the identification problem thereby enhancing the chances of convergence to the correct minimum of the criterion the algorithm is applied to live data from a system consisting of two cascaded tanks with promising results a matlab software package which implements the proposed algorithm and a set of support scripts can be freely downloaded from http www it uu se research reports

on a model of phase relaxation for the hyperbolic stefan problem in this paper we study a model of phase relaxation for the stefan problem with the cattaneo maxwell heat flux law we prove an existence and uniqueness result for the resulting problem and we show that its solution converges to the solution of the stefan problem as the two relaxation parameters go to zero provided a relation between these parameters holds

a phase field model with temperature dependent constraint this paper is concerned with a phase field model with temperature dependent constraint for the order parameter in the kinetic equation of the order parameter we take account of diffusion as well as undercooled superheated state which is modelled by means of a hysteresis input output relation the input is the temperature field and the output is the order parameter in this paper we shall show that there exists a solution globally in time in a physically reasonable sense when the coefficient of interfacial energy is sufficiently small

double shock fronts for hyperbolic systems of conservation laws in multidimensional space the existence of a unique double shock front for hyperbolic systems of conservation laws in several space variables is established extending an earlier result of metivier an example of a double shock wave arising from physical applications is given the purpose of this paper is to show the existence of a unique double shock front for hyperbolic systems of conservation laws in several space variables describing physical phenomena unlike the one dimensional case it is only recently that majda 7 using microlocal analysis established the existence of a unique stable shock front for hyperbolic systems for double shock fronts the only known result is due to metivier 8 9 where 2 x 2 systems of conservation laws in two dimensional space are considered in this paper the existence of a double shock front for hyperbolic systems of conservation laws in several space variables is established when restricted to 2 x 2 systems in two dimensional space our result is similar to that of metivier but our approach is somewhat different and perhaps simpler the notation and the basic assumptions are given in 1 the free boundary problem is transformed in 2 into one with fixed boundaries the linearized problem is considered in 3 and the nonlinear one is studied in 4 using the iteration method finally in 5 an application to isentropic gas in two dimensional space is given let g be an open subset of rn with a smooth boundary ag and let x xi xn be the generic point of g for each n tuple a a n of nonnegative integers we write n n d f dj dj 0 x la zaj j l j l received by the editors january 17 1987 and in revised form april 15 1988 1980 mathematics subject classification 1985 revision primary 35l65 35l50 secondary 76l05 work supported by nserc canada 1989 american mathematical society 0002 9947 89 1 00 25 per page

lp convergence rate to nonlinear diffusion waves for p system with damping abstract in this paper we study the p system with frictional damping and show that the solutions time asymptotically tend to the nonlinear diffusion waves governed by the classical darcy s law by introducing an approximate green function we obtain the optimal l p 2 p convergence rate of the solution which is a perturbation of the nonlinear diffusion wave to the hyperbolic system

large time behavior for the system of compressible adiabatic flow through porous media in r3 abstract the cauchy problem of the system of compressible adiabatic flow through porous media in r 3 is considered we obtain the global existence and large time behavior of the solution when the initial data is close to its equilibrium in h 3 norm this is a continuing work of 31 where due to non dissipative property of the entropy s an additional assumption that the initial data is bounded in l 1 norm plays a key role in closing the a priori energy estimates and hence establishing the global existence of the solution in this paper to remove this assumption we first decompose the solution u into high frequency part u h and low frequency part u l then by constructing delicate energy functionals for u h and optimal decay rates for u l we can get the desired energy estimates to close the a priori assumption furthermore we get the optimal l 2 l 2 decay rate of the solution and show that the velocity decays faster than the density

instrumental variable methods for identification of hammerstein systems abstract consistency conditions for instrumental variable methods applied to hammerstein models are derived and analysed it is necessary to have a model that is not over parametrized and an input that is strongly persistently exciting i e the inputs and their powers are jointly persistently exciting some specific choices of the instruments are proved to give consistency those instruments are formed as filtered inputs and powers of the input consistency can be guaranteed if either the input is white noise or if a certain transfer function is positive real the vector of instruments is in general of larger dimension than the parameter vector it is shown then that a non singular linear transformation of the instruments gives a new and still consistent estimate some simulations using the iv variants proposed in this paper are also included

abstract approach to evolution equations of phase field type and applications a general system of abstract nonlinear parabolic equations deriving from phase field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved then applications are presented to two physical situations first a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting second a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case moreover through convex analysis techniques it is performed a mathematical study of the subdifferential type relations linking the phase fields and the temperatures to the internal energies of the fluids email schimper dragon ian pv cnr it keywords phase field models subdifferential operators abstract evolution equations

the pointwise estimates of solutions for euler equations with damping in multi dimensions we study the time asymptotic behavior of solutions for the isentropic euler equations with damping in multi dimensions the global existence and pointwise estimates of the solutions are obtained furthermore we obtain the optimal lp 1 p convergence rate of the solution when it is a perturbation of a constant state our approach is based on a detailed analysis of the green function of the linearized system and some energy estimates

global existence and asymptotic behavior for one dimensional thermoelasticity of integral type with hyperbolic heat conduction abstract we consider an initial boundary value problem for thermoelasticity of integral type with hyperbolic heat conduction we show that for small initial data in h 1 norm there exist globally defined solutions which are exponentially stable in bounded spatial domains

identification of switched linear regression models using sum of norms regularization this paper proposes a general convex framework for the identification of switched linear systems the proposed framework uses over parameterization to avoid solving the otherwise combinatorially forbidding identification problem and takes the form of a least squares problem with a sum of norms regularization a generalization of the 1 regularization the regularization constant regulates the complexity and is used to trade off the fit and the number of submodels

identification of piecewise affine systems via mixed integer programming this paper addresses the problem of identification of hybrid dynamical systems by focusing the attention on hinging hyperplanes and wiener piecewise affine autoregressive exogenous models in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron in particular we provide algorithms based on mixed integer linear or quadratic programming which are guaranteed to converge to a global optimum for the special case where the estimation data only seldom switches between the different submodels we also suggest a way of trading off between optimality and complexity by using a change detection approach

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

existence of stationary supersonic flows past a pointed body in this paper we study the mathematical aspects of the stationary supersonic flow past a non axisymmetric curved pointed body the flow is described by a steady potential flow equation which is a quasilinear hyperbolic equation of second order we prove the local existence of the solution to this problem with a pointed shock attached at the tip of the pointed body provided the pointed body is a perturbation of a circular cone and the vertex angle of the approximate cone of the pointed body is less than a critical value the solution is smooth in between the shock and the surface of the body consequently such a structure of flow near the tip of the pointed body and its stability is verified mathematically

phase field systems with nonlinear coupling and dynamic boundary conditions we consider phase field systems of caginalp type on a three dimensional bounded domain the order parameter fulfills a dynamic boundary condition while the relative temperature is subject to a homogeneous boundary condition of dirichlet neumann or robin type moreover the two equations are nonlinearly coupled through a quadratic growth function here we extend several results which have been proven by some of the authors for the linear coupling more precisely we demonstrate the existence and uniqueness of global solutions then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors we also discuss the convergence of given solutions to a single equilibrium

on the complexity of piecewise affine system identification the paper provides results regarding the computational complexity of hybrid system identification more precisely we focus on the estimation of piecewise affine pwa maps from input output data and analyze the complexity of computing a global minimizer of the error previous work showed that a global solution could be obtained for continuous pwa maps with a worst case complexity exponential in the number of data in this paper we show how global optimality can be reached for a slightly more general class of possibly discontinuous pwa maps with a complexity only polynomial in the number of data however with an exponential complexity with respect to the data dimension this result is obtained via an analysis of the intrinsic classification subproblem of associating the data points to the different modes in addition we prove that the problem is np hard and thus that the exponential complexity in the dimension is a natural expectation for any exact algorithm

identification of piecewise affine systems based on statistical clustering technique this paper is concerned with the identification of a class of piecewise affine systems called a piecewise affine autoregressive exogenous pwarx model the pwarx model is composed of arx sub models each of which corresponds to a polyhedral region of the regression space under the temporary assumption that the number of sub models is known a priori the input output data are collected into several clusters by using a statistical clustering algorithm we utilize support vector classifiers to estimate the boundary hyperplane between two adjacent regions in the regression space in each cluster the parameter vector of the sub model is obtained by the least squares method it turns out that the present statistical clustering approach enables us to estimate the number of sub models based on the information criteria such as caic and mdl the estimate of the number of sub models is performed by applying the identification procedure several times to the same data set after having fixed the number of sub models to different values finally we verify the applicability of the present identification method through a numerical example of a hammerstein model

steady supersonic flow past an almost straight wedge with large vertex angle abstract this paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary it is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic this paper is devoted to generalizing this result under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small a sequence of approximate solutions constructed by a modified glimm scheme is proved to be convergent to a global weak solution of the steady problem a sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution the regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied

on a system of nonlinear pde s for phase transitions with vector order parameter and convective effect the paper deals with a system of nonlinear pde s which describes a phase field model with convection and temperature dependent constraint to the vector order parameter existence of solutions for the system under consideration is proved by the method of yosida approximation and fixed point arguments

existence and uniqueness of solutions for a non uniformly parabolic equation abstract in this paper we study a generalized burgers equation ut u2 x tuxx which is a non uniformly parabolic equation for t 0 we show the existence and uniqueness of classical solutions to the initial value problem of the generalized burgers equation with rough initial data belonging to l r

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

a posteriori error estimates for variable time step discretizations of nonlinear evolution equations we study the backward euler method with variable time steps for abstract evolution equations in hilbert spaces exploiting convexity of the underlying potential or the angle bounded condition thereby assuming no further regularity we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual these estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps we also prove that they converge to zero with an optimal rate with respect to the regularity of the solution we apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character 2000 john wiley sons inc

general boundary stabilization of memory type thermoelasticity in this paper we consider an n dimensional system of thermoelasticity where a viscoelastic dissipation is acting on a part of the boundary by using a lemma first appeared in the work of messaoudi and soufyane general decay of solutions of a wave equation with a boundary control of memory type nonlinear anal real world appl to be published we establish a general decay result from which the usual exponential and polynomial decay are only special cases our result improves earlier ones in the literature and extends the general result of messaoudi and soufyane general decay of solutions of a wave equation with a boundary control of memory type nonlinear anal real world appl 11 2896 2010 to thermoelastic systems

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection