optimal h model reduction via linear matrix inequalities continuous and discrete time cases necessary and sufficient conditions are derived for the existence of a solution to the continuous time and discrete time h model reduction problems these conditions are expressed in terms of linear matrix inequalities lmis and a coupling non convex rank constraint set in addition an explicit parametrization of all reduced order models that correspond to a feasible solution is provided in terms of a contractive matrix these results follow from the recent solution of the h control design problem using lmis particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth order h approximation problem and the convexity of this problem is demonstrated computational issues are discussed and an iterative procedure is proposed to solve the h model reduction problem using alternating projections although global convergence of the algorithm is not guaranteed

homotopy methods for solving the optimal projection equations for the h2 reduced order model problem the optimal projection approach to solving the h2 reduced order model problem produces two coupled highly nonlinear matrix equations with rank conditions as constraints due to the resemblance of these equations to standard matrix lyapunov equations they are called modified lyapunov equations the algorithms proposed herein utilize probability one homotopy theory as the main tool it is shown that there is a family of systems the homotopy that make a continuous transformation from some initial system to the final system with a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable controllable and observable one method which solves the equations in their original form requires a decomposition of the projection matrix using the drazin inverse of a matrix it is shown that the appropriate inverse is a differentiable function an effective algorithm for computing the derivative of the projection matrix that involves solving a set of sylvester equations is given another class of methods considers the equations in a modified form using a decomposition of the pseudogramians based on a contragredient transformation some freedom is left in making an exact match between the number of equations and the number of unknowns thus effectively generating a family of methods

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

strong tractability of multivariate integration using quasi monte carlo algorithms we study quasi monte carlo algorithms based on low discrepancy sequences for multivariate integration we consider the problem of how the minimal number of function evaluations needed to reduce the worst case error from its initial error by a factor of e depends on e 1 and the dimension s strong tractability means that it does not depend on s and is bounded by a polynomial in e 1 the least possible value of the power of e 1 is called the e exponent of strong tractability sloan and wozniakowski established a necessary and sufficient condition of strong tractability in weighted sobolev spaces and showed that the e exponent of strong tractability is between 1 and 2 however their proof is not constructive in this paper we prove in a constructive way that multivariate integration in some weighted sobolev spaces is strongly tractable with e exponent equal to 1 which is the best possible value under a stronger assumption than sloan and wozniakowski s assumption we show that quasi monte carlo algorithms using niederreiter s t s sequences and sobol sequences achieve the optimal convergence order o n 1 for any 0 independent of the dimension with a worst case deterministic guarantee where n is the number of function evaluations this implies that strong tractability with the best e exponent can be achieved in appropriate weighted sobolev spaces by using niederreiter s t s sequences and sobol sequences

internal positivity preserved model reduction this article studies model reduction of continuous time stable positive linear systems under the hankel norm h norm and h 2 norm performance the reduced order systems preserve the stability as well as the positivity of the original systems this is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the lyapunov matrices are decoupled with the system matrices in this way the positivity constraints in the reduced order model can be imposed in a natural way as the model reduction performances are expressed in linear matrix inequalities with equality constraints the desired reduced order positive models can be obtained by using the cone complementarity linearisation iterative algorithm a numerical example is presented to illustrate the effectiveness of the given methods

alternating convex projection methods for covariance control design the problem of designing a static state feedback or full order dynamic controller is formulated as a problem of designing an appropriate plant state covariance matrix we show that closed loop stability and multiple output norm constraints imply that the plant state covariance matrix lies at the intersection of some specified closed convex sets in the space of symmetric matrices we address the covariance feasibility problem to determine the existence and compute a covariance matrix to satisfy assignability and output norm performance constraints we address the covariance optimization problem to construct an assignable covariance matrix which satisfies output performance constraints and is as close as possible to a given desired covariance we also treat inconsistent constraints where we look for an assignable covariance matrix which best approximates desired but non achievable output performance objectives we call this the infeasible covariance optimization problem all these problems are of a conve

paper low order control design for lmi problems using alternating projection methods computational techniques based on alternating projections are proposed to solve control design problems described by linear matrix inequalities lmis in particular we concentrate on the stabilization and the suboptimal h output feedback control design problems these problems can be described by a pair of lmis and an additional coupling condition this coupling condition is convex for the full order control design problem but convexity is lost for the control problem of order strictly less than the plant order we formulate these problems as feasibility problems with matrix constraint sets of simple geometry and we utilize this geometry to obtain analytical expressions for the orthogonal projection operators onto these sets full order and low order controllers are designed using alternating projection methods for the full order controller case global convergence of the alternating projection methods to a feasible solution is guaranteed however for the low order control case only local convergence is guaranteed an example is provided to illustrate the use of these methods for the full order and the low order controller design

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

extremum properties of finite step solutions in elastoplasticity with nonlinear mixed hardening abstract the class of elastic plastic constitutive laws assumed herein can be described as follows we envisage y 1 yield criteria or modes which define the current yield surface each yield function consists of two addends an effective stress and a yield limit y 1 y the former is an order one positively homogeneous function of the difference between the stresses ii and reference stresses ii which are generally nonlinear functional of the plastic strain history so that kinematic hardening is accounted for the yield limit y is generally a nonlinear function of y nondecreasing internal variables so that another hardening mechanism which may reduce to isotropic hardening is accommodated in the model the rates of these variables play the role of plastic multipliers in the flow rule a finite step in the geometrically linear evolutive analysis of such solids is defined according to the backward difference or stepwise holonomic strategy for approximate time integration for the finite step boundary value problem thus arising various extremum characterizations of solutions are established and the underlying constitutive restrictions are pointed out and discussed

component cost analysis of large scale systems abstract a brief summary of component cost analysis cca is given cca provides a means to determine the contribution of each dynamical element of the system in the overall performance metric using cca a theory of cost equivalent realizations is developed cost equivalent realizations are realizations that have the same value of the performance metric as the original system and they can be of lower order than kalman s minimal realization

an fe mpec approach for limit load evaluation in the presence of contact and displacement constraints abstract this paper describes a mathematical programing based approach for the direct limit load evaluation of a structural system under simultaneous contact and limited displacement conditions the contact model we adopt can simulate either a classical unilateral nonassociative coulomb friction situation or a cohesive fracture idealization at the potential discontinuity interface between contacting bodies the discrete fe model is constructed using locking free mixed finite elements the main feature and novelty of our proposed approach is to compute in a single step the maximum load capacity of the structure such that both the imposed displacement limitations and the contact conditions are satisfied in essence the formulation is a nontrivial extension of classical limit analysis the analysis is cast in its most natural form namely in mixed static kinematic variables and leads to what is known in the mathematical programing literature as a mathematical program with equilibrium constraints or mpec a nonlinear programing nlp based algorithm is proposed to solve the mpec two examples are provided to illustrate application of the proposed scheme and some comments regarding the various advantages of the adopted mathematical programing framework are made

error estimates for the anova method with polynomial chaos interpolation tensor product functions we focus on the analysis of variance anova method for high dimensional function approximation using jacobi polynomial chaos to represent the terms of the expansion first we develop a weight theory inspired by quasi monte carlo theory to identify which functions have low effective dimension using the anova expansion in different norms we then present estimates for the truncation error in the anova expansion and for the interpolation error using multielement polynomial chaos in the weighted korobov spaces over the unit hypercube we consider both the standard anova expansion using the lebesgue measure and the anchored anova expansion using the dirac measure the optimality of different sets of anchor points is also examined through numerical examples

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

model reduction of discrete time systems through linear matrix inequalities in this paper model reduction of discrete time linear systems is revisited the main objective is to define a convex programming problem expressed in terms of linear matrix inequalities which provides a good suboptimal solution to the model reduction problem the quality of the proposed suboptimal solution is assessed through comparisons performed via simulation with the already classical balanced truncation method the results lead to the conclusion that the proposed method performs equivalently when the h 2 norm of the reduction error is considered and significantly better when the h norm is considered instead

adaptive surrogate modeling by anova and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation the polynomial dimensional decomposition pdd is employed in this work for the global sensitivity analysis and uncertainty quantification uq of stochastic systems subject to a moderate to large number of input random variables due to the intimate connection between the pdd and the analysis of variance anova approaches pdd is able to provide a simpler and more direct evaluation of the sobol sensitivity indices when compared to the polynomial chaos expansion pc unfortunately the number of pdd terms grows exponentially with respect to the size of the input random vector which makes the computational cost of standard methods unaffordable for real engineering applications in order to address the problem of the curse of dimensionality this work proposes essentially variance based adaptive strategies aiming to build a cheap meta model i e surrogate model by employing the sparse pdd approach with its coefficients computed by regression three levels of adaptivity are carried out in this paper 1 the truncated dimensionality for anova component functions 2 the active dimension technique especially for second and higher order parameter interactions and 3 the stepwise regression approach designed to retain only the most influential polynomials in the pdd expansion during this adaptive procedure featuring stepwise regressions the surrogate model representation keeps containing few terms so that the cost to resolve repeatedly the linear systems of the least squares regression problem is negligible the size of the finally obtained sparse pdd representation is much smaller than the one of the full expansion since only significant terms are eventually retained consequently a much smaller number of calls to the deterministic model is required to compute the final pdd coefficients

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

efficient computation of multiple solutions in quasibrittle fracture analysis we propose a novel and efficient computational scheme for capturing the multiplicity of solutions that may exist at any loading step of a quasibrittle fracture analysis formulated as a linear complementarity problem lcp the algorithm proposed is based on successively augmenting the current lcp by complementarity constraints that effectively remove previously found solutions from the solution set we focus on the single step or so called holonomic analysis for mode i behavior the performance of the proposed approach is illustrated by means of physically meaningful benchmark problems

investigation of rotating hollow cylinders of strain hardening viscoplastic materials by sequential limit analysis the paper employs sequential limit analysis to investigate rotating hollow cylinders of nonlinear isotropic strain hardening viscoplastic materials a computational optimization procedure was developed to account for hardening material properties and weakening behavior corresponding to the strain rate sensitivity and widening deformation a sequence of limit analysis problems was then conducted to seek the corresponding plastic limit angular velocities sequentially by a general algorithm incorporated with an inner and outer iterative sequence particularly analytical solutions of plastic limit angular velocities and the onset of instability were derived for rigorous comparisons the corresponding stability condition was also obtained explicitly specifically the implicit form of the onset of instability was solved by the fixed point iteration it is found that the computed limit angular velocities are rigorous upper bounds and match well with analytical solutions

technical communique a convex optimization approach to the mode acceleration problem the purpose of this note is to introduce an alternative procedure to the mode acceleration method when the underlying structure model includes damping we will show that the problem can be cast as a convex optimization problem that can be solved via linear matrix inequalities

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

convergence analysis and validation of sequential limit analysis of plane strain problems of the von mises model with non linear isotropic hardening the paper presents sequential limit analysis of plane strain problems of the von mises model with non linear isotropic hardening by using a general algorithm the general algorithm is a combined smoothing and successive approximation cssa method in the paper emphasis is placed on its convergence analysis and validation applied to sequential limit analysis involving materials with isotropic hardening by sequential limit analysis the paper treats deforming problems as a sequence of limit analysis problems stated in the upper bound formulation especially the cssa algorithm was proved to be unconditionally convergent by utilizing the cauchy schwarz inequality finally rigorous validation was conducted by numerical and analytical studies of a thick walled cylinder under pressure it is found that the computed limit loads are rigorous upper bounds and agree very well with the analytical solutions copyright 2005 john wiley sons ltd

l2 and l2 l model reduction via linear matrix inequalities necessary and sufficient conditions are derived for the existance of a solution to the continuous time and discrete time suboptimal l2 and l2 l model reduction problems these conditions are expressed in terms of linear matrix inequalities lmis and a coupling non convex rank constraint in addition explicit parametrizations of all reduced order models that correspond to a feasible solution are presented in terms of contractive matrices

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

multi element probabilistic collocation method in high dimensions we combine multi element polynomial chaos with analysis of variance anova functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity specifically we employ the multi element probabilistic collocation method mepcm 1 and so we refer to the new method as mepcm a we investigate the dependence of the convergence of mepcm a on two decomposition parameters the polynomial order m and the effective dimension n with n n and n the nominal dimension numerical tests for multi dimensional integration and for stochastic elliptic problems suggest that n m for monotonic convergence of the method we also employ mepcm a to obtain error bars for the piezometric head at the hanford nuclear waste site under stochastic hydraulic conductivity conditions finally we compare the cost of mepcm a against monte carlo in several hundred dimensions and we find mepcm a to be more efficient for up to 600 dimensions for a specific multi dimensional integration problem involving a discontinuous function

projection properties of the l 2 optimal reduced order model the paper investigates the optimal l 2 reduced order model in the context of reduced order models obtained by projection it deals first with the general case of such systems and shows that the existence and uniqueness of a projection that relates two models of different orders can be inferred from the kronecker canonical form of a certain linear pencil combining the l 2 optimality conditions into that framework gives a different point of view on the optimal reduced order model it is shown that the zeros of the optimal approximation error transfer function are the mirror image of the reduced model poles with the same multiplicity in non square systems and doubled multiplicity in square systems the latter case enhances an existing result for siso systems that the multiplicity of the error zeros exceeds that of the optimal reduced model poles by one it is also shown that the optimal projection which is obtained in some solutions of the problem is discontinuous at the optimal solution if the reduced

an mpec approach for the critical post collapse behavior of rigid plastic structures this paper presents a numerical method to identify and trace the critical post collapse response of rigid perfectly plastic structures to account for the possibility of multiple equilibrium paths the critical one is directly identified using the minimum 2nd order work criterion our proposed enhanced sequential limit analysis is formulated as an instance of the challenging class of optimization problems known as a mathematical program with equilibrium constraints mpec this mpec formulation minimizes the 2nd order work expression subject to the set of constraints describing the complete complementarity system in mixed static kinematic variables governing simultaneously the two adjacent equilibrium configurations namely the current one and its neighboring state we use a nonlinear programming based algorithm involving relaxation of the complementarity terms to solve the mpec four numerical examples are provided to illustrate application of the proposed scheme

large deformation of structures by sequential limit analysis although most structures are designed to function under small deformation their large deformation behavior can be used to estimate reliability and safety for survivorship from an accident or a natural disaster structures such as buildings bridges ships vehicles and machinery are designed with a safety factor to protect certain assets from the unexpected and unknot elements the large deformation analysis provides the rational basis for a safety factor in this paper the method of sequential limit analysis is used to compute large deformation solutions of truss and frame problems differing from the incremental method of plasticity the limit analysis method is numerically stable more effecient and requires simpler input data a duality theorem serves as the foundation of an algorithm for computing the complete static and kinematic solutions sim uhaneously and for establishing their accuracy in each step of a deformation sequence the phenom ena encountered in large deformation such as loading unloading under monotone deformation bifurcation more generally loss of uniqueness of solutions and internal contact of structural members are revealed by the sequential limit analysis presented in this paper

mathematical programming approaches for the safety assessment of semirigid elastoplastic frames abstract this paper presents two complementary mathematical programming based approaches for the accurate safety assessment of semirigid elastoplastic frames under quasistatic loads the inelastic behavior of the flexible connections and material plasticity are accommodated through piecewise linearized nonlinear yield surfaces as is necessary for this class of structures geometric nonlinearity is taken into account moreover only a 2nd order geometric approximation is included as this is sufficiently accurate for practical structures the work described has a twofold contribution first we develop an algorithm that can robustly and efficiently process the complete path dependent nonholonomic response of the structure in a stepwise path independent holonomic fashion the governing formulation is cast in mixed static kinematic variables and leads naturally to what is known in the mathematical programming literature as a mixed complementarity problem mcp the novelty of the proposed algorithm is that it processes the mcp directly without using some iterative and often cumbersome predictor corrector procedure second in the spirit of simplified analyses the classical limit analysis approach is extended to compute the limit load multiplier under the simultaneous influence of joint flexibility material and geometric nonlinearities and limited ductility our formulation is an instance of the challenging class of optimization problems known as a mathematical program with equilibrium constraints mpec various nonlinear programming based algorithms are proposed to solve the mpec finally four numerical examples concerning practical structures and benchmark cases are provided to illustrate application of the analyses as well as to validate the accuracy and robustness of the proposed schemes

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

sensitivity analysis using anchored anova expansion and high order moments computation an anchored analysis of variance anova method is proposed in this paper to decompose the statistical moments compared to the standard anova with mutually orthogonal component functions the anchored anova with an arbitrary choice of the anchor point loses the orthogonality if employing the same measure however an advantage of the anchored anova consists in the considerably reduced number of deterministic solver s computations which renders the uncertainty quantification of real engineering problems much easier different from existing methods the covariance decomposition of the output variance is used in this work to take account of the interactions between non orthogonal components yielding an exact variance expansion and thus with a suitable numerical integration method provides a strategy that converges this convergence is verified by studying academic tests in particular the sensitivity problem of existing methods to the choice of anchor point is analyzed via the ishigami case and we point out that covariance decomposition survives from this issue also with a truncated anchored anova expansion numerical results prove that the proposed approach is less sensitive to the anchor point the covariance based sensitivity indices si are also used compared to the variance based si furthermore we emphasize that the covariance decomposition can be generalized in a straightforward way to decompose higher order moments for academic problems results show the method converges to exact solution regarding both the skewness and kurtosis finally the proposed method is applied on a realistic case that is estimating the chemical reactions uncertainties in a hypersonic flow around a space vehicle during an atmospheric reentry

reweighted l1 minimization method for stochastic elliptic differential equations abstract we consider elliptic stochastic partial differential equations spdes with random coefficients and solve them by expanding the solution using generalized polynomial chaos gpc under some mild conditions on the coefficients the solution is sparse in the random space i e only a small number of gpc basis makes considerable contribution to the solution to exploit this sparsity we employ reweighted l 1 minimization to recover the coefficients of the gpc expansion we also combine this method with random sampling points based on the chebyshev probability measure to further increase the accuracy of the recovery of the gpc coefficients we first present a one dimensional test to demonstrate the main idea and then we consider 14 and 40 dimensional elliptic spdes to demonstrate the significant improvement of this method over the standard l 1 minimization method for moderately high dimensional 10 problems the combination of chebyshev measure with reweighted l 1 minimization performs well while for higher dimensional problems reweighted l 1 only is sufficient the proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources