antenna array pattern synthesis via convex optimization we show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems which can be numerically solved with great efficiency by recently developed interior point methods the synthesis problems involve arrays with arbitrary geometry and element directivity constraints on far and near field patterns over narrow or broad frequency bandwidth and some important robustness constraints we show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays

robust beamforming via worst case sinr maximization minimum variance beamforming which uses a weight vector that maximizes the signal to interference plus noise ratio sinr is often sensitive to estimation error and uncertainty in the parameters steering vector and covariance matrix robust beamforming attempts to systematically alleviate this sensitivity by explicitly incorporating a data uncertainty model in the optimization problem in this paper we consider robust beamforming via worst case sinr maximization that is the problem of finding a weight vector that maximizes the worst case sinr over the uncertainty model we show that with a general convex uncertainty model the worst case sinr maximization problem can be solved by using convex optimization in particular when the uncertainty model can be represented by linear matrix inequalities the worst case sinr maximization problem can be solved via semidefinite programming the convex formulation result allows us to handle more general uncertainty models than prior work using a special form of uncertainty model we illustrate the method with a numerical example

extreme optics and the search for earth like planets in this paper i describe a new and exciting application of optimization technology the problem is to design a space telescope capable of imaging earth like planets around nearby stars because of limitations inherent in the wave nature of light the design problem is one of diffraction control so as to provide the extremely high contrast needed to image a faint planet positioned very close to its much brighter star i will describe the mathematics behind the diffraction control problem and explain how modern optimization tools were able to provide unexpected solutions that actually changed nasa s approach to this problem

iterative image reconstruction algorithms based on cross entropy minimization the related problems of minimizing the functionals f x alpha kl y px 1 alpha kl p x and g x alpha kl px y 1 alpha kl x p respectively over the set of vectors x or 0 are considered kl a b is the cross entropy or kullback leibler distance between two nonnegative vectors a and b iterative algorithms for minimizing both functionals using the method of alternating projections are derived a simultaneous version of the multiplicative algebraic reconstruction technique mart algorithm called smart is introduced and its convergence is proved

robust perturbation algorithms for adaptive antenna arrays the generation of a perturbation sequence for an adaptive beamformer is described this perturbation sequence permits simultaneous adaption and reception by use of weight perturbations that do not obstruct the look direction constraint it is shown that this sequence is generally shorter in length than previously described sequences and offers scope for computational savings through reduction of the number of projection operations required convergence in the mean of the resulting adaptive algorithm is demonstrated experiments conducted using a four element linear array operating in the mw rf range have confirmed that the predicted results are achievable under the nonideal conditions of quantized array weights and finite word length arithmetic

robust wideband beamforming by the hybrid steepest descent method this paper uses the hybrid steepest descent method hsdm to design robust smart antennas several design criteria as well as robustness are mathematically described by a finite collection of closed convex sets in a real euclidean space desirable beamformers are defined as points of the generalized convex feasible set which is well defined even in the case of inconsistent design criteria a quadratic cost function is formed by the correlations of the incoming data and the hsdm constructs a point sequence that strongly converges to the unique minimizer of the cost function over the generalized convex feasible set numerical examples validate the proposed design

optimal narrow beam low sidelobe synthesis for arbitrary arrays the classical narrow main beam and low sidelobe synthesis problem is addressed and extended to arbitrary sidelobe envelopes in both one and two dimensional scenarios the proposed approach allows also to handle arbitrary arrays this extended basic synthesis problem is first rewritten as a convex optimization problem this problem is then transformed into either a linear program or a second order cone program that can be solved efficiently by readily available software the convex formulation is not strictly equivalent to the initial synthesis problem but arguments are given that establish that the obtained optimum is either identical or can be made as close as wanted to the desired one finally numerical applications and a comparison with a classical solution are presented to both confirm the optimality of the solution and illustrate the potentialities of the proposed approach

a joint optimization approach for the synthesis of large 4 d heterogeneous antenna arrays due to the inherent nonlinearity in the synthesis of large four dimensional 4 d heterogeneous arrays conventional pattern synthesis techniques are no longer suitable in this paper an effective joint optimization approach based on the combination of convex programming cp and differential evolution de algorithm is proposed by designing suitable time sequences of a 4 d heterogeneous array with uniform static amplitude excitations the original synthesis problem is simplified into two independent small problems which can be easily solved in two steps respectively in the first step the approach takes definite advantages from the convexity of the problem with respect to the static phase excitations and the switch on duration times these optimization variables become known and are fixed in the second step then the de algorithm is used to optimize the switch on time instants owing to the efficiency of the cp only 2 3 of the optimization variables need be solved in the first step while even fewer number of optimization variables are solved in the second step consequently the joint optimization of cp and de algorithm significantly improves the overall synthesis efficiency two numerical examples are presented to demonstrate the effectiveness of the proposed approach

global existence of solutions to the incompressible navier stokes vlasov equations in a time dependent domain in this article we prove the existence of global weak solutions for the in compressible navier stokes vlasov system in a three dimensional time dependent domain with absorption boundary conditions for the kinetic part this model arises from the study of respiratory aerosol in the human airways the proof is based on a regularization and approximation strategy designed for our time dependent framework

on the parameterization of positive real sequences and ma parameter estimation an algorithm for moving average ma parameter estimation was proposed by stoica et al see ibid vol 48 p 1999 2012 2000 its key step covariance fitting is a semidefinite programming sdp problem with two convex constraints one reflecting the real positiveness of the desired covariance sequence and the other having a second order cone form we analyze two parameterizations of a positive real sequence and show that there is a one to one correspondence between them we also show that the dual of the covariance fitting problem has a significantly smaller number of variables and thus a much reduced computational complexity we discuss in detail the formulations that are best suited for the currently available semidefinite quadratic programming packages experimental results show that the execution times of the newly proposed algorithms scale well with the ma order which are therefore convenient for large order ma signals

optimal array pattern synthesis using semidefinite programming we present algorithms for the solution of two problems in array pattern synthesis the first is the design of nonuniform arrays with a desired magnitude response the second is that of robust design i e design in the presence of uncertainties constraints such as power limitation can be addressed with both problems the algorithms that we present are based on semidefinite programming for which efficient software is readily available we present examples that illustrate the effectiveness of our approach

the mean field limit for solid particles in a navier stokes flow we propose a mathematical derivation of brinkman s force for a cloud of particles immersed in an incompressible viscous fluid specifically we consider the stokes or steady navier stokes equations in a bounded domain 3 for the velocity field u of an incompressible fluid with kinematic viscosity and density 1 brinkman s force consists of a source term 6 j where j is the current density of the particles and of a friction term 6 u where is the number density of particles these additional terms in the motion equation for the fluid are obtained from the stokes or steady navier stokes equations set in minus the disjoint union of n balls of radius e 1 n in the large n limit with no slip boundary condition the number density and current density j are obtained from the limiting phase space empirical measure frac 1 n sum 1 le k le n delta x k v k where xk is the center of the k th ball and vk its instantaneous velocity this can be seen as a generalization of allaire s result in arch ration mech anal 113 209 259 1991 who considered the case of periodically distributed xks with vk 0 and our proof is based on slightly simpler though similar homogenization arguments similar equations are used for describing the fluid phase in various models for sprays

homogenization of the navier stokes equations in open sets perforated with tiny holes i abstract framework a volume distribution of holes this paper treats the homogenization of the stokes or navier stokes equations with a dirichlet boundary condition in a domain containing many tiny solid obstacles periodically distributed in each direction of the axes for example in the three dimensional case the obstacles have a size of e3 and are located at the nodes of a regular mesh of size e a suitable extension of the pressure is used to prove the convergence of the homogenization process to a brinkman type law in which a linear zero order term for the velocity is added to a stokes or navier stokes equation

a new approach to the optimization of robust antenna array processors the authors present a method for solving a class of optimization problems with nonsmooth constraints in particular they apply the method to the design of narrowband minimum power antenna array processors which are robust in the presence of errors such as array element placement look direction misalignment and frequency offset they first show that the constrained minimum power problem has a unique global minimum provided that the constraint set is nonempty then it is shown how the design problem derived directly from considerations of the sensitivity of the antenna array processor to errors can be transformed into a quadratic programming problem with linear inequality constraints which can be solved efficiently by the standard active set strategy they also present numerical results for two types of nonsmooth constraints developed to provided robustness these results confirm the effectiveness of the method

low dimensional sdp formulation for large antenna array synthesis bounding the sidelobe and mainlobe levels of an array with complex weights is attractive in that it allows direct control of the radiation pattern in this paper we propose an efficient beamforming technique for synthesizing the patterns of large arrays with bound constraints on the sidelobe and mainlobe levels it is shown that the pattern synthesis problem can be posed as a convex semi infinite program sip which is then turned into a semi definite program sdp via a novel linear matrix inequality lmi characterization of semi infinite trigonometric polynomial constraints in contrast to existing sdp formulations which require a large number of additional variables our sdp representation only uses a minimal number of additional variables this subsequently enables the design of patterns for arrays with several hundred elements to be efficiently achieved on a standard personal computer using existing sdp solvers

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

on the exponential decay to equilibrium of the degenerate linear boltzmann equation abstract in this paper we study the decay to the equilibrium state for the solution of the linear boltzmann equation in the torus t d r d z d d n by allowing that the non negative cross section can vanish in a subregion x x t d x 0 of the domain with meas x 0 with respect to the lebesgue measure we show that the geometrical characterization of x is the key property to produce exponential decay to equilibrium

global existence and large time behavior of classical solutions to the euler maxwell vlasov fokker planck system abstract we consider global classical solutions to the cauchy problem for an euler maxwell vlasov fokker planck system which is a fluid particle model describe the evolutions of disperse two phase flows in these two phase flows the compressible euler maxwell equations modeling a dense phase fluid and the vlasov fokker planck equation for the dispersed phase particles which is usually considered from the statistical view of point they are coupled through the friction force global existence of classical solutions to the cauchy problem in r 3 is established when initial data is a small perturbation of some given equilibrium the algebraic rate of convergence of solution toward the equilibrium state is also obtained under additional conditions on initial data the proof is based on the classical energy estimates and fourier multiplier technique which are considerably complicated and some new ideas and techniques are thus required

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

tractable transmit mimo beampattern design under a constant modulus constraint multiple input multiple output mimo radar systems allow each antenna element to transmit a different waveform this waveform diversity can be exploited to enhance the beampattern design in particular effective management of radar radiation power in directions of interest we address the problem of designing a beampattern for mimo radar which in turn is determined by the transmit waveform while unconstrained design is straightforward a key open challenge is enforcing the constant modulus constraint on the radar waveform it is well known that the problem of minimizing deviation of the designed beampattern from an idealized one subject to the constant modulus constraint constitutes a hard nonconvex problem existing methods that address constant modulus invariably lead to a stiff tradeoff between analytical tractability achieved by relaxations and approximations and realistic design that exactly achieves constant modulus but is computationally burdensome a new approach is proposed in our paper which involves solving a sequence of convex equality constrained quadratic programs each of which has a closed form solution and such that constant modulus is achieved at convergence we further prove that the converged solution satisfies the karush kuhn tucker optimality conditions of the aforementioned hard nonconvex problem we evaluate the proposed successive closed forms scf algorithm against the state of the art mimo beampattern design techniques in both narrowband and wideband setups and show that the scf breaks the tradeoff between desirable performance and the associated computation cost

robust minimum variance beamforming this paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold in our method uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction we choose weights that minimize the total weighted power output of the array subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid the robust weight selection process can be cast as a second order cone program that can be solved efficiently using lagrange multiplier techniques if the ellipsoid reduces to a single point the method coincides with capon s method we describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response we form separate uncertainty ellipsoids for each component in the signal path e g antenna electronics and then determine an aggregate uncertainty ellipsoid from these we give new results for modeling the element wise products of ellipsoids we demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

an effective approach for the optimal focusing of array fields subject to arbitrary upper bounds a new approach to the optimal focusing of array fields subject to arbitrary upper bounds is presented the approach formulates the problem as the minimization of a linear function in a convex set unlike other approaches this one guarantees the achievement of the global optimum by using local optimization techniques and can moreover deal with any convex constraint on the unknowns such as near field constraints optimization is performed by two ad hoc developed solution algorithms which exploit the geometrical characteristics of the problem at hand thus leading to extremely effective and computationally efficient numerical codes an extensive numerical analysis has been performed in all cases of linear planar and circular arc arrays the enhanced performance of the proposed technique with respect to the solution algorithms available in the literature fully confirms the effectiveness of the approach

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

graph based decoding in the presence of isi we propose a new graph representation for isi channels that can be used for combined equalization and decoding by linear programming lp or iterative message passing imp decoding algorithms we derive this graph representation by linearizing the ml detection metric which transforms the equalization problem into a classical decoding problem we observe that the performance of lp and imp decoding on this model are very similar in the uncoded case while imp decoding significantly outperforms lp decoding when low density parity check ldpc codes are used in particular in the absence of coding for certain classes of channels both lp and imp algorithms always find the exact ml solution using the proposed graph representation without complexity that is exponential in the size of the channel memory this applies even to certain two dimensional isi channels however for some other channel impulse responses both decoders have nondiminishing probability of failure as snr increases we provide analytical explanations for many of these observations in addition we study the error events of lp decoding in the uncoded case and derive a measure that can be used to classify isi channels in terms of the performance of the proposed detection scheme

reconfigurable adaptive array beamforming by antenna selection traditional adaptive array beamforming with a fixed array configuration can lead to significant inefficiencies and performance loss under different scenarios as antennas become smaller and cheaper relative to front ends it becomes important to devise a reconfigurable adaptive antenna array raaa strategy to yield high signal to noise and interference ratio using fewer antennas this is achieved by selecting k from n antennas to minimize the spatial correlation coefficient scc between the desired signal and the interference the lower bound of optimum scc is formulated with two relaxation methods to give information about the suitable number of selected antennas k a correlation measurement cm method is proposed to select the optimum subarray with k antennas thereby reducing complexity we carry out performance analysis and show that a 1 k2 suboptimum solution can be guaranteed with arbitrary shaped arrays furthermore a difference of convex sets dcs method is proposed to select the optimum subarray with controlled quiescent pattern in order to reduce the effect of interference during the reconfiguration time the utility of the proposed array reconfiguration for performance improvement without increasing the cost is demonstrated using both simulated and experimental data

optimal design of efficient acoustic antenna arrays minimax optimal design of sonar transducer arrays can be formulated as a nonlinear program with many convex quadratic constraints and a nonconvex quadratic efficiency constraint the variables of this problem are a scaling and phase shift applied to the output of each sensor this problem is solved by applying lagrangian relaxation to the convex quadratic constraints extensive computational experience shows that this approach can efficiently find near optimal solutions of problems with up to 391 variables and 579 constraints

the vlasov navier stokes system in a 2d pipe existence and stability of regular equilibria in this paper we study the vlasov navier stokes system in a 2d pipe with partially absorbing boundary conditions we show the existence of stationary states for this system near small poiseuille flows for the fluid phase for which the kinetic phase is not trivial we prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations the analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase

pattern synthesis of conformal array antenna in the presence of platform using differential evolution algorithm a synthesis method based on differential evolution algorithm is proposed for conformal antenna arrays in the presence of platform with the desired phase weights determined by the scan angle and array geometry the amplitude weights of elements are optimized by differential evolution algorithm to drive down the side lobes both the effect of platform on radiation pattern of conformal array and the mutual coupling between the elements are taken into account by calculating the antenna array and platform simultaneously using the method of moment application of adaptive integral method reduces the storage and computation time needed by the method of moments

geometry and weight optimization for minimizing sidelobes in wideband planar arrays arrays are often analyzed using a narrowband or single frequency assumption for simplicity however for arrays used in wideband applications this simplification is not valid in this paper a sidelobe minimizing method is presented that efficiently determines the optimal weights given a specified beamwidth for wideband planar arrays of elements scanned to broadside this method can be executed significantly faster than optimal wideband weighting methods developed previously this speedup in computation time allows for simultaneous weight geometry optimization optimal array geometries are determined using the particle swarm optimization pso algorithm in conjunction with the optimal weights in order to minimize sidelobe levels in wideband arrays results optimal positions weights and sidelobe levels are presented for wideband arrays of 4 7 elements

applications of convex optimization in signal processing and digital communication in the last two decades the mathematical programming community has witnessed some spectacular advances in interior point methods and robust optimization these advances have recently started to significantly impact various fields of applied sciences and engineering where computational efficiency is essential this paper focuses on two such fields digital signal processing and communication in the past the widely used optimization methods in both fields had been the gradient descent or least squares methods both of which are known to suffer from the usual headaches of stepsize selection algorithm initialization and local minima with the recent advances in conic and robust optimization the opportunity is ripe to use the newly developed interior point optimization techniques and highly efficient software tools to help advance the fields of signal processing and digital communication this paper surveys recent successes of applying interior point and robust optimization to solve some core problems in these two fields the successful applications considered in this paper include adaptive filtering robust beamforming design and analysis of multi user communication system channel equalization decoding and detection throughout our emphasis is on how to exploit the hidden convexity convex reformulation of semi infinite constraints analysis of convergence complexity and performance as well as efficient practical implementation

on the optimal synthesis of sum or difference patterns of centrosymmetric arrays under arbitrary sidelobe constraints this paper deals with the optimal synthesis of sum or difference patterns radiated by an array of fixed geometry the aim is to establish the most general class of arrays for which such synthesis can be reduced to a linear programming problem extending and completing previous results regarding uniformly spaced linear and planar arrays it is shown that such a class consists of the centrosymmetric arrays whose element patterns are either equal or hermitian symmetric with respect to the center of symmetry the optimal solutions are also hermitian symmetric and the corresponding patterns are real furthermore it is shown that the above class of arrays coincides with that of the arrays capable of radiating real patterns some numerical examples are provided to give an idea of the geometries and the synthesis problems to which the achieved result successfully applies and of the achievable computational advantage

some aspects of the modeling at different scales of multiphase flows abstract we present the various levels of possible modeling for multiphase flows coupling of fluid equations in different domains with a free boundary coupling in the same domain of a fluid equation and a kinetic vlasov or vlasov boltzmann equation coupling in the same domain of two or more fluid equations we briefly present the mathematical results relative to the passage from one of these approaches to another approach and we give some ideas of how to use those different models on a specific practical example

pattern synthesis of narrowband conformal arrays using iterative second order cone programming a new design method is proposed for the power or shaped beam pattern synthesis problem of narrowband conformal arrays where only the magnitude response is specified the proposed method iteratively linearizes the non convex power pattern function to obtain a convex subproblem in the design variables which can be solved optimally using second order cone programming socp in addition a wide variety of magnitude constraints such as non convex lower bound magnitude constraints can be incorporated an efficient technique for determining a reasonably good initial guess to the problem is also proposed to further improve the reliability of the method computer simulations show that the initial guesses so obtained converge to satisfactory solutions while satisfying various prescribed magnitude constraints design results show that the performance of the proposed method is comparable to the optimal solution previously obtained for uniform linear arrays with isotropic elements moreover we show by means of examples that the proposed method is also applicable to general non convex power pattern synthesis problems involving arbitrary array geometries arbitrary polarization characteristics and mutual coupling effect