introduction to fuzzy partial differential equations this paper considers solutions to elementary fuzzy partial differential equations we follow the same strategy as in buckley and feuring fuzzy sets and systems to appear which is 1 first check to see if the buckley feuring method produces a solution and 2 if the buckley feuring method does not give a solution then see if the seikkala procedure generates a solution examples are presented showing the buckley feuring solution and the seikkala solution

h div preconditioning for a mixed finite element formulation of the diffusion problem with random data we study h div preconditioning for the saddle point systems that arise in a stochastic galerkin mixed formulation of the steady state diffusion problem with random data the key ingredient is a multigrid v cycle for an h div operator with random weight function acting on a certain tensor product space of random fields with finite variance we build on the arnold falk winther multigrid algorithm presented in 1997 by varying the spatial discretization from grid to grid whilst keeping the stochastic discretization fixed we extend the deterministic analysis to accommodate the modified h div operator and establish spectral equivalence bounds with a new multigrid v cycle operator that are independent of the spatial and stochastic discretization parameters we implement multigrid within a block diagonal preconditioner for the full saddle point problem derive eigenvalue bounds for the preconditioned system matrices and investigate the impact of all the discretization parameters on the convergence rate of preconditioned minre s

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

superconvergence analysis of yee scheme for metamaterial maxwell s equations on non uniform rectangular meshes since the development of yee scheme back in 1966 it has become one of the most popular simulation tools for modeling electromagnetic wave propagation in various situations however its rigorous error analysis on nonuniform rectangular type grids was carried out until 1994 by monk and suli they showed that the yee scheme is still second order convergent on a nonuniform mesh even though the local truncation error is only of first order in this paper we extend their results to maxwell s equations in metamaterials by a simpler proof and show the second order superconvergence in space for the true yee scheme instead of the only semi discrete form discussed in monk and suli s original work numerical results supporting our analysis are presented

computational modeling of uncertainty in time domain electromagnetics we discuss computationally efficient ways of accounting for the impact of uncertainty e g lack of detailed knowledge about sources materials shapes etc in computational time domain electromagnetics in contrast to classic statistical monte carlo based methods we explore a probabilistic approach based on high order accurate expansions of general stochastic processes we show this to be highly efficient and accurate on both one and two dimensional examples enabling the computation of global sensitivities of measures of interest e g radar cross sections rcs in scattering applications for a variety of types of uncertainties

regularity analysis of metamaterial maxwell s equations with random coefficients and initial conditions abstract in this paper we develop and analyze the stochastic collocation method for solving the time dependent metamaterial maxwell s equations subject to random coefficients and random initial conditions we provide a rigorous regularity analysis of the solution with respect to the random variables to our best knowledge this is the first theoretical results derived for the stochastic metamaterial maxwell equations the rate of convergence is proved depending on the regularity of the solution numerical results are presented to confirm the theoretical analysis we also demonstrate that the backward wave propagation phenomenon still exists in random metamaterial

the variational iteration method for a reliable treatment of the linear and the nonlinear goursat problem abstract the linear and the nonlinear goursat problems are investigated the he s variational iteration method vim is used to construct exact solutions to these problems the linear and the nonlinear structures are handled in a like manner without any need to restrictive assumptions the accuracy level of the obtained results reveal the power of this method over existing numerical methods

on fuzzy solutions for partial differential equations in this study we investigate heat wave and poisson equations as classical models of partial differential equations pdes with uncertain parameters considering the parameters as fuzzy numbers the fuzzy solution is built from fuzzification of the deterministic solution the continuity of the zadeh extension is used to obtain qualitative properties on regular a cuts of the fuzzy solution we prove the stability with respect to the initial boundary data and show that as time goes to zero the diameter of the fuzzy solution converges to zero and as a consequence to the cylindrical surface determined by the curve of the degree of membership numerical simulations are used to obtain a graphical representation of the fuzzy solution and a defuzzification of this solution is obtained using the center of gravity method we theoretically show that the surface obtained by defuzzification with the plane determined by fixing time is indeed the solution of the same initial boundary problem for this time point for the heat and poisson equations and in a particular case for the wave equation the deterministic solution and the defuzzified surface intercept are numerically compared using the euclidean distance

on the solutions to first order linear fuzzy differential equations in this paper we study different formulations of first order linear fuzzy differential equations using the concept of generalized differentiability we present sufficient conditions for the existence of solutions and obtain the general expression of these solutions which exhibit different behavior some examples are given to illustrate our results

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

hermitian completely positive matrices abstract in this paper we introduce the hermitian completely positive hcp matrix which has a hermitian completely positive hcp decomposition with all real and imaginary parts of the decomposition vectors being nonnegative some properties of the hermitian completely positive matrix are given a semidefinite algorithm is also proposed for checking whether a hermitian matrix is hcp or not if a matrix is not hcp a certificate for it can be obtained if it is an hcp decomposition can be obtained

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

solving elliptic problems with non gaussian spatially dependent random coefficients we propose a simple and effective numerical procedure for solving elliptic problems with non gaussian random coefficients assuming that samples of the non gaussian random inputs are available from a statistical model given a correlation function the karhunen loeve k l expansion is employed to reduce the dimensionality of random inputs using the kernel density estimation technique we obtain the marginal probability density functions pdfs of the random variables in the k l expansion based on which we define an auxiliary joint pdf we then implement the generalized polynomial chaos gpc method via a collocation projection according to the auxiliary joint pdf based on the observation that the solution has an analytic extension in the parametric space we ensure that the polynomial interpolation achieves point wise convergence in the parametric space regardless of the pdf where the energy norm is employed in the physical space hence we can sample the gpc solution using the joint pdf instead of the auxiliary one to obtain the correct statistics we also implement monte carlo methods to further refine the statistics using the gpc solution for variance reduction numerical results are presented to demonstrate the efficiency of the proposed approach

multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification in this paper we study multiscale finite element methods for stochastic porous media flow equations as well as applications to uncertainty quantification we assume that the permeability field the diffusion coefficient is stochastic and can be described in a finite dimensional stochastic space this is common in applications where the coefficients are expanded using chaos approximations the proposed multiscale method constructs multiscale basis functions corresponding to sparse realizations and these basis functions are used to approximate the solution on the coarse grid for any realization furthermore we apply our coarse scale model to uncertainty quantification problem where the goal is to sample the porous media properties given an integrated response such as production data our algorithm employs pre computed posterior response surface obtained via the proposed coarse scale model using fast analytical computations of the gradients of this posterior we propose approximate langevin samples these samples are further screened through the coarse scale simulation and finally used as a proposal in metropolis hasting markov chain monte carlo method numerical results are presented which demonstrate the efficiency of the proposed approach

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

initial value problems for higher order fuzzy differential equations we consider nth order fuzzy differential equations with initial value conditions we prove the existence and uniqueness of solution for nonlinearities satisfying a lipschitz condition we apply the obtained results to the particular case of linear fuzzy problems

finite element approximation of the linear stochastic wave equation with additive noise semidiscrete finite element approximation of the linear stochastic wave equation lswe with additive noise is studied in a semigroup framework optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions these are used to prove strong convergence estimates for the stochastic problem the theory presented here applies to multidimensional domains and spatially correlated noise numerical examples illustrate the theory

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

a stochastic collocation method for the second order wave equation with a discontinuous random speed in this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions the speed is piecewise smooth in the physical space and depends on a finite number of random variables the numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space this approach leads to the solution of uncoupled deterministic problems as in the monte carlo method we consider both full and sparse tensor product spaces of orthogonal polynomials we provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data in particular we show that unlike in elliptic and parabolic problems the solution to hyperbolic problems is not in general analytic with respect to the random variables therefore the rate of convergence may only be algebraic an exponential fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data we present numerical examples which confirm the analysis and show that the collocation method is a valid alternative to the more traditional monte carlo method for this class of problems

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

fubini theorem for generalized fuzzy number valued integrals in this paper we introduce the notion of generalized fuzzy numbers and generalized fuzzy mappings and give fubini theorem for integrals of generalized fuzzy mappings

error analysis of the dynamically orthogonal approximation of time dependent random pdes in this work we discuss the dynamically orthogonal do approximation of time dependent partial differential equations with random data the approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion we analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the s terms do solution by the corresponding s terms best approximation i e the truncated s terms karhunen loeve expansion at each time instant the bound is applicable on the largest time interval in which the best s terms approximation is continuously time differentiable properties of the do approximations are analyzed on simple cases of deterministic equations with random initial data numerical tests are presented that confirm the theoret

optimality conditions and finite convergence of lasserre s hierarchy lasserre s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally this paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of lasserre s hierarchy our main results are i lasserre s hierarchy has finite convergence when the constraint qualification strict complementarity and second order sufficiency conditions hold at every global minimizer under the standard archimedean condition the proof uses a result of marshall on boundary hessian conditions ii these optimality conditions are all satisfied at every local minimizer if a finite set of polynomials which are in the coefficients of input polynomials do not vanish at the input data i e they hold in a zariski open set this implies that under archimedeanness lasserre s hierarchy has finite convergence generically

numerical study of signal propagation in corrugated coaxial cables this paper is concerned with high fidelity modeling of signal propagation in corrugated coaxial cables taking advantage of the axisymmetry we reduce the 3 d problem to a 2 d problem by solving a time dependent maxwell s equations in cylindrical coordinates we develop a nodal discontinuous galerkin method for solving our model equations stability and error analysis are proved for the semi discrete scheme extensive numerical results are provided to demonstrate that our algorithm not only converges as our theoretical analysis predicts but also is very effective in solving various signal propagation problems in practical corrugated coaxial cables

linear optimization with cones of moments and nonnegative polynomials let mathcal a a be a finite subset of mathbb n n nn and mathbb r x mathcal a r x a be the space spanned by monomials x alpha x with alpha in mathcal a a let k k be a compact semialgebraic set of mathbb r n rn such that a polynomial in mathbb r x mathcal a r x a is positive on k k denote by fancyscript p mathcal a k pa k the cone of polynomials in mathbb r x mathcal a r x a that are nonnegative on k k the dual cone of fancyscript p mathcal a k pa k is fancyscript r mathcal a k ra k the set of all truncated moment sequences in mathbb r mathcal a ra that admit representing measures supported in k k first we study geometric properties of the cones fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k like interiors closeness duality memberships and construct a convergent hierarchy of semidefinite relaxations for each of them second we propose a semidefinite algorithm for solving linear optimization problems with the cones fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k and prove its asymptotic and finite convergence third we show how to check whether fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k intersect affine subspaces if they do we show how to get a point in the intersections if they do not we prove certificates for the empty intersection

semidefinite representations for finite varieties we consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities when the polynomial equations have a finite set of complex solutions we can reformulate this problem as a semidefinite programming problem our semidefinite representation involves combinatorial moment matrices which are matrices indexed by a basis of the quotient vector space x 1 xn i where i is the ideal generated by the polynomial equations in the problem moreover we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by lasserre semidefinite approximations can be constructed by considering truncated combinatorial moment matrices rank conditions are given in a grid case that ensure that the approximation solves the original problem to optimality

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

on fuzzy solutions for heat equation based on generalized hukuhara differentiability abstract in this paper a fuzzy heat equation with fuzzy initial values is considered the concept of generalized hukuhara differentiation is interpreted thoroughly in the univariate and multivariate cases and also several properties for generalized hukuhara differentiability are obtained on the topics such as switching point the univariate and multivariate fuzzy chain rules fuzzy mean value theorem among others the objective of this paper is to prove the uniqueness of a solution for a fuzzy heat equation and show that a fuzzy heat equation can be modeled as two systems of fuzzy differential equations by considering the type of differentiability of solutions finally some examples show the behavior of the solutions obtained

a fuzzy solution of heat equation under generalized hukuhara differentiability by fuzzy fourier transform in this paper an analytical solution of a fuzzy heat equation under generalized hukuhara partial differentiability by fuzzy fourier transform is investigated a fuzzy fourier transform is introduced and the related theorems and properties are provided in detail some examples show the effectiveness and efficiency of the approach

algebraic degree of polynomial optimization consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials under some genericity assumptions we prove that the optimality conditions always hold on optimizers and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials we also give a general formula for the algebraic degree of the optimal coordinates the derivation of the algebraic degree is equivalent to counting the number of all complex critical points as special cases we obtain the algebraic degrees of quadratically constrained quadratic programming qcqp second order cone programming socp and p th order cone programming pocp in analogy to the algebraic degree of semidefinite programming j nie k ranestad and b sturmfels the algebraic degree of semidefinite programming math programm to appear

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

a stochastic multi symplectic scheme for stochastic maxwell equations with additive noise abstract in this paper we investigate a stochastic multi symplectic method for stochastic maxwell equations with additive noise based on the stochastic version of variational principle we find a way to obtain the stochastic multi symplectic structure of three dimensional 3 d stochastic maxwell equations with additive noise we propose a stochastic multi symplectic scheme and show that it preserves the stochastic multi symplectic conservation law and the local and global stochastic energy dissipative properties which the equations themselves possess numerical experiments are performed to verify the numerical behaviors of the stochastic multi symplectic scheme

the truncated moment problem via homogenization and flat extensions abstract this paper studies how to solve the truncated moment problem tmp via homogenization and flat extensions of moment matrices we first transform tmp to a homogeneous tmp htmp and then use semidefinite programming sdp techniques to solve htmp our main results are 1 a truncated moment sequence tms is the limit of a sequence of tms admitting measures on r n if and only if its homogenized tms htms admits a measure supported on the unit sphere in r n 1 2 an htms admits a measure if and only if the optimal values of a sequence of sdp problems are nonnegative 3 under some conditions that are almost necessary and sufficient by solving these sdp problems a representing measure for an htms can be explicitly constructed if one exists

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

on the complexity of testing attainment of the optimal value in nonlinear optimization we prove that unless hbox p hbox np there exists no polynomial time or even pseudo polynomial time algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low degree polynomials is attained if the degrees of these polynomials are fixed our results along with previously known frank wolfe type theorems imply that exactly one of two cases can occur either the optimal value is attained on every instance or it is strongly np hard to distinguish attainment from non attainment we also show that testing for some well known sufficient conditions for attainment of the optimal value such as coercivity of the objective function and closedness and boundedness of the feasible set is strongly np hard as a byproduct our proofs imply that testing the archimedean property of a quadratic module is strongly np hard a property that is of independent interest to the convergence of the lasserre hierarchy finally we give semidefinite programming sdp based sufficient conditions for attainment of the optimal value in particular a new characterization of coercive polynomials that lends itself to an sdp hierarchy

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

an exact jacobian sdp relaxation for polynomial optimization given polynomials f x gi x hj x we study how to minimize f x on the set s left x in mathbb r n h 1 x cdots h m 1 x 0 g 1 x geq 0 ldots g m 2 x geq 0 right let fmin be the minimum of f on s suppose s is nonsingular and fmin is achievable on s which are true generically this paper proposes a new type semidefinite programming sdp relaxation which is the first one for solving this problem exactly first we construct new polynomials varphi 1 ldots varphi r by using the jacobian of f hi gj such that the above problem is equivalent to begin gathered underset x in mathbb r n min f x hfill rm s t h i x 0 varphi j x 0 1 leq i leq m 1 1 leq j leq r hfill quad g 1 x nu 1 cdots g m 2 x nu m 2 geq 0 quad forall nu in 0 1 m 2 hfill end gathered second we prove that for all n big enough the standard n th order lasserre s sdp relaxation is exact for solving this equivalent problem that is its optimal value is equal to fmin some variations and examples are also shown

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

computation of the structured singular value via moment lmi relaxations the structured singular value ssv provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties unfortunately computing the ssv is an np hard problem and the polynomial time algorithms available in the literature are only able to provide except for some special cases upper and lower bounds on the exact value of the ssv in this work we present a new algorithm to compute an upper bound on the ssv in case of mixed real complex uncertainties the underlying idea of the developed approach is to formulate the ssv computation as a nonconvex polynomial optimization problem which is relaxed into a sequence of convex optimization problems through moment based relaxation techniques two heuristics to compute a lower bound on the ssv are also discussed the analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the robust control toolbox in matlab and it provides in most of the cases coincident lower and upper bounds on the structured singular value

certifying convergence of lasserre s hierarchy via flat truncation consider the optimization problem of minimizing a polynomial function subject to polynomial constraints a typical approach for solving it globally is applying lasserre s hierarchy of semidefinite relaxations based on either putinar s or schmudgen s positivstellensatz a practical question in applications is how to certify its convergence and get minimizers in this paper we propose flat truncation as a certificate for this purpose assume the set of global minimizers is nonempty and finite our main results are 1 putinar type lasserre s hierarchy has finite convergence if and only if flat truncation holds under some generic assumptions the same conclusion holds for the schmudgen type one under weaker assumptions 2 flat truncation is asymptotically satisfied for putinar type lasserre s hierarchy if the archimedean condition holds the same conclusion holds for the schmudgen type one if the feasible set is compact 3 we show that flat truncation can be used as a certificate to check exactness of standard sos relaxations and jacobian sdp relaxations

the hierarchy of local minimums in polynomial optimization this paper studies the hierarchy of local minimums of a polynomial in the vector space mathbb r n rn for this purpose we first compute h h minimums for which the first and second order necessary optimality conditions are satisfied to compute each h h minimum we construct a sequence of semidefinite relaxations based on optimality conditions we prove that each constructed sequence has finite convergence under some generic conditions a procedure for computing all local minimums is given when there are equality constraints we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums

tensor eigenvalue complementarity problems this paper studies tensor eigenvalue complementarity problems basic properties of standard and complementarity tensor eigenvalues are discussed we formulate tensor eigenvalue complementarity problems as constrained polynomial optimization when one tensor is strictly copositive the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity when no tensor is strictly copositive we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process the complementarity eigenvalues can be computed sequentially the formulated polynomial optimization can be solved by lasserre s hierarchy of semidefinite relaxations we show that it has finite convergence for generic tensors numerical experiments are presented to show the efficiency of proposed methods

fuzzy solutions to partial differential equations adaptive approach a new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations pdes is presented the design objective is to find a fuzzy solution to satisfy precisely the pdes with boundary conditions according to the adaptive scheme of fuzzy logic systems a fuzzy solution with adjustable parameters for the pde is first described then a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function in addition an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived furthermore the convergence of error equations in mesh points is also discussed from the energy perspective in this paper we show that the proposed method can solve a variety of pdes encountered in engineering comparisons are also made with solutions obtained by the finite element method

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

the solvability of fuzzy fractional partial differential equations under caputo gh differentiability the foundation of the concepts of fuzzy fractional integral and caputo gh partial for fuzzy valued multivariable functions is defined as a result fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved we present two new results on the existence of two kinds of gh weak solutions of these problems the first result is based on the banach fixed point theorem with the lipschitz condition of functions on the right hand side of equations the second result is based on the nonlinear alternative of the schauder type for fuzzy valued continuous functions without the lipschitz condition moreover we indicate the boundedness and continuous dependence of solutions on the initial data of the problems our models are embedded in the sense of caputo gh differentiability some examples are presented to illustrate the results

distinguished representations of non negative polynomials abstract let g 1 g r r x 1 x n such that the set k g 1 0 g r 0 in r n is compact we study the problem of representing polynomials f with f k 0 in the form f s 0 s 1 g 1 s r g r with sums of squares s i with particular emphasis on the case where f has zeros in k assuming that the quadratic module of all such sums is archimedean we establish a local global condition for f to have such a representation vis a vis the zero set of f in k this criterion is most useful when f has only finitely many zeros in k we present a number of concrete situations where this result can be applied as another application we solve an open problem from s kuhlmann et al positivity sums of squares and the multi dimensional moment problem ii adv geometry in press on one dimensional quadratic modules

monotonically positive matrices abstract a matrix a is monotonically positive mp if there exists a matrix u such that a u u t and each column of u is monotonically nonincreasing or nondecreasing following the approach in j nie 2014 15 we propose a semidefinite algorithm for checking whether or not a matrix is mp if it is not mp a certificate for it can be obtained if it is mp an mp decomposition can be obtained some computational experiments are presented to show how to do this

analysis and application of the nodal discontinuous galerkin method for wave propagation in metamaterials in this paper we develop a nodal discontinuous galerkin method for solving the time dependent maxwell s equations when metamaterials are involved both semi and fully discrete schemes are constructed numerical stability and error estimate are proved for both schemes numerical results are presented to demonstrate that the method is not only efficient but also very effective in solving metamaterial maxwell s equations

quasi monte carlo finite element methods for elliptic pdes with lognormal random coefficients in this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in mathbb r d rd d 1 2 3 d 1 2 3 with diffusion coefficient a varvec x omega a x given as a lognormal random field i e a varvec x omega exp z varvec x omega a x exp z x where varvec x x is the spatial variable and z varvec x cdot z x is a gaussian random field the analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 0 or infty over all possible realizations of a a focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem we give a rigorous error analysis for methods constructed from 1 standard continuous and piecewise linear finite element approximation in physical space 2 truncated karhunen loeve expansion for computing realizations of a a leading to a possibly high dimensional parametrized deterministic diffusion problem and 3 lattice based quasi monte carlo qmc quadrature rules for computing integrals over parameter space which define the expected values the paper contains novel error analysis which accounts for the effect of all three types of approximation the qmc analysis is based on a recent result on randomly shifted lattice rules for high dimensional integrals over the unbounded domain of euclidean space which shows that under suitable conditions the quadrature error decays with mathcal o n 1 delta o n 1 with respect to the number of quadrature points n n where delta 0 0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

the automorphism group of a completely positive cone and its lie algebra abstract given a closed cone c in r n we consider the corresponding completely positive convex cone k generated by uu t u c in s n under certain conditions on c we describe the automorphism group of k and its corresponding lie algebra in terms of those of c c and or c in particular we show that when c is a closed convex proper cone the automorphism groups of c and k are isomorphic and their corresponding lie algebras are isomorphic

exact algorithms for linear matrix inequalities let a x a 0 x 1a 1 cdots x na n be a linear matrix or pencil generated by given symmetric matrices a 0 a 1 ldots a n of size m with rational entries the set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron described by a linear matrix inequality we design an exact algorithm that up to genericity assumptions on the input matrices computes an exact algebraic representation of at least one point in the spectrahedron or decides that it is empty the algorithm does not assume the existence of an interior point and the computed point minimizes the rank of the pencil on the spectrahedron the degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil we provide explicit bounds for the complexity of our algorithm proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a mu

a multilevel analysis of the lasserre hierarchy abstract this paper analyzes the relation between different orders of the lasserre hierarchy for polynomial optimization pop although for some cases solving the semidefinite programming relaxation corresponding to the first order of the hierarchy is enough to solve the underlying pop other problems require sequentially solving the second or higher orders until a solution is found for these cases and assuming that the lower order semidefinite programming relaxation has been solved we develop prolongation operators that exploit the solutions already calculated to find initial approximations for the solution of the higher order relaxation we can prove feasibility in the higher order of the hierarchy of the points obtained using the operators as well as convergence to the optimal as the relaxation order increases furthermore the operators are simple and inexpensive for problems where the projection over the feasible set is easy to calculate for example integer 0 1 and 1 1 pops our numerical experiments show that it is possible to extract useful information for real applications using the prolongation operators in particular we illustrate how the operators can be used to increase the efficiency of an infeasible interior point method by using them as an initial point we use this technique to solve quadratic integer 0 1 problems as well as max cut and integer partition problems

an existence and uniqueness result for fuzzy goursat partial differential equation abstract in this paper we study a fuzzy partial differential equation the so called goursat problem under generalized differentiability concept we give necessary and sufficient conditions for the existence of a unique solution we also provide some sufficient conditions for the coincidence of the second order mixed derivatives of a fuzzy function which is useful for the study of higher order fuzzy partial differential equations