on conservation at grid interfaces this paper considers the solution of hyperbolic systems of conservation laws on discontinuous grids in particular we consider what happens to conservation at grid interfaces a procedure is presented to derive conservative difference approximations at the grid interfaces for two dimensional grids which overlap in an arbitrary configuration the same procedures are applied to compute interface formulas for grids which are refined in space and or time and for continuous grids where a switch in the scheme causes the discontinuity

adjoint recovery of superconvergent functionals from pde approximations motivated by applications in computational fluid dynamics a method is presented for obtaining estimates of integral functionals such as lift or drag that have twice the order of accuracy of the computed flow solution on which they are based this is achieved through error analysis that uses an adjoint pde to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy numerical results are presented for the poisson equation in one and two dimensions and for the nonlinear quasi one dimensional euler equations the theory is equally applicable to nonlinear equations in complex multi dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

constrained lqr problems in elliptic distributed control systems with point observations in this paper we study bound constrained lqr problems in distributed control systems governed by the elliptic equation with point observations which are motivated by problems in corrosion engineering and contemporary smart materials several regularity and characterization theorems have been established in particular three decomposition formulas are obtained to characterize the optimal control and optimal layer density and are used to direct the numerical computations these results cannot be obtained by the traditional galerkin variational method in the process several useful lemmas are established which are of independent interest we point out that the classical lagrangian multiplier method lmm may fail to provide a reliable numerical algorithm based on our characterization results and the boundary element method two algorithms are proposed to carry out numerical computations it has been shown by our numerical experiments that both algorithms are efficient and insensitive to the partition number of the boundary an adaptive local refinement scheme has also been designed to handle the rough behavior of the optimal solution around sensor locations

new elliptic projections and a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations in this paper we discuss a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions both semidiscrete and fully discrete schemes are considered based on some new elliptic projections we derive a priori error estimates for the control variable the state variables and the adjoint state variables the related a priori error estimates for the new projections error are also established a numerical example is given to demonstrate the theoretical results

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

adaptive finite element methods for optimal control of partial differential equations basic concept a new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations the lagrangian formalism yields the first order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive galerkin finite element method the mesh design in the resulting reduced models is controlled by residual based a posteriori error estimates these are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error in this case the computed state and costate variables can be used as sensitivity factors multiplying the local cell residuals in the error estimators this results in a generic and simple algorithm for mesh adaptation within the optimization process this method is developed and tested for simple boundary control problems in semiconductor models

boundary velocity control of incompressible flow with an application to viscous drag reduction an optimal boundary control problem for the navier stokes equations is presented the control is the velocity on the boundary which is constrained to lie in a closed convex subset of h 1 2 of the boundary a necessary condition for optimality is derived computations are done when the control set is actually finite dimensional resulting in an application to viscous drag reduction

a posteriori error estimates for control problems governed by stokes equations in this paper we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the stokes equations we obtain a posteriori error estimators for both the state and the control approximation in the l2 norm and the h1 norm these estimates can be used to construct reliable adaptive finite element approximation for the control problems

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

ritz galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions this paper deals with ritz galerkin approximations of the following two problems i boundary value problems with l 2 boundary data given in the form of dirichlet boundary conditions ii time optimal control problems for parabolic systems with control acting on the boundary in both cases optimal rates of convergence of approximation are obtained also the specialization is considered to approximating with nonconformal elements which are not required to satisfy boundary conditions corresponding rates of convergence are also proved

theory of minimum spanning trees ii exact graphical methods and perturbation expansion at the percolation threshold continuing the program begun by the authors in a previous paper we develop an exact low density expansion for the random minimum spanning tree mst on a finite graph and use it to develop a continuum perturbation expansion for the mst on critical percolation clusters in space dimension d the perturbation expansion is proved to be renormalizable in d 6 dimensions we consider the fractal dimension d sub p of paths on the latter mst our previous results lead us to predict that d sub p 2 for d d sub c 6 using a renormalization group approach we confirm the result for d 6 and calculate d sub p to first order in epsilon 6 d for d 6 using the connection with critical percolation with the result d sub p 2 epsilon 7 o epsilon sup 2

transition between strong and weak disorder regimes for the optimal path abstract we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path l opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight i exp ar i where r i is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finite a there is a crossover network size n a such that for n n a the scaling behavior of l opt is in the strong disorder regime while for n n a the scaling behavior is in the weak disorder regime we derive the scaling relation between n a and a with the help of simulations and also present an analytic derivation of the relation

optimal path in random networks with disorder a mini review abstract we review the analysis of the length of the optimal path l opt in random networks with disorder i e random weights on the links in the case of strong disorder in which the maximal weight along the path dominates the sum we find that l opt increases dramatically compared to the known small world result for the minimum distance l min for erd s renyi er networks l opt n 1 3 while for scale free sf networks with degree distribution p k k we find that l opt scales as n 3 1 for 3 4 and as n 1 3 for 4 thus for these networks the small world nature is destroyed for 2 3 our numerical results suggest that l opt scales as ln 1 n we also find numerically that for weak disorder l opt ln n for er models as well as for sf networks we also study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path l opt in er and sf networks

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a high accuracy nfem for constrained optimal control problems governed by elliptic equations in this paper a new nonconforming finite element method nfem is proposed for the constrained optimal control problems ocps governed by elliptic equations the state and co state are approximated by the nonconforming eq 1 rot element and the control is approximated by the piecewise constant element respectively the superclose properties for the above three variables are obtained moreover for the state and co state the superconvergence results are achieved in a broken energy norm by using the post processing technique lastly some numerical examples are provided to verify the theoretical analysis

optimal paths in disordered media scaling of the crossover from self similar to self affine behavior we study optimal paths in disordered energy landscapes using energy distributions of the type p mathrm log 10 e mathrm const that lead to the strong disorder limit if we truncate the distribution so that p mathrm log 10 e mathrm const only for e mathrm min l el e mathrm max and p mathrm log 10 e 0 otherwise we obtain a crossover from self similar strong disorder to self affine moderate disorder behavior at a path length mathcal l ifmmode times else texttimes fi we find that mathcal l ifmmode times else texttimes fi ensuremath propto mathrm log 10 e mathrm max e mathrm min ensuremath kappa where the exponent ensuremath kappa has the value ensuremath kappa 1 60 ifmmode pm else textpm fi 0 03 both in d 2 and d 3 we show how the crossover can be understood from the distribution of local energies on the optimal paths

transport in weighted networks partition into superhighways and roads transport in weighted networks is dominated by the minimum spanning tree mst the tree connecting all nodes with the minimum total weight we find that the mst can be partitioned into two distinct components having significantly different transport properties characterized by centrality the number of times a node or link is used by transport paths one component superhighways is the infinite incipient percolation cluster for which we find that nodes or links with high centrality dominate for the other component roads which includes the remaining nodes low centrality nodes dominate we find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law with an exponent smaller than that for the entire mst the significance of this finding is that one can improve significantly the global transport by improving a tiny fraction of the network the superhighways

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

automatic adaptive grid refinement for the euler equations a method of adaptive grid refinement for the solution of the steady euler equations for transonic flow is presented the algorithm automatically decides where the coarse grid accuracy is insufficient and creates locally uniform refined grids in these regions this typically occurs at the leading and trailing edges the solution is then integrated to steady state using the same integrator flo52 in the interior of each grid the boundary conditions needed on the fine grids are examined and the importance of treating the fine coarse grid interface conservatively is discussed numerical results indicate substantial computational savings for the same solution accuracy can be achieved

transport and percolation theory in weighted networks we study the distribution p of the equivalent conductance for erdos renyi er and scale free sf weighted resistor networks with n nodes each link has conductance geax where x is a random number taken from a uniform distribution between 0 and 1 and the parameter a represents the strength of the disorder we provide an iterative fast algorithm to obtain p and compare it with the traditional algorithm of solving kirchhoff equations we find both analytically and numerically that p for er networks exhibits two regimes i a low conductance regime for eapc where pc 1 k is the critical percolation threshold of the network and k is the average degree of the network in this regime p is independent of n and follows the power law p where 1 k a ii a high conductance regime for eapc in which we find that p has strong n dependence and scales as pf apc n 1 3 for sf networks with degree distribution pkk kminkkmax we find numerically also two regimes similar to those found for er networks

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

universality of the optimal path in the strong disorder limit we study numerically the optimal paths in two and three dimensions on various disordered lattices in the limit of strong disorder we find that the length l of the optimal path scales with geometric distance r as l approximately r d opt with d opt 1 22 0 01 for d 2 and 1 44 0 02 for d 3 independent of whether the optimization is on a path of weighted bonds or sites and independent of the lattice or its coordination number our finding suggests that the exponent d opt is universal depending only on the dimension of the system

scaling of optimal path lengths distribution in complex networks we study the distribution of optimal path lengths in random graphs with random weights associated with each link disorder with each link i we associate a weight i expari where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we suggest in an analogy with the average length of the optimal path that the distribution of optimal path lengths has a universal form that is controlled by the expression 1 pc a where is the optimal path length in strong disorder a and pc is the percolation threshold this relation is supported by numerical simulations for erd s renyi and scale free graphs we explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network

quasi norm local error estimators for p laplacian in this paper we extend the quasi norm techniques used in a priori error estimation of finite element approximation of degenerate nonlinear systems in order to carry out an improved a posteriori error analysis for the p laplacian we derive quasi norm a posteriori error estimators of residual type which are shown to provide improved upper and lower bounds on the discretization error for sufficiently regular solutions these estimators are further shown to be equivalent on the discretization error in a quasi norm numerical results demonstrating these a posteriori estimators are also presented

local a posteriori error estimates for convex boundary control problems in this paper we derive some improved a posteriori error estimates for finite element approximation of neumann boundary control problems we first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems we then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications

effect of disorder strength on optimal paths in complex networks we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight ti expsarid where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finitea there is a crossover network size n sad at which the transition occurs for n n sad the scaling behavior of opt is in the strong disorder regime with opt n 1 3 for er networks and for sf networks with lo 4 and opt n sl 3d sl 1d for sf networks with 3 l 4 for n n sad the scaling behavior is in the weak disorder regime with opt ln n for er networks and sf networks with l 3 in order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder we propose a scaling ansatz for this measure and demonstrate its validity we proceed to derive the scaling relation between n sad and a we find thatn sa d a 3 for er networks and for sf networks with lo 4 and n sa d a sl 1d sl 3d for sf networks with 3 l 4

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

optimal paths in complex networks with correlated weights the worldwide airport network we study complex networks with weights wij associated with each link connecting node i and j the weights are chosen to be correlated with the network topology in the form found in two real world examples a the worldwide airport network and b the e coli metabolic network here wijxijkikj where ki and kj are the degrees of nodes i and j xij is a random number and represents the strength of the correlations the case 0 represents correlation between weights and degree while 0 represents anticorrelation and the case 0 reduces to the case of no correlations we study the scaling of the lengths of the optimal paths opt with the system size n in strong disorder for scale free networks for different we find two different universality classes for opt in strong disorder depending on i if 0 then for 2 the scaling law optn 1 3 where is the power law exponent of the degree distribution of scale free networks and ii if 0 then opt nopt with opt identical to its value for the uncorrelated case 0 we calculate the robustness of correlated scale free networks with different and find the networks with 0 to be the most robust networks when compared to the other values of we propose an analytical method to study percolation phenomena on networks with this kind of correlation and our numerical results suggest that for scale free networks with 0 the percolation threshold pc is finite for 3 which belongs to the same universality class as 0 we compare our simulation results with the real worldwide airport network and we find good agreement

a posteriori error estimates for some model boundary control problems in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal boundary control problems we derive a posteriori error estimates for both the state and the control approximation on polygonal domains such estimates which are apparently not available in the literature can be used to construct reliable adaptive finite element approximation schemes for the control problems

grid embedding technique using cartesian grids for euler solutions grid embedding techniques are presently applied to the solutions of the euler equations on cartesian grids for the naca 0012 airfoil and the multielement skf1 1 airfoil at transonic speeds based on comparisons with solutions on fine grids it is shown that the present scheme when used in conjunction with the runge kutta time stepping scheme as well as cartesian grids yields reduced memory requirements and a substantial reduction in computational time without loss of accuracy