iterative regularization for elliptic inverse problems elliptic inverse problems can be formulated using coefficient dependent energy least squares functionals resulting in a smooth convex objective functional a variational inequality emerges as a necessary and sufficient optimality condition the principle of iterative regularization when coupled with the auxiliary problem principle results in a strongly convergent scheme for the solution of elliptic inverse problems

convergence rates for tikhonov regularization of a two coefficient identification problem in an elliptic boundary value problem we investigate the convergence rates for tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the neumann problem for the elliptic equation rm div q nabla u au f rm in omega q partial u partial n g on the boundary partial omega omega subset mathbb r d d geq 1 when u is imprecisely given by z delta in h 1 omega u z delta h 1 omega le delta delta 0 we regularize this problem by minimizing the strictly convex functional of q a begin array lll int limits omega left q nabla u q a z delta 2 a u q a z delta 2 right dx quad rho left q q 2 l 2 omega a a 2 l 2 omega right end array over the admissible set k where 0 is the regularization parameter and q a is an a priori estimate of the true pair q a which is identified and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem we obtain the convergence rate mathcal o sqrt delta as 0 and for the regularized solutions under the simple and weak source condition rm there is a function w in v rm such that u prime q dagger a dagger w q dagger q a dagger a with q dagger a dagger being the q a minimum norm solution of the coefficient identification problem u the frechet derivative of u v the sobolev space on which the boundary value problem is considered our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill posed problems furthermore some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces

finite volume lattice boltzmann schemes in two and three dimensions simple and practical finite volume schemes for the lattice boltzmann equation are derived in two and three dimensions through the application of modern finite volume methods the schemes use a finite volume vortextype formulation based on quadrilateral elements in two dimensions and trilinear hexahedral elements in three dimensions it is shown that the schemes are applicable to domains with irregular boundaries of arbitrary shape in two and three dimensions s1063 651x 99 10809 2

edge deletion problems if pi is a property on graphs or digraphs the edge deletion problem can be stated as follows find the minimum number of edges whose deletion results in a subgraph or subdigraph satisfying property pi several well studied graph problems can be formulated as edge deletion problems in this paper we show that the edge deletion problem is np complete for the following properties 1 without cycles of specified length l or of any length leqq l 2 connected and degree constrained 3 outerplanar 4 transitive digraph 5 line invertible 6 bipartite 7 transitively orientable for problems 5 6 7 we determine the best possible bounds on the node degrees for which the problems remain np complete

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

a variational method for numerical differentiation a new method is given for effecting numerical differentiation by means of an optimization procedure the method is shown to be effective for the differentiation of noisy functions and stability and convergence results for a steepest descent implementation are proved

error estimates for the numerical identification of a variable coefficient error estimates are derived for the approximate identification of an unknown transmissivity coefficient in a partial differential equation describing a model problem in groundwater now the approximation scheme considered determines the coefficient by least squares fitting of the observed pressure data 1 introduction in this paper we wish to present an error analysis of a common numerical scheme used in the identification of parameters in distributed systems specifically we shall concern ourselves here with a model problem in groundwater flow the problem is to identify a spatially varying transmissivity coefficient a x from observations of the piezometric head u x in a two dimensional static aquifer s where a and u are related by the equations 1 div avw ins2

probe order backtracking the algorithm for constraint satisfaction problems probe order backtracking has an average running time much faster than any previously analyzed algorithm under conditions where solutions are common the algorithm uses a probing assignment a preselected test assignment to unset variables to help guide the search for a solution if the problem is not satisfied when the unset variables are temporarily set to the probing assignment the algorithm selects one of the relations which is not satisfied by the probing assignment and selects an unset variable which affects the value of that relation it then does a backtracking splitting step where it generates subproblems by setting the selected variable each possible way each subproblem is simplified and then solved recursively for random problems with v variables t clauses and probability p that a literal appears in a clause the average time for probe order backtracking is no more than v n when p geq ln t v plus lower order terms the best previous result was p geq sqrt ln t v when the algorithm is combined with an algorithm of franco that makes selective use of resolution the average time for solving random problems is no more than v n for all values of p when t leq o n 1 3 v ln v 2 3 the best previous result was t leq o n 1 3 v ln v 1 6 probe order backtracking also runs in polynomial average time when p leq 1 v compared with the best previous result of p leq 1 2v with probe order backtracking the range of p that leads to more than polynomial time is much smaller than that for previously analyzed algorithms

one point recovery of a parabolic equation abstract we recover the shape of the domain and the diffusion coefficient of a parabolic equation from observations of an infinite sequence of solutions taken at an arbitrary single point inside or on the boundary of the unknown domain the method is to reconstruct the principal eigenfunction of the associated elliptic operator from the observed data

volumetric formulation of the lattice boltzmann method for fluid dynamics basic concept a lattice boltzmann algorithm based on a volumetric representation is formulated for achieving properties consistent with the standard form at finer resolution in contrast to pointwise interpolation schemes this approach can be applied to arbitrary meshes without compromising exact conservation or equilibrium properties

on the recovery of multiple flow parameters from transient head data the problem of estimating groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system we consider here a new method that allows for the simultaneous computation of multiple parameters as the unique minimum of a convex functional

approximation and intractability results for the maximum cut problem and its variants the maximum cut problem is known to be an important np complete problem with many applications the authors investigate this problem which they call the normal maximum cut problem and a variant of it which is referred to as the connected maximum cut problem they show that any n vertex e edge graph admits a cut with at least the fraction 1 2 1 2n of its edges thus improving the ratio 1 2 2 e known before it is shown that it is np complete to decide if a given graph has a normal maximum cut with at least a fraction 1 2 epsilon of its edges where the positive constant epsilon can be taken smaller than any value chosen the authors present an approximation algorithm for the normal maximum cut problem on any graph that runs in o e log e n log n p log p log n parallel time using p 1

detailed analysis of the lattice boltzmann method on unstructured grids the lattice boltzmann method has become a standard for efficiently solving problems in fluid dynamics while unstructured grids allow for a more efficient geometrical representation of complex boundaries the lattice boltzmann method is often implemented using regular grids here we analyze two implementations of the lattice boltzmann method on unstructured grids the standard forward euler method and the operator splitting method we derive the evolution of the macroscopic variables by means of the chapman enskog expansion and we prove that it yields the navier stokes equation and is first order accurate in terms of the temporal discretization and second order in terms of the spatial discretization relations between the kinetic viscosity and the integration time step are derived for both the euler method and the operator splitting method finally we suggest an improved version of the bounce back boundary condition we test our implementations in both standard benchmark geometries and in the pore network of a real sample of a porous rock

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

acceleration of lattice bgk schemes with grid refinement abstract the lbgk method with local grid refinement has been shown to be an efficient and accurate tool for the simulation of incompressible viscous flows over complex geometries in the present study further improvement of this concept is proposed enabling the use of smaller amount of time steps on refined grids without impairing the spatial or temporal accuracy this extension of the lbgk method has been proved by analytical and numerical investigations the gain in computational time was found to be significant

on implementation of boundary conditions in the application of finite volume lattice boltzmann method a new implementation of boundary condition based on the half covolume and bounce back rule for the non equilibrium distribution function for the finite volume lbm is proposed here the numerical simulation results for the expansion channel flow and driven cavity problem indicate that this method is workable for arbitrary meshes in addition the fourth order runge kutta scheme is found to be a practical way in the lbm to accelerate the calculation speed

an explicit update scheme for inverse parameter and interface estimation of piecewise constant coefficients in linear elliptic pdes we introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities for this purpose we use an output least squares approach with level set and augmented lagrangian methods our formulation incorporates the inherent nature of the piecewise constant coefficients which eliminates the need for a complicated nonlinear solve at every iteration instead we obtain an explicit update formula and therefore vastly speed up computation we employ our approach to the example problems of poisson s equation and linear elasticity and provide the combination of simultaneously recovering coefficients and interface

some progress in lattice boltzmann method part i nonuniform mesh grids a new lattice boltzmann algorithm is proposed to simulate the navier stokes equation on arbitrary nonuniform mesh grids the new algorithm retains the advantages of the lattice boltzmann method parallel of algorithm ease of programming and ability to incorporate microscopic interactions a simulation of flow in a two dimensional symmetric channel with sudden expansion is carried out using the new algorithm on a nonuniform mesh the results of the simulation are in excellent agreement with previous experimental and numerical results

a wavelet multiscale method for the inverse problem of a nonlinear convection diffusion equation abstract this paper is concerned with the problem of identifying the diffusion parameters in a nonlinear convection diffusion equation which arises as the saturation equation in the fractional flow formulation of the two phase porous media flow equations the forward problem is discretized using finite difference methods and the inverse problem is formulated as a minimization problem with regularization terms in order to overcome disturbance of local minimum a wavelet multiscale method is applied to solve this parameter identification inverse problem this method works by decomposing the inverse problem into multiple scales with wavelet transform so that the original inverse problem is reformulated to a set of sub inverse problems relying on scale variables and successively solving these sub inverse problems according to the size of scale from the smallest to the largest the stable and fast regularized gauss newton method is applied to each scale numerical simulations show that the proposed algorithm is widely convergent computationally efficient and has the anti noise and de noising abilities

complexity of partial satisfaction a conjunctive normal form expression cnf is said to be 2 satisfiable if and only if any two of its clauses are simultaneously satisfiable it is shown that every 2 satisfiable cnf has a truth assignment that satisfies at least the fraction h of its clauses where h sqrt 5 1 2 sim 0 618 the reciprocal of the golden ratio the proof is constructive in that it provides a polynomial time algorithm that will find for any 2 satisfiable cnf a truth assignment satisfying at least the fraction h of its clauses furthermore this result is optimal in that the constant h is as large as possible it is shown that for any rational h h the set of all 2 satisfiable cnfs that have truth assignments satisfying at least the fraction h of their clauses is an np complete set

the lattice boltzmann equation on irregular lattices a general framework to extend the lattice boltzmann equation to arbitrary lattice geometries is presented and numerically demonstrated for the case of a two dimensional poiseuille flow the new scheme considerably extends the range of applicability of the boltzmann method to problems requiring the use of nonuniform grids

a new bound for 3 satisfiable maxsat and its algorithmic application let f be a cnf formula with n variables and m clauses f is 3 satisfiable if for any 3 clauses in f there is a truth assignment which satisfies all of them lieberherr and specker 1982 and later yannakakis 1994 proved that in each 3 satisfiable cnf formula at least 23 of its clauses can be satisfied by a truth assignment we improve this result by showing that every 3 satisfiable cnf formula f contains a subset of variables u such that some truth assignment t will satisfy at least 23m 13m u rn clauses where m is the number of clauses of f m u is the number of clauses of f containing a variable from u n is the total number of variables in clauses not containing a variable in u and r is a positive absolute constant both u and t can be found in polynomial time we use our result to show that the following parameterized problem is fixed parameter tractable and moreover has a kernel with a linear number of variables in 3 s maxsat ae we are given a 3 satisfiable cnf formula f with m clauses and asked to determine whether there is an assignment which satisfies at least 23m k clauses where k is the parameter

uniqueness for an elliptic inverse problem for the elliptic equation nabla cdot p x nabla v lambda q x v f x in omega subset bbr n the problem of determining when one or more of the coefficient functions p q and f are defined uniquely by a knowledge of one or more of the solution functions v v p q f lambda is considered

node deletion problems on bipartite graphs a set of problems which has attracted considerable interest recently is the set of node deletion problems the general node deletion problem can be stated as follows given a graph find the minimum number of nodes whose deletion results in a subgraph satisfying property r in ly this problem was shown to be np complete for a large class of properties the class of properties that are hereditary on induced subgraphs using a small number of reduction schemes from the node cover problem since the node cover problem becomes polynomial on bipartite graphs it might be hoped that this is the case with other node deletion problems too in this paper we characterize those properties for which the bipartite restriction of the node deletion problem is polynomial and those for which it remains np complete similar results follow for analogous problems on other structures such as families of sets hypergraphs and 0 1 matrices for example in the case of matrices our result states that ifm is a class of 0 1 matrices which is closed under permutation and deletion of rows and columns then finding the largest submatrix inm of a matrix is polynomial if the matrices ofm have bounded rank and np complete otherwise 1 introduction a common approach when faced with an np complete problem c k gj is to restrict its input domain with the hope that the problem may become solvable in polynomial time under the restriction in fact it turns out that sometimes this is the case for example the node cover problem np complete on general graphs k can be solved efficiently when restricted to bipartite graphs this follows from k6nig s theorem that the node covering number of a bipartite graph equals the number of edges in a maximum matching and moreover a minimum node cover can be found efficiently from a maximum matching when the graph is bipartite see for example la in other cases however the node cover problem continues to be np complete under restriction for example the node cover problem on planar graphs gjs much work has been done recently kd ly ldl on a class of problems called the node deletion or maximum subgraph problems the general node deletion problem can be stated as follows given a graph or digraph g find a set of nodes of minimum cardinality whose deletion results in a graph or digraph satisfying a property r several of the known np complete problems such as the node cover the max clique the feedback node set k as well as some polynomial problems such as the connectivity of a graph e it can be formulated in an obvious way as node deletion problems by specifying appropriately the property r

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

on the inverse problem of identifying lame coefficients in linear elasticity an output least squares type functional is employed to identify the lame parameters in linear elasticity to be able to identify even the discontinuous lame parameters the regularization is performed by the bv seminorm finite element discretization is used and convergence analysis is given numerical examples are given to show the feasibility of the approach

parameter identification for elliptic problems abstract the problem of computing a principal coefficient function p in the differential equation p x u f x r m m 1 on a bounded region from a knowledge of the solution function u and the right hand side f where u f are known only approximately and p may have mild discontinuities is solved by minimization of an associated functional

lattice boltzmann method on irregular meshes a finite volume scheme for the lattice boltzmann method fvlbm is described the scheme uses a finite volume formulation based on triangular elements in two dimensions and is implemented assuming that velocity space is isotropically discretized at each node in position space the accuracy of the finite volume scheme is numerically demonstrated by comparing the computed macroscopic velocity field with the exact solution of the navier stokes equations for the flow of an incompressible fluid between two relatively rotating cylinders the fvlbm scheme is applicable to irregular two dimensional regions which contain both exterior and interior boundaries of arbitrary shape thus the range of applicability of systems to which the fvlbm may be applied is seen to be significantly extended

implications of forbidden structures for extremal algorithmic problems abstract we develop and analyze p optimal approximation algorithms for various generalized satisfiability problems and determine the corresponding p optimal thresholds the most novel aspect of the paper is that we develop new p optimal algorithms for restricted generalized satisfiability problems which are defined by forbidden subformulas all our algorithms run in linear time on a ram

descent gradient methods for nonsmooth minimization problems in ill posed problems descent gradient methods are the most frequently used algorithms for computing regularizers of inverse problems they are either directly applied to the discrepancy term which measures the difference between operator evaluation and data or to a regularized version incorporating suitable penalty terms in its basic form gradient descent methods converge slowly we aim at extending different optimization schemes which have been recently introduced for accelerating these approaches by addressing more general penalty terms in particular we use a general setting in infinite hilbert spaces and examine accelerated algorithms for regularization methods using total variation or sparsity constraints to illustrate the efficiency of these algorithms we apply them to a parameter identification problem in an elliptic partial differential equation using total variation regularization

finite volume lattice boltzmann method we present a finite volume formulation for the lattice boltzmann method fvlbm based on standard bilinear quadrilateral elements in two dimensions the accuracy of this scheme is demonstrated by comparing the velocity field with the analytical solution of the navier stokes equations for time dependent rotating couette flow and taylor vortex flow to demonstrate the flexibility of the scheme we have also simulated a modified rotating couette flow where the inner cylinder has an elliptical shape the results agree with those obtained from the traditional marker and cell method the fvlbm scheme is applicable to arbitrarily shaped two dimensional regions and thus the range of applicability of the lattice boltzmann method has been significantly extended

finite volume scheme for the lattice boltzmann method on unstructured meshes department of mathematics and statistics bowling green state university bowling green ohio 43403 received 9 september 1998 a nite volume scheme for the lattice boltzmann method is developed for unstructured triangular meshes intwo dimensions the accuracy of this new scheme is demonstrated by comparing the numerical results with theexact solutions to the navier stokes equations for taylor vortex ow shear ow between two parallel plates shear ow between two rotating cylinders and poiseuille ow the agreement between the numerical andanalytical results is very good for each of these tests s1063 651x 99 07804 6 pacs number s 47 10 1g 47 11 1j 05 20 ddi introduction

an inverse problem for the steady state diffusion equation we consider the problem of identifying a spatially varying diffusion coefficient using an observed solution to the forward problem under appropriate conditions this problem can be treated as a first order hyperbolic equation in the unknown coefficient we develop some continuous dependence results for this problem and also propose a particularly advantageous set of test conditions for observing the forward solution

convergence rates for total variation regularization of coefficient identification problems in elliptic equations ii abstract we investigate the convergence rates for total variation regularization of the problem of identifying i the coefficient q in the neumann problem for the elliptic equation div q u f in q u n g on ii the coefficient a in the neumann problem for the elliptic equation u a u f in u n g on r d d 1 when u is imprecisely given by z h 1 u z h 1 0 we regularize these problems by correspondingly minimizing the strictly convex functionals 1 2 q u q z 2 d x 1 2 q l 2 2 q and 1 2 u a z 2 d x 1 2 a u a z 2 d x 1 2 a l 2 2 a over admissible sets where u q u a is the solution of the first second neumann boundary value problem 0 is the regularization parameter taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems we obtain the convergence rates of them to the solution of the inverse problem in the sense of the bregman distance and in the l 2 norm under relatively simple source conditions without the smallness requirement on the source functions