convergence of boundary integral method for a free boundary system abstract boundary integral method has been implemented successfully in practice for simulating problems with free boundaries though the method produces accurate and efficient numerical results its convergence study is usually limited to numerical demonstrations by successively reducing time step and increasing resolution for a test problem in this paper we present a rigorous convergence and error analysis of the boundary integral method for a free boundary system we focus our study on a nonlinear tumor growth problem the boundary integral formulation yields a fredholm type integral equation with moving boundaries we show that in two dimensions the convergence of the scheme in the l norm has first order accuracy on the time direction and on the spatial direction

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

a three dimensional steady state tumor system abstract the growth of tumors can be modeled as a free boundary problem involving partial differential equations we consider one such model and compute steady state solutions for this model these solutions include radially symmetric solutions where the free boundary is a sphere and nonradially symmetric solutions linear and nonlinear stability for these solutions are determined numerically

small covering designs by branch and cut a branch and cut algorithm for finding covering designs is presented its originality resides in the use of isomorphism pruning of the enumeration tree a proof that no 4 10 5 1 covering design with less than 51 sets exists is obtained together with all non isomorphic 4 10 5 1 covering designs with 51 sets

polyhedral characterization of discrete dynamic programming many interesting combinatorial problems can be optimized efficiently using recursive computations often termed discrete dynamic programming in this paper we develop a paradigm for a general class of such optimizations that yields a polyhedral description for each model in the class the elementary concept of dynamic programs as shortest path problems in acyclic graphs is generalized to one seeking a least cost solution in a directed hypergraph sufficient conditions are then provided for binary integrality of the associated hyperflow problem given a polynomially solvable dynamic program the result is a linear program in polynomially many variables and constraints having the solution vectors of the dynamic program as its extreme point optima that is the linear program provides a succinct characterization of the solutions to the underlying optimization problem expressed through an appropriate change of variables we also discuss projecting this formulation to recover constraints on the original variables and illustrate how some important dynamic programming solvable models fit easily into our paradigm a classic multiechelon lot sizing problem in production and a host of optimization problems on recursively defined classes of graphs are included

bifurcation from stability to instability for a free boundary problem modeling tumor growth by stokes equation we consider a free boundary problem modeling tumor growth in fluid like tissue the model equations include a diffusion equation for the nutrient concentration and the stokes equation with a source which represents the proliferation of tumor cells the proliferation rate and the cell to cell adhesiveness which keeps the tumor intact are two parameters which characterize the aggressiveness of the tumor for any positive radius r there exists a unique radially symmetric stationary solution with radius r r for a sequence mn r there exist symmetry breaking bifurcation branches of solutions with free boundary r r eyn 0 o e 2 n even 2 for small e w hereyn 0 is the spherical harmonic of mode n 0 furthermore the smallest mn r s aymn r is such that n n r as r in this paper we prove that the radially symmetric stationary solution with r rs is linearly stable if n rs w heren rs mn rs and we prove that strict inequality holds if is small or if is large the biological implications of these results are discussed at the end of the paper

completeness of solutions of bethe s equations we consider the bethe equations for the isotropic spin 1 2 heisenberg quantum spin chain with periodic boundary conditions we formulate a conjecture for the number of solutions with pairwise distinct roots of these equations in terms of numbers of so called singular or exceptional solutions using homotopy continuation methods we find all such solutions of the bethe equations for chains of length up to 14 the numbers of these solutions are in perfect agreement with the conjecture we also discuss an indirect method of finding solutions of the bethe equations by solving the baxter t q equation we briefly comment on implications for thermodynamical computations based on the string hypothesis

polytopes associated with symmetry handling this paper investigates a polyhedral approach to handle symmetries in mixed binary programs we study symretopes i e the convex hulls of all binary vectors that are lexicographically maximal in their orbit with respect to the symmetry group these polytopes turn out to be quite complex for practical use we therefore develop an integer programming formulation with ternary coefficients based on knapsack polytopes arising from a single lexicographic order enforcing inequality we show that for these polytopes the optimization as well as the separation problem of minimal cover inequalities can be solved in almost linear time we demonstrate the usefulness of this approach by computational experiments showing that it is competitive with state of the art methods and is considerably faster for specific problem classes

extended formulations for packing and partitioning orbitopes we give compact extended formulations for the packing and partitioning orbitopes with respect to the full symmetric group described and analyzed in kaibel and pfetsch kaibel v m e pfetsch 2008 packing and partitioning orbitopes math programming ser a114 1 1 36 these polytopes are the convex hulls of all 0 1 matrices with lexicographically sorted columns and at most respectively exactly one 1 entry per row they are important objects for symmetry reduction in certain integer programs using the extended formulations we also derive a rather simple proof of the fact established in the paper mentioned above that basically shifted column inequalities suffice to describe those orbitopes linearly

reformulations in mathematical programming automatic symmetry detection and exploitation if a mathematical program has many symmetric optima solving it via branch and bound techniques often yields search trees of disproportionate sizes thus finding and exploiting symmetries is an important task we propose a method for automatically finding the formulation group of any given mixed integer nonlinear program and for reformulating the problem by means of static symmetry breaking constraints the reformulated problem which is likely to have fewer symmetric optima can then be solved via standard branch and bound codes such as cplex for linear programs and couenne for nonlinear programs our computational results include formulation group tables for the miplib3 miplib2003 globallib and minlplib instance libraries and solution tables for some instances in the aforementioned libraries

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

an a posteriori error estimator for anisotropic refinement besides an algorithm for local refinement an a posteriori error estimator is the basic tool of every adaptive finite element method using information generated by such an error estimator the refinement of the grid is controlled for 2nd order elliptic problems we present an error estimator for anisotropically refined grids like n d cuboidal and 3 d prismatic grids that gives correct information about the size of the error additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way numerical examples are presented for 2 d rectangular and 3 d prismatic grids

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

packing and partitioning orbitopes we introduce orbitopes as the convex hulls of 0 1 matrices that are lexicographically maximal subject to a group acting on the columns special cases are packing and partitioning orbitopes which arise from restrictions to matrices with at most or exactly one 1 entry in each row respectively the goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints e g for a well known formulation of the graph coloring problem we provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group our main results are complete linear inequality descriptions of these polytopes by facet defining inequalities for the cyclic group case the descriptions turn out to be totally unimodular while for the symmetric group case both the description and the proof are more involved the associated separation problems can be solved in linear time

vibrations of thin plates with small clamped patches eigenvalues of fourth order elliptic operators feature prominently in stability analysis of elastic structures this paper considers out of plane modes of vibration of a thin elastic plate perforat

computing steady state solutions for a free boundary problem modeling tumor growth by stokes equation we consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the stokes equation for the proliferation of tumor cells for any positive radius r it is known that there exists a unique radially symmetric stationary solution the proliferation rate m and the cell to cell adhesiveness c are two parameters for characterizing aggressiveness of the tumor we compute symmetry breaking bifurcation branches of solutions by studying a polynomial discretization of the system by tracking the discretized system we numerically verified a sequence of m c symmetry breaking bifurcation branches furthermore we study the stability of both radially symmetric and radially asymmetric stationary solutions

adaptive multiprecision path tracking this article treats numerical methods for tracking an implicitly defined path the numerical precision required to successfully track such a path is difficult to predict a priori and indeed it may change dramatically through the course of the path in current practice one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved to avoid unnecessary computational cost it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path we present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates we then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations

network formulations of mixed integer programs we consider mixed integer sets described by system of linear inequalities in which the constraint matrix a is totally unimodular the right hand side is arbitrary vector and a subset of the variables is required to be integer we show that the problem of checking nonemptiness of a set of this type is np complete even in the case in which the linear system describes mixed integer network flows with half integral requirement on the nodes this is in contrast to the case in which a is totally unimodular and contains at most two nonzeros per row in this case we provide an extended formulation for the convex hull of solutions whose constraint matrix is a dual network matrix with an integral right hand side vector the size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of the convex hull of the given set when this number is polynomial in the dimension of a the extended formulation is of polynomial size if in addition the corresponding list of fractional parts can be computed efficiently then our result provides a polynomial algorithm for the optimization problem over these sets we show that there are instances for which this list is of exponential size and we also give conditions under which it is short and can be efficiently computed finally we show that these results provide a unified framework leading to polynomial size extended formulations for several generalizations of mixing sets and lot sizing sets studied in the last few years

boundary element method for a free third boundary problem modeling tumor growth with spectral accuracy abstract by boundary element method we present a numerical iterative process for solving a free third boundary problem modeling tumor growth with spectral accuracy the piecewise quadratic curves are fitted to maintain local smoothness of the boundary at every node the double layer and single layer potentials with weakly singular kernels are evaluated with spectral accuracy the method of characteristics is employed to transform interfacial velocity pde into discrete odes the numerical integral formula for weakly singular operator with logarithmic singularity is deduced and the convergence and error are presented the nonradially symmetric solutions of the free boundary problem on a perturbed boundary are provided to test the accuracy and effectiveness of the numerical method

boundary element method for a free boundary problem modeling three dimensional tumor growth abstract in the mathematical model of tumor growth the initial boundary surface of the three dimensional tumor domain is known but the tumor domain and its boundary surface change over time in a way that is unknown in advance people are concerned whether the tumor is likely to spread or shrink and how the tumor will change in this article by the boundary element method the free boundary problem in the three dimensional tumor domain will be solved via the integration on a two dimensional boundary surfaces at the expense of singularity in the green s functions we will numerically compute and graphically show the changing boundary surfaces of the three dimensional tumor domain over time we will numerically analyze the trend of tumor growth with varying proliferation rate our numerical approach professor bei hu proposed is new and our numerical experiments contribute to the prediction of tumor growth in clinical medicine

exploiting orbits in symmetric ilp this paper describes components of a branch and cut algorithm for solving integer linear programs having a large symmetry group it describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group pruning and orbit computations are performed by backtracking procedures using a schreier sims table for representing the symmetry group applications to hard set covering problems generation of covering designs and error correcting codes are given

on optimal triangular meshes for minimizing the gradient error construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed using an appropriate linear coordinate transformation rigorously optimal meshes for controlling the error in quadratic data functions are constructed it is shown that the transformation can be generated as a curvilinear coordinate transformation for anyc data function with nonsingular hessian matrix using this transformation a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed in particular it is shown that equilibration of errors is not a sufficient condition for optimality a comparison of meshes generated under several different criteria is made and their equilibrating properties illustrated

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

pruning by isomorphism in branch and cut the paper presents a branch and cut for solving 0 1 integer linear programs having a large symmetry group the group is used for pruning the enumeration tree and for generating cuts the cuts are non standard cutting integer feasible solutions but leaving the optimal value of the problem unchanged pruning and cut generation are performed by backtracking procedures using a schreier sims table for representing the group applications to hard set covering problems and to the generation of covering designs and error correcting codes are presented

a bootstrapping approach for computing multiple solutions of differential equations discretizing systems of nonlinear algebraic differential equations yields polynomial systems when using a fine discretization the resulting polynomial system is often too large to solve using a direct solving approach our approach for solving such systems is to utilize a homotopy continuation based method arising from domain decomposition this method solves polynomial systems arising from subdomains and then uses homotopy continuation to build solutions of the original polynomial system we illustrate this approach on both one and two dimensional problems

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner