the maximum entropy inference of solutions to pdes we use the maximum entropy variational technique to infer solutions to second order partial differential equations we restrict ourselves to problems with dirichlet boundary conditions first we construct a basis of moment functions in terms of which we set up the constraints for a maximum entropy inversion for the inference of solutions to elliptic equations then we extend the scheme to the inference of time dependence in evolution equations this extension is done by solving a small system of first order initial value differential equations for the moments of the solution vector

approximations of a ginzburg landau model for superconducting hollow spheres based on spherical centroidal voronoi tessellations in this paper the numerical approximations of the ginzburg landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal voronoi tessellations a reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit we present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations properties of the spherical centroidal voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

a fully discrete evolving surface finite element method in this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface in earlier papers using a suitable variational formulation in time dependent sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces ima j numer anal 25 2007 pp 385 407 math comp to appear optimal order l 2 gamma t and h 1 gamma t error bounds were proved for linear finite elements in this work we prove optimal order error bounds for a backward euler time discretization

solving differential equations by a maximum entropy minimum norm method with applications to fokker planck equations the method of maximum entropy minimum norm is utilized to produce a general method of solving differential equations the technique is a generalization and extension of previous work performed by baker jarvis j math phys 30 302 1989 it is found that introducing an additional constraint on the norm of the solution vector produces a probability distribution that is integrable over the entire real axis a number of simplifications occur in this extended method the lagrange multipliers and solution vector can be solved for explicitly thus eliminating the necessity of solving systems of nonlinear equations for the lagrange multipliers as was required in the previous approach it is shown that the solution obtained is equivalent to a minimum norm approximation the maximum entropy solution of differential equations with fourier moments is shown to be identical to a fourier series solution additionally the new method is applied to solving the random walk and fokker planck equations

meshfree methods on manifolds for hydrodynamic flows on curved surfaces a generalized moving least squares gmls approach abstract we utilize generalized moving least squares gmls to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds we use exterior calculus to formulate incompressible hydrodynamic equations in the stokesian regime and handle the divergence free constraints via a generalized vector potential this provides less coordinate centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth order governing equations in terms of a system of second order elliptic operators using a hodge decomposition we develop methods for manifolds having spherical topology we show the methods exhibit high order convergence rates for solving hydrodynamic flows on curved surfaces the methods also provide general high order approximations for the metric curvature and other geometric quantities of the manifold and associated exterior calculus operators the approaches also can be utilized to develop high order solvers for other scalar valued and vector valued problems on manifolds

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

on the theorem of uniform recovery of random sampling matrices we consider two theorems from the theory of compressive sensing mainly a theorem concerning uniform recovery of random sampling matrices where the number of samples needed in order to recover an s sparse signal from linear measurements with high probability is known to be m s ln s 3 lnn we present new and improved constants together with what we consider to be a more explicit proof a proof that also allows for a slightly larger class of m n matrices by considering what is called effective sparsity we also present a condition on the so called restricted isometry constants s ensuring sparse recovery via l 1 minimization we show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type

finite element approximation of the cahn hilliard equation on surfaces abstract in this paper we consider the phase separation on general surfaces by solving the nonlinear cahn hilliard equation using a finite element method a fully discrete approximation scheme is introduced and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time this in turn leads to convergence and optimal error estimates of the discretization scheme numerical examples are also provided to substantiate the theoretical results

geodesic curvature driven surface microdomain formation abstract lipid bilayer membranes are not uniform and clusters of lipids in a more ordered state exist within the generally disorder lipid milieu of the membrane these clusters of ordered lipids microdomains are now referred to as lipid rafts recent reports attribute the formation of these microdomains to the geometrical and molecular mechanical mismatch of lipids of different species on the boundary here we introduce the geodesic curvature to characterize the geometry of the domain boundary and develop a geodesic curvature energy model to describe the formation of these microdomains as a result of energy minimization our model accepts the intrinsic geodesic curvature of any binary lipid mixture as an input and will produce microdomains of the given geodesic curvature as demonstrated by three sets of numerical simulations our results are in contrast to the surface phase separation predicted by the classical surface cahn hilliard equation which tends to generate large domains as a result of the minimizing line tension our model provides a direct and quantified description of the structure inhomogeneity of lipid bilayer membrane and can be coupled to the investigations of biological processes on membranes for which such inhomogeneity plays essential roles

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

an energy stable and convergent finite difference scheme for the phase field crystal equation we present an unconditionally energy stable finite difference scheme for the phase field crystal equation the method is based on a convex splitting of a discrete energy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present local in time error estimates that ensure the convergence of the scheme while this paper is primarily concerned with the phase field crystal equation most of the theoretical results hold for the related swift hohenberg equation as well

on the best constants in noncommutative khintchine type inequalities we obtain new proofs with improved constants of the khintchine type inequality with matrix coefficients in two cases the first case is the pisier and lust piquard noncommutative khintchine inequality for p 1 where we obtain the sharp lower bound of 12 in the complex gaussian case and for the sequence of functions ei2nt n 1 the second case is junge s recent khintchine type inequality for subspaces of the operator space r c which he used to construct a cb embedding of the operator hilbert space oh into the predual of a hyperfinite factor also in this case we obtain a sharp lower bound of 12 as a consequence it follows that any subspace of a quotient of r c is cb isomorphic to a subspace of the predual of the hyperfinite factor of type iii1 with cb isomorphism constant 2 in particular the operator hilbert space oh has this property

constrained centroidal voronoi tessellations for surfaces centroidal voronoi tessellations are useful for subdividing a region in euclidean space into voronoi regions whose generators are also the centers of mass with respect to a prescribed density function of the regions their extensions to general spaces and sets are also available for example tessellations of surfaces in a euclidean space may be considered in this paper a precise definition of such constrained centroidal voronoi tessellations ccvts is given and a number of their properties are derived including their characterization as minimizers of an energy deterministic and probabilistic algorithms for the construction of ccvts are presented and some analytical results for one of the algorithms are given computational examples are provided which serve to illustrate the high quality of ccvt point sets finally ccvt point sets are applied to polynomial interpolation and numerical integration on the sphere

numerical simulation of dealloying by surface dissolution via the evolving surface finite element method in this article we formulate a numerical method for the simulation of dealloying of a binary alloy by the selective removal of one component via electrochemical dissolution such that there is phase separation of the other component at the solid electrolyte interface the evolution of the interface is modelled by a forced mean curvature flow strongly coupled to the solution of a cahn hilliard equation modelling surface phase separation the method is based on a triangulated hypersurface whose evolution is calculated as well as the solution of the cahn hilliard equation by the evolving surface finite element method esfem the numerical experiments simulate complex morphology and concentration patterns providing evidence that the mathematical model may describe the formation of nanoporosity

stability of multicomponent biological membranes equilibrium equations and stability conditions are derived for a general class of multicomponent biological membranes the analysis is based on a generalized helfrich energy that accounts for geometry through the stretch and curvature the composition and the interaction between geometry and composition the use of nonclassical differential operators and related integral theorems in conjunction with appropriate composition and mass conserving variations simplify the derivations we show that instabilities of multicomponent membranes are significantly different from those in single component membranes as well as those in systems undergoing spinodal decomposition in flat spaces this is due to the intricate coupling between composition and shape as well as the nonuniform tension in the membrane specifically critical modes have high frequencies unlike single component vesicles and stability depends on system size unlike in systems undergoing spinodal decomposition in flat space an important implication is that small perturbations may nucleate localized but very large deformations we show that the predictions of the analysis are in qualitative agreement with experimental observations

an adaptive finite element method for the laplace beltrami operator on implicitly defined surfaces we present an adaptive finite element method for approximating solutions to the laplace beltrami equation on surfaces in mathbb r 3 which may be implicitly represented as level sets of smooth functions residual type a posteriori error bounds which show that the error may be split into a residual part and a geometric part are established in addition implementation issues are discussed and several computational examples are given

lipid membranes with an edge consider an open lipid membrane with a free exposed edge the energy describing this membrane is quadratic in the extrinsic curvature that describing the edge is proportional to its length we determine the boundary conditions satisfied by the equilibria of the membrane on this edge the derivation is free of any assumptions on the symmetry of the membrane geometry with respect to the axially symmetric case there is an additional boundary condition that is identically satisfied in that limit by considering the balance of the forces operating at the edge a physical interpretation for the boundary conditions is provided the effect of the addition of a gaussian rigidity term for the membrane is also considered

anisotropic centroidal voronoi tessellations and their applications in this paper we introduce a novel definition of the anisotropic centroidal voronoi tessellation acvt corresponding to a given riemann metric tensor a directional distance function is used in the definition to simplify the computation we provide algorithms to approximate the acvt using the lloyd iteration and the construction of anisotropic delaunay triangulation under the given riemannian metric the acvt is applied to the optimization of two dimensional anisotropic delaunay triangulation to the generation of surface cvt and high quality triangular mesh on general surfaces various numerical examples demonstrating the effectiveness of the proposed method are presented

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

analysis of an iterative scheme of fractional steps type associated to the reaction diffusion equation endowed with a general nonlinearity and cauchy neumann boundary conditions abstract the paper studies the existence uniqueness regularity and the approximation of solutions to the reaction diffusion equation endowed with a general nonlinear regular potential and cauchy neumann boundary conditions the convergence and error estimate results for an iterative scheme of fractional steps type associated to the nonlinear parabolic equation are also established the advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

an efficient time adaptivity based on chemical potential for surface cahn hilliard equation using finite element approximation abstract we present numerical simulations for the surface cahn hilliard equation which describes phase separation phenomenon occurred on general surfaces the spatial discretization is based on surface finite element method while the temporal discretization methods are first and second order stabilized semi implicit schemes which guarantee the free energy decay an efficient and parameter free adaptive time stepping strategy is proposed based on the numerical energy stability generated by stabilized semi implicit scheme the main idea is to use discrete chemical potential the byproduct of stabilized semi implicit scheme to estimate the variation of numerical energy that is used as a indicator to update the time step the operation avoid calculating numerical energy with gauss integral in each time step and reduce the calculated cost in addition optimal error estimate of first order stabilized semi implicit scheme in the case of curved surface are provided finally numerical experiments are presented to demonstrate the stability accuracy and efficiency of the proposed algorithms

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

an eulerian space time finite element method for diffusion problems on evolving surfaces in this paper we study numerical methods for the solution of partial differential equations on evolving surfaces the evolving hypersurface in bbb r d defines a d dimensional space time manifold in the space time continuum bbb r d 1 we derive and analyze a variational formulation for a class of diffusion problems on the space time manifold for this variational formulation new well posedness and stability results are derived the analysis is based on an inf sup condition and involves some natural but nonstandard anisotropic function spaces based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous galerkin dg space time finite element discretization this dg space time method is explained and results of numerical experiments are presented that illustrate its properties

higher order meshing of implicit geometries part ii approximations on manifolds abstract a new concept for the higher order accurate approximation of partial differential equations on manifolds is proposed where a surface mesh composed by higher order elements is automatically generated based on level set data thereby it enables a completely automatic workflow from the geometric description to the numerical analysis without any user intervention a master level set function defines the shape of the manifold through its zero isosurface which is then restricted to a finite domain by additional level set functions it is ensured that the surface elements are sufficiently continuous and shape regular which is achieved by manipulating the background mesh the numerical results show that optimal convergence rates are obtained with a moderate increase in the condition number compared to handcrafted surface meshes

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

solution to boundary value problems using the method of maximum entropy the method of maximum entropy is used to solve a class of linear boundary value problems the method is based on using various moments of the differential equation as constraints when maximizing the entropy various examples are presented and compared to exact solutions for varying numbers of moments it is found that the maximum entropy approximation is in many cases better than a fourier series solution for a given number of expansion terms and moments the method is very general and will find applications in many areas of physics a comparison of the amount of work necessary for the maximum entropy solution versus finite difference techniques is presented and it is found that the maximum entropy technique shows promise as an alternative solution technique

restricted isometry of fourier matrices and list decodability of random linear codes we prove that a random linear code over mathbb f q with probability arbitrarily close to 1 is list decodable at radius 1 1 q epsilon with list size l o 1 epsilon 2 and rate r omega q epsilon 2 log 3 1 epsilon up to the polylogarithmic factor in 1 epsilon and constant factors depending on q this matches the lower bound l omega q 1 epsilon 2 for the list size and upper bound r o q epsilon 2 for the rate previously only existence and not abundance of such codes was known for the special case q 2 guruswami et al 2002 in order to obtain our result we employ a relaxed version of the well known johnson bound on list decoding that translates the average hamming distance between codewords to list decoding guarantees we furthermore prove that the desired average distance guarantees hold for a code provided that a natural complex matrix encoding the codewords satisfies the restricted isometry property with respect to the euclidean norm for the case of random binary

an error bound for the finite element approximation of the cahn hilliard equation with logarithmic free energy an error bound is proved for a fully practical piecewise linear finite element approximation using a backward euler time discretization of the cahn hilliard equation with a logarithmic free energy

coarsening kinetics from a variable mobility cahn hilliard equation application of a semi implicit fourier spectral method an efficient semi implicit fourier spectral method is implemented to solve the cahn hilliard equation with a variable mobility the method is orders of magnitude more efficient than the conventional forward euler finite difference method thus allowing us to simulate large systems for longer times we studied the coarsening kinetics of interconnected two phase mixtures using a cahn hilliard equation with its mobility depending on local compositions in particular we compared the kinetics of bulk diffusion dominated and interface diffusion dominated coarsening in two phase systems results are compared with existing theories and previous computer simulations

a geometric theory on the elasticity of bio membranes the purpose of this paper is to study the shapes and stabilities of bio membranes within the framework of exterior differential forms after a brief review of the current status of theoretical and experimental studies on the shapes of bio membranes a geometric scheme is proposed to discuss the shape equation of closed lipid bilayers the shape equation and boundary conditions of open lipid bilayers and two component membranes the shape equation and in plane strain equations of cell membranes with cross linking structures and the stabilities of closed lipid bilayers and cell membranes the key point of this scheme is to deal with the variational problems on surfaces embedded in three dimensional euclidean space by using exterior differential forms

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

the lumped mass finite element method for surface parabolic problems error estimates and maximum principle abstract the lumped mass method is extended to the surface finite element method for solving the surface parabolic equations the main purpose of the proposed method is to overcome the difficulty that the surface finite element method does not guarantee the maximal principle of the surface heat equation optimal error estimates are given for the semi discrete and fully discrete schemes of the proposed method respectively the maximum principle is shown for surface heat equations and its preservation by the lumped mass surface finite element under the delaunay type triangulation moreover some results of positivity and monotonicity are derived for nonlinear parabolic equations finally some numerical experiments are displayed to illustrate the validity and numerical performance of the proposed method

evolving surface finite element method for the cahn hilliard equation we use the evolving surface finite element method to solve a cahn hilliard equation on an evolving surface with prescribed velocity we start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting the finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity we go on to show a rigorous well posedness result for the continuous equations by showing convergence along a subsequence of the finite element scheme we conclude the paper by deriving error estimates and present various numerical examples

a local riemann hypothesis i daniel bump1 kwok kwong choi2 par kurlberg3 jeffrey vaaler4 1 department of mathematics stanford university stanford ca 94305 2125 usa e mail bump math stanford edu 2 department of mathematics the university of hong kong pokfulam road hong kong hong kong e mail choi maths hku hk 3 department of mathematics university of georgia athens ga 30602 usa e mail kurlberg math uga edu 4 university of texas at austin department of mathematics rlm 8 100 austin tx 78712 1082 usa e mail vaaler math utexas edu

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

lumped finite elements for reaction cross diffusion systems on stationary surfaces all the authors am is cv mf thank the isaac newton institute for mathematical sciences for its hospitality during the programme coupling geometric pdes with physics for cell morphology motility and pattern formation epsrc ep k032208 1 this work am has received funding from the european union s horizon 2020 research and innovation programme under the marie sklodowska curie grant agreement no 642866 am and cv acknowledge support from the engineering and physical sciences research council ep j016780 1 on modelling analysis and simulation of spatial patterning on evolving biological surfaces and the leverhulme trust research project grant rpg 2014 149 on unravelling new mathematics for 3d cell migration am was partially supported by a fellowship from the simons foundation am is a royal society wolfson research merit award holder generously funded by the wolfson foundation

matrix infinitely divisible series tail inequalities and their applications in this paper we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible i d series which is a finite sum of fixed matrices weighted by i d random variables we obtain several types of tail inequalities including bennett type and bernstein type inequalities this allows us to further bound the expectation of the spectral norm of a matrix i d series moreover by developing a new lower bound function for q s s 1 log s 1 s that appears in the bennett type inequality we derive a tighter tail inequality of the largest eigenvalue of the matrix i d series than the bernstein type inequality when the matrix dimension is high the resulting lower bound function is of independent interest and can improve any bennett type concentration inequality that involves the function q s the class of i d probability distributions is large and includes gaussian and poisson distributions among many others therefore our results encompass the existing work on matrix gaussian series as a special case lastly we show that the tail inequalities of a matrix i d series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints in addition we also use the resulting tail bounds to show that random matrices constructed from i d random variables satisfy the restricted isometry property rip when it acts as a measurement matrix in compressed sensing

phase field model for three dimensional dendritic growth with fluid flow we study the effect of fluid flow on three dimensional 3d dendrite growth using a phase field model on an adaptive finite element grid in order to simulate 3d fluid flow we use an averaging method for the flow problem coupled to the phase field method and the semi implicit approximated projection method siapm we describe a parallel implementation for the algorithm using the charm fem framework and demonstrate its efficiency we introduce an improved method for extracting dendrite tip position and tip radius facilitating accurate comparison to theory we benchmark our results for 2d dendrite growth with solvability theory and previous results finding them to be in good agreement the physics of dendritic growth with fluid flow in three dimensions is very different from that in two dimensions and we discuss the origin of this behavior

phase separation patterns for diblock copolymers on spherical surfaces a finite volume method we explore phase separation on spherical surfaces by solving the cahn hilliard equation modified for diblock copolymers using a finite volume method the spherical surface is discretized into almost uniform triangles by employing successive dyadic refinements of the spherical icosahedron a methodology that avoids potential mathematical and numerical problems related to the poles in spherical coordinates the finite volume method is based on averaging voronoi cells built from triangular meshes to calculate the laplace beltrami operator on the curved surface which greatly improves both the accuracy and speed of calculation as compared to the conventional finite difference method by using this method we simulate the phase separation of diblock copolymers on a spherical surface it is found that stable and intrinsic defects which would not occur in a flat space after sufficient annealing appear in the periodic arrangement of the domains on the curved surface due to the distinct euler characteristic of the surface

restricted isometry constants where ell p sparse recovery can fail for 0 p leq 1 this paper investigates conditions under which the solution of an underdetermined linear system with minimal lscrp norm 0 p les 1 is guaranteed to be also the sparsest one matrices are constructed with restricted isometry constants ric delta2m arbitrarily close to 1 radic2 ap 0 707 where sparse recovery with p 1 fails for at least one m sparse vector as well as matrices with delta2m arbitrarily close to one where lscr1 minimization succeeds for any m sparse vector this highlights the pessimism of sparse recovery prediction based on the ric and indicates that there is limited room for improving over the best known positive results of foucart and lai which guarantee that lscr1 minimization recovers all m sparse vectors for any matrix with delta2m 2 3 radic2 7 ap 0 4531 these constructions are a by product of tight conditions for lscrp recovery 0 les p les 1 with matrices of unit spectral norm which are expressed in terms of the minimal singular values of 2m column submatrices compared to lscr1 minimization lscrp minimization recovery failure is shown to be only slightly delayed in terms of the ric values furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used it is shown that when lscrp optimization is attempted using an iterative reweighted lscr1 scheme failure can still occur for delta2m arbitrarily close to 1 radic2

on the use of local maximum entropy approximants for cahn hilliard phase field models in 2d domains and on surfaces abstract we apply a local maximum entropy lme approximation scheme to the fourth order phase field model for the traditional cahn hilliard theory the discretization of the cahn hilliard equation by galerkin method requires at least c 1 continuous basis functions this requirement can be fulfilled by using lme shape functions which are c continuous in this case the primal variational formulations of the fourth order partial differential equation are well defined and integrable hence there is no need to split the fourth order partial differential equation into two second order partial differential equations this splitting scheme is a common practice in mixed finite element formulations with c 0 continuous lagrange shape functions furthermore we use a general and simple numerical method such as statistical manifold learning technique that allows dealing with general point set surfaces avoiding a global parameterization which can be applied to tackle surfaces of complex geometry and topology and to solve cahn hilliard equation on general surfaces finally the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples

statistical treatment of autonomous systems with divergencelless flows abstract statistical mechanics sm has been quite successful in providing one with exact or at least approximate descriptions of the time dependent solutions of the liouville or of the von neumann equation within the context of hamiltonian dynamics hd here we investigate how to extend some of its ideas to more general dynamical systems we find that for systems that retain from hd just the divergenceless character of the phase space flow many ideas borrowed from the maximum entropy principle approach to sm apply in such a generalized context as well

a surface phase field model for two phase biological membranes we study vesicles formed by lipid bilayers that are governed by an elastic bending energy and on which the lipids laterally separate forming two different phases the energy laden phase interfaces may be modeled as curves on the hypersurface representing the membrane sharp interface model the phase field methodology is another powerful tool to model such phase separation phenomena where thin layers describe the interfaces diffuse interface model for both approaches we characterize equilibrium shapes in terms of the euler lagrange equations of the total membrane energy subject to constraints on the area of the two phases and the volume we further show by matching appropriate formal asymptotic expansions that the sharp interface model is obtained from the diffuse interface model as the thickness of the phase interface tends to zero the essential challenge lies in the fact that also the geometry of the membrane is unknown and depends on a small parameter representing the interface thickness

analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits we propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science the primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method in particular by focusing on the dependence of the error bounds on the parameter e known as the measure of the interface thickness optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size h and the time step size k in particular it is shown that all error bounds depend on 1 a only in some lower polynomial order for small e the cruxes of the analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of chen and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect finally as a nontrivial byproduct the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients the sharp interface limits include the classical stefan problem the generalized stefan problems with surface tension and surface kinetics the motion by mean curvature flow and the hele shaw model

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

a meshfree generalized finite difference method for surface pdes abstract in this paper we propose a novel meshfree generalized finite difference method gfdm approach to discretize pdes defined on manifolds derivative approximations for the same are done directly on the tangent space in a manner that mimics the procedure followed in volume based meshfree gfdms as a result the proposed method not only does not require a mesh it also does not require an explicit reconstruction of the manifold in contrast to some existing methods it avoids the complexities of dealing with a manifold metric while also avoiding the need to solve a pde in the embedding space a major advantage of this method is that all developments in usual volume based numerical methods can be directly ported over to surfaces using this framework we propose discretizations of the surface gradient operator the surface laplacian and surface diffusion operators possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced and a few practical applications are presented

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments