an integral equation method for epitaxial step flow growth simulations in this paper we describe an integral equation approach for simulating diffusion problems with moving interfaces the solutions are represented as moving layer potentials where the unknowns are only defined on the interfaces the resulting integro differential equation ide system is solved using spectral deferred correction sdc techniques developed for general differential algebraic equations daes and the time dependent potentials are evaluated efficiently using fast convolution algorithms the numerical solver is applied to the bcf model for the epitaxial step flow growth of crystals for which the solutions are calculated accurately instead of using quasi static approximations numerical results in 1 1 dimensions are compared with available results in the literature

observability conditions for multiple signal direction finding and array sensor localization three observability issues which arise in narrowband signal direction finding in the presence of array sensor location errors are described necessary and sufficient conditions for the phase delay ambiguity and the rank 2 ambiguity are presented however the characterization of the rank n ambiguity is still an open problem and only a conjecture is given barring these ambiguities the minimal numbers of known and unknown signals required for uniquely localizing the sources and sensors are determined under the assumption of no measurement noises

detection estimation of more uncorrelated gaussian sources than sensors in nonuniform linear antenna arrays part iii detection estimation nonidentifiability for pt ii see ibid vol 51 no 6 p 1492 507 2003 we investigate nonidentifiability conditions for the detection estimation problem with multiple uncorrelated plane waves incident upon a nonuniform sparse linear antenna array specifically we define conditions under which a given hermitian covariance matrix has a nonunique multiple decomposition into an admissible number of dyads weighted by the source powers and a white noise identity matrix our method is based on the proukakis manikas technique of generating ambiguity generator sets agss and allows ambiguous sets of sources associated with a given sparse antenna geometry to be determined

high order accurate methods for the evaluation of layer heat potentials we discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries one of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel which is both weakly singular in time and rapidly decaying in space we show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy which we refer to as geometrically induced stiffness but that rules based on product integration of the full heat kernel in time are robust when combined with previously developed fast algorithms for the evolution of the history part of layer potentials diffusion processes in complex moving geometries can be computed accurately and in nearly optimal time

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

investigative study of planar array ambiguities based on hyperhelical parameterization in this paper a novel approach is proposed for the calculation of ambiguous generator sets of directions for planar arrays this approach is based on the concept of hyperhelical curves and their equivalent linear arrays which provide a vehicle for investigating planar array ambiguities by adopting and extending techniques proposed for linear arrays thus ambiguous generator sets of directions of constant azimuth and different elevation are initially calculated and the concept of ambiguous generator lines is introduced then by using cone angle parametrization ambiguities associated with directions of different azimuth and different elevation are investigated

manifold studies of nonlinear antenna array geometries the manifold of a general two or three dimensional 2 d or 3 d array of sensors is studied using differential geometry by considering the azimuth and elevation angles as the parameters of interest a manifold surface is formed embedded in a multidimensional complex space initially this surface is investigated by establishing a number of differential geometry parameters associated with the array manifold then the concept of development is proposed for mapping manifold surfaces embedded in a multidimensional complex space on the real 2 d parameter plane the proposed mapping preserves certain differential geometry characteristics of the manifold surface and provides a simplified representation for the analysis the potential benefits of this mapping as well as of the proposed parameters are demonstrated in the analysis of 3 d and planar arrays of omnidirectional sensors and in a number of potential applications varying from array design to handling the array ambiguity problem

a fast algorithm for the evaluation of heat potentials a fast algorithm for the evaluation of heat potentials to appear in communications and pure and applied mathematics l greengard j strain may 1989 abstract numerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials when m points are used in the discretization of the boundary and n time steps are computed an amount of work of the order o n 2 m 2 has traditionally been required in this paper we present an algorithm which requires an amount of work of the order o n m and we observe speedups of five orders of magnitude for large scale problems thus the method makes it possible to solve the heat equation by potential theory in practical situations keywords heat equation potential theory fast algorithms integral equations introduction a classical approach to the solution of the heat equation u t u q t in a space time domain t t 0 t see fig 1 is through the use of heat potentials 3 7 given zero initial conditions one seeks a representation of u as a single layer heat potential z t z s x t k x x 0 t t 0 x 0 t 0 dx 0 dt 0 t 0 or a double layer heat potential d x t z t z t 0 k x x 0 t t 0 x 0 t 0 dx 0 dt 0 n 0 where k is a fundamental solution of the heat equation in some region containing q n 0 denotes the unit t outward normal to t t at x and is a surface density defined on t t 0 t yale university and courant institute of mathematical sciences new york university the work of this author was supported in part by the office of naval research under grant n00014 89 j 1527 in part by ibm under grant p00038437 and in part by a nsf mathematical sciences postdoctoral fellowship courant institute of mathematical sciences new york university the work of this author was supported by darpa afosr contract no f 49620 87 c 0065

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

efficient thermal field computation in phase field models we solve the phase field equations in two dimensions to simulate crystal growth in the low undercooling regime the novelty is the use of a fast solver for the free space heat equation to compute the thermal field this solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient free space heat equation with a smooth source term the computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed this allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field which can result in significant computational savings in the low undercooling regime

a lagrangian particle level set method we present a novel particle level set method for capturing interfaces the level set equation is solved in a lagrangian frame using particles that carry the level set information a key aspect of the method involves a consistent remeshing procedure for the regularization of the particle locations when the particle map gets distorted by the advection field the lagrangian description of the level set method is inherently adaptive and exact in the case of solid body motions the efficiency and accuracy of the method is demonstrated in several benchmark problems in two and three dimensions involving pure advection and curvature induced motion of the interface the simplicity of the particle description is shown to be well suited for real time simulations of surfaces involving cutting and reconnection as in virtual surgery environments

sparse array design for azimuthal direction of arrival estimation we propose a novel approach to the design of array geometries for azimuthal direction of arrival doa estimation the proposed array design is related to that of minimum redundancy arrays but the array sensors are not required to lie on a uniform grid based on the proposed array geometry design we develop a subspace based doa estimation technique which allows to estimate the doas of more sources than sensors using only second order statistics of the received data this doa estimation technique is related to the covariance augmentation technique but in contrast to the latter technique it provides nonambiguous doa estimates for the full 360 azimuth field of view

rank one subspaces of bimodules over maximal abelian selfadjoint algebras spaces of operators that are left and right modules over maximal abelian selfadjoint algebras masa bimodules for short are natural generalizations of algebras with commutative subspace lattices this paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules it is shown that every finite rank operator of a norm closed masa bimodule m is in the trace norm closure of the rank one subspace of m an important consequence is that the rank one subspace of a strongly reflexive masa bimodule that is one which is the reflexive hull of its rank one operators is dense in the module in the weak operator topology however in contrast to the situation for algebras it is shown that such density need not hold in the ultraweak topology a new method of representing masa bimodules is introduced this uses a novel concept of an topology with the appropriate notion of support a correspondence is established between reflexive masa bimodules and their supports it is shown that if a c2 closed masa bimodule contains a trace class operator then it must contain rank one operators indeed every such operator is in the c2 norm closure of the rank one subspace of the module consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive however the trace norm closure of the rank one subspace need not contain all trace class operators of the module also it is shown that there exists a csl algebra which contains no trace class operators yet contains an operator belonging to cpfor allp 1 from this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in cpfor 1 p as an application it is shown that there exists a commutative subspace lattice l such that l is non synthetic but every weakly closed algebra which contains a masa and has invariant lattice l coincides with alg l

a study of the rank ambiguity issues in direction of arrival estimation we first extend a theorem on linear independence of steering vectors proposed by godara and cantoni to include more array sensor scenarios we then show that an array can have pairwise linearly independent steering vectors even when all its intersensor spacings are more than spl lambda 2 where spl lambda is the wavelength of the signals we next propose a theorem for characterizing rank 2 ambiguity which is applicable to direction of arrival estimation applications where the array sensor locations are fixed and known subsequently we identify a class of three sensor arrays and a class of uniform circular arrays that have pairwise linearly independent steering vectors and are free of rank 2 ambiguity we also show that collinearity of sensors uniformity in intersensor spacings the dimensions of intersensor spacings or a combination of some or all of these may give rise to linearly dependent steering vectors in particular we demonstrate that for a m sensor array m linearly dependent steering vectors exist if the aperture is greater than m 1 2 spl lambda 2 or when at least m 1 2 1 sensors are collinear

numerical approximation of a thermally driven interface using finite elements a two dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh the model tracks the position of the sharp solid liquid interface using a set of marker points placed on the interface the simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it at the interface the heat flux is discontinuous due to the release of latent heat during the solidification melting process two ways to calculate the temperature gradients at the interface evaluating their interpolants at gauss points were proposed using known one and two dimensional solutions to stable solidification problems the stefan problem it was shown that the method converges with second order accuracy when applied to the unstable solidification of a crystal into an undercooled liquid it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface i e different approximations and different meshes can yield different solutions the cause of these difficulties is examined the effect of different types of interpolation on the simulations is investigated and the necessary criteria to ensure converged solutions are established copyright 2003 john wiley sons ltd

on unique localization of multiple sources by passive sensor arrays necessary and sufficient conditions are derived for the unique localization of narrowband sources having the same known center frequency by passive sensor arrays the conditions specify the maximum number of sources that can be uniquely localized by a general array that satisfies some mild geometric constraints the conditions are expressed in terms of the number of sensors and the rank of the correlation matrix of the sources two different conditions are presented the first guarantees uniqueness for every batch of sampled data the second which is weaker guarantees uniqueness for almost every batch of sampled data with the exception of a set of batches of measure zero it is shown that a condition that is slightly weaker than the second one is also necessary

operator holder zygmund functions abstract it is well known that a lipschitz function on the real line does not have to be operator lipschitz we show that the situation changes dramatically if we pass to holder classes namely we prove that if f belongs to the holder class r with 0 1 then f a f b const a b for arbitrary self adjoint operators a and b we prove a similar result for functions f in the zygmund class 1 r for arbitrary self adjoint operators a and k we have f a k 2 f a f a k const k we also obtain analogs of this result for all holder zygmund classes r 0 then we find a sharp estimate for f a f b for functions f of class def f f const for an arbitrary modulus of continuity in particular we study moduli of continuity for which f a f b const a b for self adjoint a and b and for an arbitrary function f in we obtain similar estimates for commutators f a q q f a and quasicommutators f a q q f b finally we estimate the norms of finite differences j 0 m 1 m j m j f a j k for f in the class m that is defined in terms of finite differences and a modulus continuity of order m we also obtain similar results for unitary operators and for contractions

shape instabilities and pattern formation in solidification a new method for numerical solution of the moving boundary problem abstract when a solidification front is advancing into a region of supercooled liquid its shape is subject to instabilities which can lead to complex modes of growth the evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension and its full development is a highly nonlinear process a new method for solving the two phase heat diffusion problem under these conditions is described the method is based on a single domain treatment weak formulation of the moving boundary problem with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite element technique calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation

a fast semi lagrangian contouring method for moving interfaces abstract general moving interface problems are solved by a new approach evaluating an explicit semi lagrangian advection formula with efficient geometric algorithms and extracting the moving interface with a fast new contouring technique the new approach decouples spatial and temporal resolutions and grid free adaptive refinement of the interface increases accuracy dramatically a modular implementation with a fast new intrinsic geometry module computes highly accurate solutions to geometric moving interface problems involving merging anisotropy and faceting with a high order embedded geometry module it solves second order problems involving curvature dynamic topology and nonlocal interactions

multiscale finite difference diffusion monte carlo method for simulating dendritic solidification abstract we present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field the first key feature of this method is an efficient multiscale diffusion monte carlo dmc algorithm which allows off lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid liquid interface as a result the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution the second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface inside this layer and in the solid the diffusion equation is solved using a standard finite difference algorithm that is interfaced with the dmc algorithm using the local conservation law for the diffusing quantity here we combine this algorithm with a previously developed phase field formulation of the interface dynamics and demonstrate that it can accurately simulate three dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

shortening three dimensional curves via two dimensional flows in this paper a curve evolution approach for the computation of geodesic curves on 3d surfaces is presented the algorithm is based on deforming via the curve shortening flow an arbitrary initial curve ending at two given surface points the 3d curve shortening flow is first transformed into an equivalent 2d one this 2d flow is implemented using an efficient numerical algorithm for curve evolution with fixed end points

efficient symmetric discretization for the poisson heat and stefan type problems with robin boundary conditions we present a novel and efficient method for solving the poisson equation the heat equation and stefan type problems with robin boundary conditions over potentially moving arbitrarily shaped domains the method utilizes a level set framework thus it has all of the benefits of a sharp implicitly represented interface such as the ease of handling complex topological changes this method is straightforward to implement and leads to a linear system that is symmetric and positive definite which can be inverted efficiently with standard iterative methods this approach is second order accurate for both the poisson and heat equations and first order accurate for the stefan problem we demonstrate the accuracy in the l 1 and l norms

boundary integral formulation of the two dimensional symmetric model of dendritic growth abstract selection of steady needle crystals in the symmetric model of dendritic growth is considered the diffusion equation and associated kinematic and thermodynamic boundary condition are recast into a nonlinear equation through the use of boundary integral methods the equation is solved numerically and for the range of peclet numbers considered it is found that a smooth solution exists only if anisotropy is included in the capillarity term of the gibbs thompson condition the behavior of the selected velocity and tip radius as a function of undercooling is also examined a linear integral equation is also derived to examine the stability of the steady state

koplienko spectral shift function on the unit circle the note shows that for every pair of contractions a trace formula of koplienko holds with a summable spectral shift function see theorem 1 below this answers a question by f gesztesy a pushnitski b simon see zh mat fiz anal geom 4 1 63 107 202 2008 open question 11 2

absolute continuity of spectral shift abstract in this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions dissipative operators unitary operators and self adjoint operators to establish the absolute continuity of spectral shift we use the sz nagy theorem on the absolute continuity of the spectral measure of the minimal unitary dilation of a completely nonunitary contraction we also give a construction of an intermediate contraction for a pair of contractions with trace class difference

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

fast potential theory ii layer potentials and discrete sums we present three new families of fast algorithms for classical potential theory based on ewald summation and fast transforms of gaussians and fourier series ewald summation separates the green function for a cube into a high frequency localized part and a rapidly converging fourier series each part can then be evaluated efficiently with appropriate fast transform algorithms our algorithms are naturally suited to the use of green functions with boundary conditions imposed on the boundary of a cube rather than free space green functions our first algorithm evaluates classical layer potentials on the boundary of a d dimensional domain with d equal to two or three the quadrature error is o hm e where h is the mesh size on the boundary and m is the order of quadrature used the algorithm evaluates the discretized potential using n elements at o n points in o n log n arithmetic operations the constant in o n log n depends logarithmically on the desired error tolerance our second scheme evaluates a layer potential on the domain itself with the same accuracy it produces md values using n boundary elements in o n md log m arithmetic operations our third method evaluates a discrete sum of values of the green function of the type which occur in particle methods it attains error at a cost o na log n where a 2 1 dd and d is the hausdorff dimension of the set where the sources concentrate in the limit n thus it is o n log n when the sources do not cluster too much and close to o n log n in the important practical case when the points are uniformly distributed over a hypersurface we also sketch an o n log n algorithm based on special functions two dimensional numerical results are presented for all three algorithms layer potentials are evaluated to second order accuracy in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation discrete sum calculations are speeded up astronomically our algorithm reduces the cpu time required for a calculation with 40 000 points from six months to one hour

front tracking finite element method for dendritic solidification 765 abstract a two dimensional finite element method capable of tracking sharp interfaces is developed the method is based on a fixed mesh of bilinear isoparametric elements and is extremely simple and easy to use the interfaces are tracked with a set of marker points that define their position at all times several different approaches to finding the interface position and velocity are discussed and their effect on the convergence rate is examined it is shown through numerical examples that the approximation to the interface position converges with second order accuracy in two dimensional problems as opposed to previously developed finite difference algorithms that are only first order accurate the approximation to the interface velocity is shown to be of first order a number of examples are examined including several that provide a careful comparison with previously published results

adaptive mesh computation of polycrystalline pattern formation using a renormalization group reduction of the phase field crystal model we implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes our numerical approach uses the systematic renormalization group formulation of a phase field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field a quantity that is spatially uniform except near topological defects grain boundaries and other lattice imperfections our algorithm employs a hybrid formulation of the amplitude equations combining cartesian and polar decompositions of the complex amplitude we show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions

on the two phase stefan problem with interfacial energy and entropy abstract the classical stefan theory for the melting of a solid or the freezing of a liquid is too simplistic to describe phenomena such as supercooling in which a liquid supports temperatures below its freezing point or superheating the analog for solids or dendritic growth in which simple shapes evolve to complicated tree like structures this paper develops a general theory for two phase phenomena of this type it develops partial differential equations satisfied in the phase regions and free boundary conditions satisfied on the interface between phases and gives arguments which indicate that the resulting boundary value problems predict the formation of dendrites keywords equilibrium liapunoo functions mathematical physics

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

numerical simulation of dendritic solidification with convection two dimensional geometry a front tracking method is presented for simulations of dendritic growth of pure substances in the presence of flow the liquid solid interface is explicitly tracked and the latent heat released during solidification is calculated using the normal temperature gradient near the interface a projection method is used to solve the navier stokes equations the no slip condition on the interface is enforced by setting the velocities in the solid phase to zero the method is validated through a comparison with an exact solution for a stefan problem a grid refinement test and a comparison with a solution obtained by a boundary integral method three sets of two dimensional simulations are presented a comparison with the simulations of beckermann et al j comput phys 154 468 1999 a study of the effect of different flow velocities and a study of the effect of the prandtl number on the growth of a group of dendrites growing together the simulations show that on the upstream side the dendrite tip velocity is increased due to the increase in the temperature gradient and the formation of side branches is promoted the flow has the opposite effect on the downstream side the results are in good qualitative agreement with published experimental results even though only the two dimensional aspects are examined here

modeling and estimation of ambiguities in linear arrays the problem of ambiguities inherent in the manifold of any linear array structure is investigated ambiguities which can be classified into trivial and nontrivial depending on the ease of their identification arise when an array cannot distinguish between two different sets of directional sources initially the new concept of an ambiguous generator set is introduced it represents generates an infinite number of ambiguous sets of directions then by uniformly nonuniformly partitioning the array manifold curve of a linear array different ambiguous generator sets ran be calculated and as a direct result a sufficient condition for the presence of ambiguities is obtained the theoretical aspects of the investigation are followed by the proposal of an innovative approach that calculates not only all such ambiguities existing in a linear array of arbitrary geometry but the rank of ambiguity in each case as well the main results presented in the paper are supported by a number of representative examples

on fully operator lipschitz functions abstract let a d be the disc algebra of all continuous complex valued functions on the unit disc d holomorphic in its interior functions from a d act on the set of all contraction operators a 1 on hilbert spaces it is proved that the following classes of functions from a d coincide 1 the class of operator lipschitz functions on the unit circle t 2 the class of operator lipschitz functions on d and 3 the class of operator lipschitz functions on all contraction operators a similar result is obtained for the class of operator c 2 lipschitz functions from a d

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

linear stability of planar solidification fronts abstract we review extend and modify the classical linear stability theory of planar solidification fronts in langer s moving symmetric model using a new integral equation for the front position we compute an exact linear stability equation and solve it exactly for an important special case the classical theory is seen to be the long time limit of the exact solution finally we show how to treat a general planar front using a short time approximation our conclusions generalize those of previous analyses by treating initial temperature field perturbations and the resulting transient growth phenomena

a detailed derivation of arrays free of higher rank ambiguities since linear dependence of steering vectors would lead to ambiguous direction of arrival doa estimates it is crucial to employ array with linearly independent steering vectors in doa estimation applications in fact it is well known that in order to determine uniquely the doas of spl nu uncorrelated signals one requires an array that is free of up to rank spl nu ambiguities i e an array whose every spl nu 1 steering vectors with distinct doas are linearly independent however the question as to the sensor arrangement that gives rise to an array that is free of up to rank spl nu ambiguities where spl nu spl isin 3 remains unanswered we construct a class of cross arrays that are free of up to rank spl nu ambiguities where spl nu spl isin 1 through this study we derive a theorem for characterizing inherent ambiguities associated with cross arrays we also provide counterexamples to the conjecture for characterizing higher rank ambiguity proposed by lo and marple see ibid vol 40 p 2641 50 1992 and that proposed by wang et al see icassp 9 vol iv p 509 12 1994

velocity effects in unstable solidification the supercooled stefan problem with surface tension is considered as a model of unstable solidification with a general anisotropic curvature and velocity dependent boundary condition on the moving interface the problem is reformulated in terms of a history dependent singular integral equation for the velocity of the boundary using this equation a new linear stability analysis of a flat interface is carried out and the smoothing role of velocity dependence and the destabilizing effect of anisotropy are demonstrated the results disagree with previous analyses because transient effects due to the initial temperature field are included and numerical results are presented that confirm the analysis presented here it is found that velocity effects cannot increase the range of unstable modes beyond that permitted by surface tension

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

level sets of viscosity solutions some applications to fronts and rendez vous problems the authors treat some applications of hamilton jacobi equations to the study of a flame front propagation model and the rendez vows problem the solution of both problems requires the determination of the level sets of the viscosity solution for the corresponding equation in the flame front propagation model described here it is assumed that the evolution is driven by a vector field satisfying a transversality condition at time t 0 the evolution in the normal direction with variable velocity c x geq 0 is considered as a special case this approach is constructive permitting the numerical solution of such problems

an efficient boundary integral method for the mullins sekerka problem we use a boundary integral technique to study the two space dimensional mullins sekerka free boundary problem which originates from a study of solidification and liquidation of materials of negligible specific heat this is an area preserving and curve shortening motion evolution equations for the free boundaries are derived in terms of the tangent angle and total arclength which makes a small scale decomposition possible and the fourier transform a powerful tool in numerical calculations with this formulation implicit schemes can be implemented to avoid the difficult numerical stiffness associated with explicit schemes we can compute solutions up to the time when there is a topological change i e when particles touch or break up our numerical results for systems of a single particle or multi particles provide some valuable information in the particle dynamics such as the circularization of each individual particle and the mass transfer between different particles during particle interactions

the chebyshev fast gauss and nonuniform fast fourier transforms and their application to the evaluation of distributed heat potentials we present a method for the fast and accurate computation of distributed heat potentials in two dimensions the distributed source is assumed to be given in terms of piecewise space time chebyshev polynomials we discretize uniformly in time whereas in space the polynomials are defined on the leaf nodes of a quadtree the quadtree can vary at each time step we combine a product integration rule with fast algorithms fast heat potentials nonuniform fft fast gauss transform to obtain a high order accurate method with optimal complexity if n is the number of time steps m is the maximum number of leaf nodes over all the time steps and the input contains a qth order polynomial representation of f then our method requires o q 3mnlogm work to evaluate the heat potential at arbitrary mn space time target locations the overall convergence rate of the method is of order q we present numerical experiments for q 4 8 and 16 and we verify the theoretical convergence rate of the method when the solution is sufficiently smooth the 16th order variant results in significant computational savings even in the case in which we require only a few digits of accuracy

an efficient direct bem numerical scheme for phase change problems using fourier series abstract due to the moving boundary nature of the solidification and melting problems the application of the standard time domain boundary element formulation of the problem presents the difficulty of the numerical evaluation of the history dependence representing more and more work as time proceeds in that regard we present an algorithm based on a green s function which allows a fourier representation of the domain integral in the reinitialization scheme of the direct boundary element method formulation of the problem from which it is possible to carry out the history integration in a recursive manner to obtain the reinitialization approach in the integral representation formula the green s function corresponding to zero temperature in a box containing the original domain will be used instead of using the classical free space fundamental solution this green s function is given in terms of the original fundamental solution plus a regular solution of the heat equation inside the domain in consideration it can therefore be used in the integral representation formula of the heat equation direct formulation to obtain the solution of a heat problem in such domain the mentioned green s function can be obtained by the images method and the resulting source series can also be rewritten in terms of a double fourier series that will be used in the domain integral of the integral representation formula to transform such integral into equivalent surface integrals in this work a new reinitialization scheme has been applied to solve moving boundary problems to detect the location of the phase change interface in solidification and melting problems the numerical results were compared with analytical solutions in the case where they were available showing a good agreement between the results giving special emphasis to the interface location and the temperature distribution