fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

a level set based eulerian method for paraxial multivalued traveltimes we apply the level set methodology to compute multivalued solutions of the paraxial eikonal equation in both isotropic and anisotropic metrics this paraxial equation is obtained from 2d stationary eikonal equations by using one of the spatial directions as the artificial evolution direction the advection velocity field used to move level sets is obtained by the method of characteristics therefore the motion of level sets is defined in a phase space and the zero level set yields the location of bicharacteristic strips in the reduced phase space the multivalued traveltime is obtained from solving another advection equation with a source term the complexity of the algorithm is o n3 logn in the worst case and o n3 in the average case where n is the number of the sampling points along one of the spatial directions numerical experiments including the well known marmousi synthetic model illustrate the accuracy and the efficiency of the eulerian method

redistancing by flow of time dependent eikonal equation construction of signed distance to a given interface is a topic of special interest to level set methods there are currently however few algorithms that can efficiently produce highly accurate solutions we introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations we provide operation counts and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy comparison with the standard level set reinitialization algorithm shows ours is superior in terms of predictability and local construction which for example are critical in local level set methods we further apply the same ideas to extension of values off interfaces together our proposed approaches can be used to advance the level set method for fast and accurate computations of the latest scientific problems

non linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method abstract the incremental harmonic balance method is improved by i reversing the order of increment and linearization to find the bifurcated solution branches ii using a fast fourier transform to compute the solution increments and the floquet matrix and iii finding the secondary path after a bifurcation by an improved arc length method a system of two duffing oscillators linked with a linear stiffness is taken as a numerical example the qualitative change of solution due to the change of the linking stiffness is studied in detail

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

an efficient method for switching branches of period doubling bifurcations of strongly non linear autonomous oscillators with many degrees of freedom abstract popular algorithms for switching branches at a bifurcation point of strongly non linear oscillators are generally quite involved as they require the computation of the tangent of a new branch and second derivatives in this paper a simple but efficient algorithm is presented by using a perturbation incremental method for switching branches at a period doubling bifurcation of strongly non linear autonomous oscillators with many degrees of freedom to switch to a new branch at a bifurcation point a parameter is simply turned on from zero to a small positive value so as to obtain an initial solution on the emanating branch for subsequent continuation the parametric value at a period doubling bifurcation can also be determined accurately furthermore limit cycles of period 2 k k 1 can be calculated to any desired degree of accuracy

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

enriched performance measure approach for reliability based design optimization an enriched performance measure approach is presented for reliability based design optimization to substantially improve computational efficiency when applied to large scale applications in the enriched performance measure approach four improvements are made over the original performance measure approach as a way to launch reliability based design optimization at a deterministic optimum design as a new enhanced hybrid mean value method as an efficient probabilistic feasibility check and as a fast reliability analysis under the condition of design closeness it is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability based design optimization iterations in reliability based design optimization a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first order method and by carrying out the refined reliability analysis using the enhanced hybrid mean value method for e active and violated constraints the enhanced hybrid mean value method is developed to handle nonlinear and or nonmonotonic constraints in reliability analysis the fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness moreover two numerical examples are provided to compare the enriched performance measure approach to existing reliability based design optimization methods from a numerical efficiency and stability point of view

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

fast sweeping methods for static hamilton jacobi equations we propose a new sweeping algorithm which discretizes the legendre transform of the numerical hamiltonian using an explicit formula this formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in our sweeping scheme can either be solved analytically or numerically we illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

dynamical systems that perform the singular value decomposition abstract a dynamical system is presented whose solutions are shown to converge exponentially to the singular value decomposition of a given matrix provided that its singular values are distinct this system has potential application to certain problems in adaptive control and adaptive filtering it is one of several techniques that employ dynamical systems to perform tasks done conventionally with the techniques of numerical linear algebra

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method abstract the periodic motions of a non linear geared rotor bearing system are investigated by the incremental harmonic balance ihb method a path following procedure using arc length continuation technique is used to trace the bifurcation diagrams the system exhibits a period doubling route and a quasiperiodic route to chaos in different regions of excitation frequency the chaotic motions are investigated by numerical integration and the lyapunov exponents are computed the periodic solutions and subharmonic solutions obtained by the ihb method compare very well with those obtained by numerical integration

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

a reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows abstract this paper presents an accurate and robust reinitialization equation for the conservative level set that does not significantly deform stationary surfaces the compression and diffusion term of the reinitialization equation are reformulated to use a distance level set directly mapped from the conservative level set the normals are calculated using a distance level set reconstructed from the interface using a fast marching method increasing robustness and allowing the use of high order non tvd transport schemes using this new reinitialization equation we present results for canonical test cases such as zalesak s disk and spurious currents which show significant improvement a simulation of a liquid gas jet with r e 5000 is also presented to demonstrate the volume conservation properties of the method in more complex flows

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

a new incompressibility discretization for a hybrid particle mac grid representation with surface tension we take a particle based approach to incompressible free surface flow motivated by the fact that an explicit representation of the interface geometry and internal deformations gives precise feedback to an implicit solver for surface tension methods that enforce incompressibility directly on the particles are typically numerically inefficient compared to those that utilize a background grid however background grid discretizations suffer from inaccuracy near the free surface where they do not properly capture the interface geometry therefore our incompressibility discretization utilizes a particle based projection near the interface and a background mac grid based projection for efficiency in the vast interior of the liquid domain as well as a novel method for coupling these two disparate projections together we show that the overall coupled elliptic solver is second order accurate and remains second order accurate when used in conjunction with an appropriate temporal discretization for parabolic problems a similar second order accurate discretization is derived when the mac grid unknowns are located on faces as opposed to cell centers so that navier stokes viscosity can be solved for implicitly as well finally we present a fully implicit approach to surface tension that is robust enough to achieve a steady state solution in a single time step beyond stable implicit surface tension for our novel hybrid discretization we demonstrate preliminary results for both standard front tracking and the particle level set method

accuracy improvement of the most probable point based dimension reduction method using the hessian matrix summary this paper proposes a most probable point mpp based dimension reduction method drm using the hessian matrix called hedrm to improve accuracy of reliability analysis in existing mpp based drm methods conventional mpp based drms contain two types of errors 1 error due to eliminating cross terms of a performance function by using the univariate drm 2 error because of dependency of an axis direction after a rotational transformation the proposed method minimizes the aforementioned errors by utilizing the hessian matrix of a performance function by performing an orthogonal transformation using the eigenvectors of the hessian matrix the cross term effect of the performance function is minimized and the axis direction that results in the most accurate calculation is obtained because the gaussian quadrature points for numerical integration are arranged along the eigenvector directions in this way the error incurred by exiting mpp based drms can be reduced that leads to more accurate probability of failure estimation in addition this paper proposes to allocate the gaussian quadrature points using the magnitude of the eigenvalues of the hessian matrix this allocation makes it possible to predetermine the number of function evaluations required to estimate the probability of failure accurately and efficiently copyright 2016 john wiley sons ltd

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

a univariate approximation at most probable point for higher order reliability analysis this paper presents a new univariate method employing the most probable point as the reference point for predicting failure probability of structural and mechanical systems subject to random loads material properties and geometry the method involves novel decomposition at the most probable point that facilitates a univariate approximation of a general multivariate function response surface generation of the univariate function and monte carlo simulation in addition to the effort of identifying the most probable point the method requires a small number of exact or numerical evaluations of the performance function at selected input results of four numerical examples involving elementary mathematical functions and structural solid mechanics problems indicate that the proposed method provides accurate and computationally efficient estimates of probability of failure finally the fatigue failure of lever arm in a wheel loader has been evaluated demonstrating the ability of the new method in solving industrial scale fatigue reliability problems

stochastic sensitivity a computable lagrangian uncertainty measure for unsteady flows uncertainties in velocity data are often ignored when computing lagrangian particle trajectories of fluids modeling these as noise in the velocity field leads to a random deviation from each traje

a simple explicit operator splitting method for effective hamiltonians understanding effective hamiltonians quantitatively is essential for the homogenization of hamilton jacobi equations we propose in this article a simple efficient operator splitting method for computing effective hamiltonians when the hamiltonian is either convex or nonconvex in the gradient variable to speed up our lie scheme based operator splitting method we further propose a cascadic initialization strategy so that the steady state of the underlying time dependent hamilton jacobi equation can be reached more rapidly extensive numerical examples demonstrate the efficiency and accuracy of the new algorithm

a locally extended finite element method for the simulation of multi fluid flows using the particle level set method abstract the simulation of immiscible two phase flows on eulerian meshes requires the use of special techniques to guarantee a sharp definition of the evolving fluid interface this work describes the combination of two distinct technologies with the goal of improving the accuracy of the target simulations first of all a spatial enrichment is employed to improve the approximation properties of the eulerian mesh this is done by injecting into the solution space new features to make it able to correctly resolve the solution in the vicinity of the moving interface then the lagrangian particle level set pls method is employed to keep trace of the evolving solution and to improve the mass conservation properties of the resulting method while the local enrichment can be understood in the general context of the xfem we employ an element local variant which allows preserving the matrix graph and hence highly improving the computational efficiency

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

effect of smoothing piecewise linear oscillators on their stability predictions the effects of smoothing non smooth function on estimation of dynamical stability of the periodic response determined in the frequency domain are considered in this paper for that purpose a simple single degree of freedom system with piecewise linear force displacement relationship subjected to a harmonic force excitation is analyzed stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the floquet liapounov theorem results obtained are verified in the time domain by the method of piecing the exact solutions and runge kutta integration routine

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

inverse analysis method using mpp based dimension reduction for reliability based design optimization of nonlinear and multi dimensional systems abstract there are two commonly used analytical reliability analysis methods linear approximation first order reliability method form and quadratic approximation second order reliability method sorm of the performance function the reliability analysis using form could be acceptable in accuracy for mildly nonlinear performance functions whereas the reliability analysis using sorm may be necessary for accuracy of nonlinear and multi dimensional performance functions even though the reliability analysis using sorm may be accurate it is not as much used for probability of failure calculation since sorm requires the second order sensitivities moreover the sorm based inverse reliability analysis is rather difficult to develop this paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second order sensitivities for reliability based design optimization rbdo of nonlinear and multi dimensional systems for the inverse reliability analysis the most probable point mpp based dimension reduction method drm is developed since the form based reliability index is inaccurate for the mpp search of the nonlinear performance function a three step computational procedure is proposed to improve accuracy of the inverse reliability analysis probability of failure calculation using constraint shift reliability index update and mpp update using the three steps a new drm based mpp is obtained which estimates the probability of failure of the performance function more accurately than form and more efficiently than sorm the drm based mpp is then used for the next design iteration of rbdo to obtain an accurate optimum design even for nonlinear and or multi dimensional system since the drm based rbdo requires more function evaluations the enriched performance measure approach pma with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations

error analysis of the dynamically orthogonal approximation of time dependent random pdes in this work we discuss the dynamically orthogonal do approximation of time dependent partial differential equations with random data the approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion we analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the s terms do solution by the corresponding s terms best approximation i e the truncated s terms karhunen loeve expansion at each time instant the bound is applicable on the largest time interval in which the best s terms approximation is continuously time differentiable properties of the do approximations are analyzed on simple cases of deterministic equations with random initial data numerical tests are presented that confirm the theoret

projection like retractions on matrix manifolds this paper deals with constructing retractions a key step when applying optimization algorithms on matrix manifolds for submanifolds of euclidean spaces we show that the operation consisting of taking a tangent step in the embedding euclidean space followed by a projection onto the submanifold is a retraction we also show that the operation remains a retraction if the projection is generalized to a projection like procedure that consists of coming back to the submanifold along admissible directions and we give a sufficient condition on the admissible directions for the generated retraction to be second order this theory offers a framework in which previously proposed retractions can be analyzed as well as a toolbox for constructing new ones illustrations are given for projection like procedures on some specific manifolds for which we have an explicit easy to compute expression

a fast marching like algorithm for geometrical shock dynamics we develop a new algorithm for the computation of the geometrical shock dynamics gsd model the method relies on the fast marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a cartesian grid the proposed algorithm is based on a first order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure reference solutions are built for a smooth radial configuration and for the 2d riemann problem the link between the gsd model and p systems is given numerical experiments demonstrate the efficiency of the scheme and its ability to handle singularities

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

hybrid control synthesis for eventuality specifications using level set methods this paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications given a hybrid system modelled by a hybrid automaton and a target set of states the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time due to the existence of set valued disturbance inputs the problem is studied in a game theoretic framework having shown that a least restrictive solution does not exist we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property to implement the algorithm reachable sets of pursuit evasion differential games need to be computed for that reason level set methods are employed where the boundary of the reachable set is characterized as the zero level set of a hamilton jacobi equation the procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology finally to demonstrate the practical character of our results a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology

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

computational study of fast methods for the eikonal equation a computational study of the fast marching and the fast sweeping methods for the eikonal equation is given it is stressed that both algorithms should be considered as direct as opposed to iterative methods on realistic grids fast sweeping is faster than fast marching for problems with simple geometry for strongly nonuniform problems and or complex geometry the situation may be reversed finally fully second order generalizations of methods of this type for problems with obstacles are proposed and implemented

stability and accuracy of the iterative differential quadrature method in this paper the stability and accuracy of an iterative method based on differential quadrature rules will be discussed the method has already been proposed by the author in a previous work where its good performance has been shown nevertheless discussion about stability and accuracy remained open an answer to this question will be provided in this paper where the conditional stability of the method will be pointed out in addition to an examination of the possible errors which arise under certain conditions the discussion will be preceded by an overview of the method and its foundations i e the differential quadrature rules and followed by a numerical case which shows how the method behaves when applied to reduce continuous systems to two degree of freedom systems in the non linear range in particular here the case of oscillators coupled in non linear terms will be treated the numerical results used to draw poincare maps will be compared with those obtained by using the runge kutta method with a high precision goal copyright 2003 john wiley sons ltd

dislocation dynamics short time existence and uniqueness of the solution we study a mathematical model describing dislocation dynamics in crystals we consider a single dislocation line moving in its slip plane the normal velocity is given by the peach koehler force created by the dislocation line itself the mathematical model is an eikonal equation with a velocity which is a non local quantity depending on the whole shape of the dislocation line we study the special case where the dislocation line is assumed to be a graph or a closed loop in the framework of discontinuous viscosity solutions for hamilton jacobi equations we prove the existence and uniqueness of a solution for small time we also give physical explanations and a formal derivation of the mathematical model finally we present numerical results based on a level sets formulation of the problem these results illustrate in particular the fact that there is no general inclusion principle for this model

discrete time high order schemes for viscosity solutions of hamilton jacobi bellman equations a general method for constructing high order approximation schemes for hamilton jacobi bellman equations is given the method is based on a discrete version of the dynamic programming principle we prove a general convergence result for this class of approximation schemes also obtaining under more restrictive assumptions an estimate in l infty of the order of convergence and of the local truncation error the schemes can be applied in particular to the stationary linear first order equation in bbb r n we present several examples of schemes belonging to this class and with fast convergence to the solution

vials an eulerian tool based on total variation and the level set method for studying dynamical systems abstract we propose a new eulerian tool to study complicated dynamical systems based on the average growth in the surface area of a family of level surfaces represented implicitly by a level set function since this proposed quantity determines the temporal variation of the averaged surface area of all level surfaces we name the quantity the variation of the integral over area of level surfaces vials numerically all these infinitely many level surfaces are advected according to the given dynamics by solving one single linear advection equation to develop a computationally efficient approach we apply the coarea formula and rewrite the surface area integral as a simple integral relating the total variation tv of the level set function the proposed method can be easily incorporated with a recent eulerian algorithm for efficient computation of flow maps to speed up our approach we will also prove that the proposed vials is closely related to the computation of the finite time lyapunov exponent ftle in the lagrangian coherent structure lcs extraction this connects our proposed eulerian approach to widely used lagrangian techniques for understanding complicated dynamical systems

dynamically orthogonal numerical schemes for efficient stochastic advection and lagrangian transport quantifying the uncertainty of lagrangian motion can be performed by solving a large number of ordinary differential equations with random velocities or equivalently a stochastic transport partial differential equation pde for the ensemble of flow maps the dynamically orthogonal do decomposition is applied as an efficient dynamical model order reduction to solve for such stochastic advection and lagrangian transport its interpretation as the method that applies instantaneously the truncated svd on the matrix discretization of the original stochastic pde is used to obtain new numerical schemes fully linear explicit central advection schemes stabilized with numerical filters are selected to ensure efficiency accuracy stability and direct consistency between the original deterministic and stochastic do advections and flow maps various strategies are presented for selecting a time stepping that accounts for the curvature of the fixed rank manifold and the error related to closely singular coefficient matrices efficient schemes are developed to dynamically evolve the rank of the reduced solution and to ensure the orthogonality of the basis matrix while preserving its smooth evolution over time finally the new schemes are applied to quantify the uncertain lagrangian motions of a 2d double gyre flow with random frequency and of a stochastic flow past a cylinder

reduction of ordering effect in reliability based design optimization using dimension reduction method in reliability based design optimization problems with correlated input variables a joint cumulative distribution function needs to be used to transform the correlated input variables into independent standard gaussian variables for the inverse reliability analysis to obtain a true joint cumulative distribution function a very large number of data if not infinite needs to be used which is impractical in industry applications in this paper a copula is proposed to model the joint cumulative distribution function using marginal cumulative distribution functions and correlation parameters obtained from samples using the joint cumulative distribution function modeled by the copula the transformation and the first order reliability method can be carried out however the first order reliability method may yield different reliability analysis results for different transformation ordering of input variables thus the most probable point based dimension reduction method which is more accurate than the first order reliability method and more efficient than the second order reliability method is proposed for the inverse reliability analysis to reduce the effect of transformation ordering

an eulerian approach for computing the finite time lyapunov exponent we propose efficient eulerian methods for approximating the finite time lyapunov exponent ftle the idea is to compute the related flow map using the level set method and the liouville equation there are several advantages of the proposed approach unlike the usual lagrangian type computations the resulting method requires the velocity field defined only at discrete locations no interpolation of the velocity field is needed also the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain the computational complexity of the algorithm is o dx d 1 with d the dimension of the physical space since there are the same number of mesh points in the x t space the computational complexity of the proposed eulerian approach is optimal in the sense that each grid point is visited for only o 1 time we also extend the algorithm to compute the ftle on a co dimension one manifold the resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold

pollution release tied to invariant manifolds a case study for the coast of florida high resolution ocean velocity data has become readily available since the introduction of very high frequency vhf radar technology the vast amount of data generated so far however remains largely unused in environmental prediction in this paper we use vhf data of the florida coastline to locate lagrangian coherent structures lcs hidden in ocean surface currents such structures govern the spread of organic contaminants and passive drifters that stay confined to the ocean surface we use the lagrangian structures in a real time pollution release scheme that reduces the effect of industrial contamination on the coastal environment

poincare maps of duffing type oscillators and their reduction to circle maps ii methods and numerical results bifurcation diagrams and plots of lyapunov exponents in the r plane for duffing type oscillators exhibit a regular pattern of repeating self similar tongues with complex internal structure we demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a poincare map of the system in action angle variables in the limit of large driving force or equivalently small driving frequency this map approaches the one dimensional form as derived in paper i this second paper describes our approach to calculating the various constants and functions introduced in paper i it gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of duffing type equations finally we show how to extend the range in the parameter space where the map is applicable

a boundary piecewise constant level set method for boundary control of eigenvalue optimization problems this paper investigates optimization of the least eigenvalue of d with the constraint of one dimension hausdorff measure of dirichlet boundary we propose the boundary piecewise constant level set bpcls method based on the regularity technique to combine two types of boundary conditions into a single robin boundary condition we derive the first variation of the least eigenvalue w r t the bpcls function and propose a penalty bpcls algorithm and an augmented lagrangian bpcls algorithm numerical results are reported for experiments on ellipse and l shape domains

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

convergence of semi lagrangian approximations to convex hamilton jacobi equations under very large courant numbers we consider a class of semi lagrangian high order approximation schemes for convex hamilton jacobi equations in this framework we prove that under certain restrictions on the relationship between delta x and delta t the sequence of approximate solutions is uniformly lipschitz continuous and hence by consistency that it converges to the exact solution the argument is suitable for most reconstructions of interest including high order polynomials and eno reconstructions

non linear dynamics of a two dimensional airfoil by incremental harmonic balance method periodic oscillations and bifurcations of a two dimensional airfoil in plunge and pitching motions with cubic pitching stiffness in incompressible flow is investigated using the incremental harmonic balance ihb method the bifurcations are obtained with the parametric continuation technique and the stability of the periodic motions is investigated using the floquet theory the autonomous non linear system of the airfoil undergoes initial hopf bifurcation leading to limit cycle oscillation as the airspeed parameter is increased further increase in the airspeed causes symmetry breaking saddle node and period doubling bifurcations leading to chaos the frequency of the limit cycle oscillation is also determined in the ihb method

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

optimization of nonlinear structural resonance using the incremental harmonic balance method abstract we present an optimization procedure for tailoring the nonlinear structural resonant response with time harmonic loads a nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis an optimization procedure based on a gradient based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities we consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super harmonic resonant responses

factored singularities and high order lax friedrichs sweeping schemes for point source traveltimes and amplitudes in the high frequency regime the geometrical optics approximation for the helmholtz equation with a point source results in an eikonal equation for traveltime and a transport equation for amplitude because the point source traveltime field has an upwind singularity at the source point all formally high order finite difference eikonal solvers exhibit first order convergence and relatively large errors in this paper we propose to first factor out the singularities of traveltimes takeoff angles and amplitudes and then we design high order lax friedrichs sweeping schemes for point source traveltimes takeoff angles and amplitudes numerical examples are presented to demonstrate the performance of our new method

deformable pedal curves and surfaces hybrid geometric active models for shape recovery in this paper we propose significant extensions to the snake pedal model a powerful geometric shape modeling scheme introduced in vemuri and guo 1998 the extension allows the model to automatically cope with topological changes and for the first time introduces the concept of a compact global shape into geometric active models the ability to characterize global shape of an object using very few parameters facilitates shape learning and recognition in this new modeling scheme object shapes are represented using a parameterized function called the generator which accounts for the global shape of an object and the pedal curve surface of this global shape with respect to a geometric snake to represent any local detail traditionally pedal curves surfaces are defined as the loci of the feet of perpendiculars to the tangents of the generator from a fixed point called the pedal point local shape control is achieved by introducing a set of pedal points lying on a snake for each point on the generator the model dubbed as a snake pedal allows for interactive manipulation via forces applied to the snake in this work we replace the snake by a geometric snake and derive all the necessary mathematics for evolving the geometric snake when the snake pedal is assumed to evolve as a function of its curvature automatic topological changes of the model may be achieved by implementing the geometric snake in a level set framework we demonstrate the applicability of this modeling scheme via examples of shape recovery from a variety of 2d and 3d image data

simulating non linear coupled oscillators by an iterative differential quadrature method abstract an iterative method based on differential quadrature rules is proposed as a new unified frame of resolution for non linear two degree of freedom systems dynamical systems with duffing type non linearity have been considered differential quadrature rules have been applied with a careful distribution of sampling points to reduce the governing equation of motion to two second order non linear non autonomous ordinary differential equations and to solve the time domain problem the time domain of the problem is discretized by means of time intervals with the same distribution of sampling points used to discretize the space domain which can be seen as a single interval it will be shown that accurate solutions depend not only on the choice of the distribution of sampling points but also on the length of the time interval one refers to in the computations the numerical results utilized to draw poincare maps are successfully compared with those obtained using the runge kutta method

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

a uniformly second order fast sweeping method for eikonal equations a uniformly second order method with a local solver based on the piecewise linear discontinuous galerkin formulation is introduced to solve the eikonal equation with dirichlet boundary conditions the method utilizes an interesting phenomenon referred as the superconvergence phenomenon that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on both its value and gradient when the solution is smooth this phenomenon greatly simplifies the local solver based on the discontinuous galerkin formulation by reducing its local degrees of freedom from two 1 d or three 2 d or four 3 d to one with the information of the gradient frozen when considering the eikonal equation with point source conditions we further utilize a factorization approach to resolve the source singularities of the eikonal by decomposing it into two parts either multiplicatively or additively one part is known and captures the source singularities the other part serves as a correction term that is differentiable at the sources and satisfies the factored eikonal equations we extend the second order method to solve the factored eikonal equations to compute the correction term with second order accuracy then recover the eikonal with second order accuracy numerical examples are presented to demonstrate the performance of the method

new high resolution semi discrete central schemes for hamilton jacobi equations we introduce a new high resolution central scheme for multidimensional hamilton jacobi equations the scheme retains the simplicity of the non oscillatory central schemes developed by c t lin and e tadmor in press siam j sci comput yet it enjoys a smaller amount of numerical viscosity independent of 1 t by letting t 0 we obtain a new second order central scheme in the particularly simple semi discrete form along the lines of the new semi discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws fully discrete versions are obtained with appropriate runge kutta solvers the smaller amount of dissipation enables efficient integration of convection diffusion equations where the accumulated error is independent of a small time step dictated by the cfl limitation the scheme is non oscillatory thanks to the use of nonlinear limiters here we advocate the use of such limiters on second discrete derivatives which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme

computation of nonclassical solutions to hamilton jacobi problems this paper is devoted to the construction of numerical methods for the approximation of nonclassical solutions to multidimensional hamilton jacobi equations for both scalar and vectorial problems recent theoretical results have yielded existence of solutions in many cases for which the usual viscosity approach was ill suited or not applicable the selection criterion used here is based on a viscoelasticity capillarity approach common in solid mechanics numerical methods adapted to this framework are built consistency of the model equation with the given selection criterion is essential it is achieved here through the use of high order finite difference schemes by considering applications to potential well problems the convergence of the methods are investigated