generalized information entropy analysis of financial time series abstract generalized information entropy has been widely applied to analyzing complex systems in this paper we propose the weighted multiscale renyi permutation entropy mswrpe based on the weight assigned to each vector as a novel technique to consider the amplitude information renyi permutation entropy rpe has a parameter q for non extensivity compared to shannon permutation entropy pe hence we speculate that rpe has a better sensitivity to patterns extracted from signals containing amplitude information and a better robustness to noise compared to pe firstly we perform the multiscale renyi permutation entropy msrpe and mswrpe methods on synthetic data we find that mswrpe suits better signals containing considerable amplitude information and is successful to consider the multiple time scales inherent in the financial systems the finding is also verified in four different stock markets then we make a comparison between mswrpe and weighted multiscale permutation entropy mswpe on different stock markets the conclusion is that the mswrpe method has a better characterization than mswpe for q 1 different markets have the same law on mswrpe while hsi can be distinguished from the other markets for q 1 which is more obvious when m 7

error estimates of stochastic optimal neumann boundary control problems we study mathematically and computationally optimal control problems for stochastic partial differential equations with neumann boundary conditions the control objective is to minimize the expectation of a cost functional and the control is of the deterministic boundary value type mathematically we prove the existence of an optimal solution and of a lagrange multiplier we represent the input data in terms of their karhunen loeve expansions and deduce the deterministic optimality system of equations computationally we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces

distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach in this paper we introduce a multiscale symbolic information theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems more precisely we show that the multiscale complexity entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system s dynamics numerical simulations obtained from the well known and widely used mackey glass oscillator operating in a high dimensional chaotic regime were used as test beds the effect of an increased amount of observational white noise was carefully examined the results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles finally several experimental and natural time series were analyzed in order to show the applicability of this scale dependent symbolic approach in practical situations

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

on monotonicity preserving stabilized finite element approximations of transport problems the aim of this work is to design monotonicity preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear shock capturing stabilization as linear stabilization we consider and analyze a novel symmetric projection stabilization technique based on a local scott zhang projector next we design a weighting of the aforementioned linear stabilization such that when combined with a finite element discretization enjoying a discrete maximum principle usually attained via nonlinear stabilization it does not spoil these monotonicity properties then we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes for the nodal scheme we prove a discrete maximum principle for time dependent multidimensional transport problems numerical experiments support the numerical analysis and we show that the resulting methods prov

control of the stochastic burgers model of turbulence we consider a control problem for a stochastic burgers equation this problem is motivated by a model from the control of turbulence see choi et al j fluid mech 253 1993 pp 509 543 we study a sequence of approximated hamilton jacobi equations by using dynamic programming

a penalized neumann control approach for solving an optimal dirichlet control problem for the navier stokes equations we introduce a penalized neumann boundary control approach for solving an optimal dirichlet boundary control problem associated with the two or three dimensional steady state navier stokes equations we prove the convergence of the solutions of the penalized neumann control problem the suboptimality of the limit and the optimality of the limit under further restrictions on the data we describe the numerical algorithm for solving the penalized neumann control problem and report some numerical results

on the design of general purpose flux limiters for finite element schemes i scalar convection the algebraic flux correction afc paradigm is extended to finite element discretizations with a consistent mass matrix it is shown how to render an implicit galerkin scheme positivity preserving and remove excessive artificial diffusion in regions where the solution is sufficiently smooth to this end the original discrete operators are modified in a mass conserving fashion so as to enforce the algebraic constraints to be satisfied by the numerical solution a node oriented limiting strategy is employed to control the raw antidiffusive fluxes which consist of a convective part and a contribution of the consistent mass matrix the former offsets the artificial diffusion due to upwinding of the spatial differential operator and lends itself to an upwind biased flux limiting the latter eliminates the error induced by mass lumping and calls for the use of a symmetric flux limiter the concept of a target flux and a new definition of upper lower bounds make it possible to combine the advantages of algebraic fct and tvd schemes introduced previously by the author and his coworkers unlike other high resolution schemes for unstructured meshes the new algorithm reduces to a consistent high order galerkin scheme in smooth regions and is designed to provide an optimal treatment of both stationary and time dependent problems its performance is illustrated by application to the linear advection equation for a number of 1d and 2d configurations

stability and error estimates of a new high order compact adi method for the unsteady 3d convection diffusion equation abstract in this paper a new high order compact adi method for the unsteady convection diffusion equation in three dimension 3d is considered collecting the truncation error of the finite difference operator by the recursion method we derive a new high order compact finite difference scheme for the unsteady 1d convection diffusion equation firstly then based on the adi method applying a correction technique to reduce the error of the splitting term a high order compact adi method for the unsteady 3d convection diffusion equation is proposed in which we solve a series of 1d problems with strictly diagonal dominant tri diagonal structures instead of the high dimensional ones the scheme is proved to be unconditional stable moreover the regularity and error estimate of the numerical solution are derived finally some numerical examples are performed which confirm the theoretical prediction and show that much better computational accuracy results can be got by applying the new scheme

comparison of approaches for random pde optimization problems based on different matching functionals in this article we consider an optimal control problem for an elliptic partial differential equation with random inputs to determine an applicable deterministic control f x we consider the four cases which we compare for efficiency and feasibility we prove the existence of optimal states adjoint states and optimality conditions for each cases we also derive the optimality systems for the four cases the optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space respectively the numerical experiments are performed for their efficiency and feasibility

explicit runge kutta schemes and finite elements with symmetric stabilization for first order linear pde systems we analyze explicit runge kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first order linear differential operator in space of friedrichs type for the time discretization we consider explicit second and third order runge kutta schemes we identify a general set of properties on the space stabilization encompassing continuous and discontinuous finite elements under which we prove stability estimates using energy arguments then we establish l 2 norm error estimates with quasi optimal convergence rates for smooth solutions in space and time these results hold under the usual cfl condition for third order runge kutta schemes and any polynomial degree in space and for second order runge kutta schemes and first order polynomials in space for second order runge kutta schemes and higher polynomial degrees in space a tightened 4 3 cfl condition is required numerical results are presented for smooth and rough solutions the case of finite volumes is briefly discussed

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

nonlinear diffusion and discrete maximum principle for stabilized galerkin approximations of the convection diffusion reaction equation we investigate stabilized galerkin approximations of linear and nonlinear convectiondiffusion reaction equations we derive nonlinear streamline and cross wind diffusion methods that guarantee a discrete maximum principle for strictly acute meshes and first order polynomial interpolation for pure convectiondiffusion problems the discrete maximum principle is achieved using a nonlinear cross wind diffusion factor that depends on the angle between the discrete solution and the flow velocity for convectiondiffusion reaction problems two methods are considered residual based isotropic diffusion and the previous nonlinear cross wind diffusion factor supplemented by additional isotropic diffusion scaling as the square of the mesh size practical versions of the present methods suitable for numerical implementation are compared to previous discontinuity capturing schemes lacking theoretical justification numerical results are investigated in terms of both solution quality violation of maximum principle smearing of internal layers and computational costs

stabilized galerkin approximation of convection diffusion reaction equations discrete maximum principle and convergence we analyze a nonlinear shock capturing scheme for h1 conforming piecewise affine finite element approximations of linear elliptic problems the meshes are assumed to satisfy two standard conditions a local quasi uniformity property and the xu zikatanov condition ensuring that the stiffness matrix associated with the poisson equation is an m matrix a discrete maximum principle is rigorously established in any space dimension for convection diffusion reaction problems we prove that the shock capturing finite element solution converges to that without shock capturing if the cell peclet numbers are sufficiently small moreover in the diffusion dominated regime the difference between the two finite element solutions super converges with respect to the actual approximation error numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

self similarity of the 1 f noise called music abstract suggestions have been made that computer musicians should attempt to compose fractal music and questions have been raised whether there is such a thing as fractal music voss and clark observed that music is scaling or 1 f noise as analyzed on the basis of the amplitude loudness of the audio signals they failed to find a fractal distribution of acoustic frequencies music notes in music analyzing bach s and mozart s compositions we have shown that the incidence of the frequency intervals or of the changes of acoustic frequency has a fractal geometry fractal phenomena are characterized by scale independency the purpose of this investigation is to demonstrate the self similarity of music and to explore its implications

complexity entropy causality plane a useful approach for distinguishing songs nowadays we are often faced with huge databases resulting from the rapid growth of data storage technologies this is particularly true when dealing with music databases in this context it is essential to have techniques and tools able to discriminate properties from these massive sets in this work we report on a statistical analysis of more than ten thousand songs aiming to obtain a complexity hierarchy our approach is based on the estimation of the permutation entropy combined with an intensive complexity measure building up the complexity entropy causality plane the results obtained indicate that this representation space is very promising to discriminate songs as well as to allow a relative quantitative comparison among songs additionally we believe that the here reported method may be applied in practical situations since it is simple robust and has a fast numerical implementation

flux correction tools for finite elements flux correction in the finite element context is addressed criteria for positivity of the numerical solution are formulated and the low order transport operator is constructed from the discrete high order operator by adding modulated dissipation so as to eliminate negative off diagonal entries the corresponding antidiffusive terms can be decomposed into a sum of genuine fluxes rather than element contributions which represent bilateral mass exchange between individual nodes thereby essentially one dimensional flux correction tools can be readily applied to multidimensional problems involving unstructured meshes the proposed methodology guarantees mass conservation and makes it possible to design both explicit and implicit fct schemes based on a unified limiting strategy numerical results for a number of benchmark problems illustrate the performance of the algorithm

a maximum principle preserving finite element method for scalar conservation equations this paper introduces a first order viscosity method for the explicit approximation of scalar conservation equations with lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension provided the lumped mass matrix is positive definite the method is shown to satisfy the local maximum principle under a usual cfl condition the method is independent of the cell type for instance the mesh can be a combination of tetrahedra hexahedra and prisms in three space dimensions

an efficient gradient projection method for stochastic optimal control problems in this work we propose a simple yet effective gradient projection algorithm for a class of stochastic optimal control problems we first reduce the optimal control problem to an optimization problem for a convex functional by means of a projection operator then we propose a convergent iterative scheme for the optimization problem the key issue in our iterative scheme is to compute the gradient of the objective functional by solving the adjoint equations that are given by backward stochastic differential equations bsdes the euler method is used to solve the resulting bsdes rigorous convergence analysis is presented and it is shown that the entire numerical algorithm admits a first order rate of convergence several numerical examples are carried out to support the theoretical finding

an optimization based domain decomposition method for pdes with random inputs an optimization based domain decomposition method for stochastic elliptic partial differential equations is presented the main idea of the method is a constrained optimization problem for which the minimization of an appropriate functional forces the solutions on the two subdomains to agree on the interface the constraints are the stochastic partial differential equations the existence of optimal solutions for the stochastic optimal control problem is shown as is the convergence to the exact solution of the given problem we prove the existence of a lagrange multiplier and derive an optimality system from which solutions of the domain decomposition problem may be determined finite element approximations to the solutions of the optimality system are defined and analyzed with respect to both spatial and random parameter spaces then the results of some numerical experiments are given to confirm theoretical error estimate results

blending low order stabilised finite element methods a positivity preserving local projection method for the convection diffusion equation in this work we propose a nonlinear blending of two low order stabilisation mechanisms for the convection diffusion equation the motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions the approach is to blend a first order artificial diffusion method which will be active only in the vicinity of layers and extrema with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution we prove existence of discrete solutions as well as convergence under appropriate assumptions on the nonlinear terms and on the exact solution numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle while the layers are not smeared significantly

a trust region algorithm with adaptive stochastic collocation for pde optimization under uncertainty the numerical solution of optimization problems governed by partial differential equations pdes with random coefficients is computationally challenging because of the large number of deterministic pde solves required at each optimization iteration this paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse grid collocation for the discretization of the pde in the stochastic space and a trust region framework for optimization and fidelity management of the stochastic discretization the overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem it frequently uses few collocation points initially and increases the number of collocation points only as necessary thereby keeping the number of deterministic pde solves low while guaranteeing convergence currently an error indicator is used to estimate gradient errors due to adaptive

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part ii analysis for p1 and q1 finite elements abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers a review and a comparison of the most methods which have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers are given in part i v john p knobloch on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review comput methods appl mech engrg 196 2007 2197 2215 in the present paper the most promising of these sold methods are investigated in more detail for p 1 and q 1 finite elements in particular the dependence of the results on the mesh the data of the problems and parameters of the methods are studied analytically and numerically furthermore the numerical solution of the nonlinear discrete problems is discussed and the capability of adaptively refined grids for reducing spurious oscillations is examined our conclusion is that also for simple problems any of the sold methods generally provides solutions with non negligible spurious oscillations

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

a second order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations this paper proposes an explicit at least second order maximum principle sat isfying lagrange finite element method for solving nonlinear scalar conservation equations the technique is based on a new viscous bilinear form introduced in guermond and nazarov com put methods appl mech engrg 272 2014 pp 198 213 a high order entropy viscosity method and the boris book zalesak flux correction technique the algorithm works for arbitrary meshes in any space dimension and for all lipschitz fluxes the formal second order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems

linearity preserving monotone local projection stabilization schemes for continuous finite elements abstract this paper presents some new tools for enforcing discrete maximum principles and or positivity preservation in continuous piecewise linear finite element approximations to convection dominated transport problems using a linear first order advection equation as a model problem we construct element level bilinear forms associated with first order artificial diffusion operators and their two scale counterparts the underlying design philosophy is similar to that behind local projection stabilization lps techniques and variational multiscale vms methods the difference lies in the structure of the local stabilization operator and in the way in which the resolved scales are detected the proposed stabilization term penalizes the difference between the nodal values and cell averages of the finite element solution in a manner which guarantees monotonicity and linearity preservation the value of the stabilization parameter is determined using a multidimensional limiter function designed to prevent unresolvable fine scale effects from creating undershoots or overshoots the result is a nonlinear high resolution scheme capable of resolving moving fronts and internal boundary layers as sharp localized nonoscillatory features the use of variational gradient recovery makes it possible to add high order background dissipation leading to improved approximation properties in smooth regions the numerical behavior of the constrained schemes is illustrated by a grid convergence study for stationary and time dependent test problems in two space dimensions

approximating infinity dimensional stochastic darcy s equations without uniform ellipticity we consider a stochastic darcy s pressure equation whose coefficient is generated by a white noise process on a hilbert space employing the ordinary rather than the wick product a weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf sup condition and well posedness of the problem we generalize the numerical approximations proposed in benth and theting stochastic anal appl 20 2002 pp 1191 1223 for wick stochastic partial differential equations to the ordinary product stochastic pressure equation we establish discrete inf sup conditions and provide a priori error estimates for a wide class of norms the proposed numerical approximation is based on wiener chaos finite element methods and yields a positive definite symmetric linear system we also improve and generalize the approximation results of benth and gjerde stochastics stochastics rep 63 1998 pp 313 326 and cao stochastics 78 2006 pp 179 187 when a generalized process is truncated by a finite wiener chaos expansion finally we present numerical experiments to validate the results

a domain decomposition algorithm for optimal control problems governed by elliptic pdes with random inputs abstract in this study we apply a domain decomposition algorithm based on optimization technique to optimal control problems governed by an elliptic partial differential equation with random inputs the domain decomposition is implemented by introducing an auxiliary optimal control problem which results in a multi objective optimization problem we prove the existence of a solution to the resulting optimization problem as well as the convergence to the optimal solution of original control problem solutions of the domain decomposition problem are determined from an optimality system and error estimates for finite element approximations are analyzed finally some numerical experiments are provided to confirm theoretical results

a petrov galerkin finite element method for convection dominated flows an accurate upwinding technique for satisfying the maximum principle abstract in this paper we present a new petrov galerkin method for convection dominated flow problems which is conservative and satisfies the discrete maximum principle this method also possesses no spurious crosswind diffusion and gives very accurate solutions the method is representative of a class of methods which may be described as fixed mesh adaptive

universality in the tail of musical note rank distribution abstract although power laws have been used to fit rank distributions in many different contexts they usually fail at the tails languages as sequences of symbols have been a popular subject for ranking distributions and for this purpose music can be treated as such here we show that more than 1800 musical compositions are very well fitted by the first kind two parameter beta distribution which arises in the ranking of multiplicative stochastic processes the parameters a and b are obtained for classical jazz and rock music revealing interesting features specially we have obtained a clear trend in the values of the parameters for major and minor tonal modes finally we discuss the distribution of notes for each octave and its connection with the ranking of the notes

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

edge based nonlinear diffusion for finite element approximations of convection diffusion equations and its relation to algebraic flux correction schemes for the case of approximation of convection diffusion equations using piecewise affine continuous finite elements a new edge based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle the diffusion operator is shown to be lipschitz continuous and linearity preserving using these properties we provide a full stability and error analysis which in the diffusion dominated regime shows existence uniqueness and optimal convergence then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters the performance of the method is illustrated on some numerical test cases in two space dimensions

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

a discontinuity capturing crosswind dissipation for the finite element solution of the convection diffusion equation abstract to avoid the local oscillations that still remain using the streamline upwind petrov galerkin formulation for the scalar convection diffusion equation the introduction of a nonlinear crosswind dissipation is proposed it is shown that the method is less overdiffusive than other discontinuity capturing techniques and has better numerical behavior the design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases

analysis of algebraic flux correction schemes a family of algebraic flux correction afc schemes for linear boundary value problems in any space dimension is studied these methods main feature is that they limit the fluxes along each one of the edges of the triangulation and we suppose that the limiters used are symmetric for an abstract problem the existence of a solution existence and uniqueness of the solution of a linearized problem and an a priori error estimate are proved under rather general assumptions on the limiters for a particular but standard in practice choice of the limiters it is shown that a local discrete maximum principle holds the theory developed for the abstract problem is applied to convection diffusion reaction equations where in particular an error estimate is derived numerical studies show its sharpness

taylor approximation and variance reduction for pde constrained optimal control under uncertainty abstract in this work we develop a scalable computational framework for the solution of pde constrained optimal control problems under high dimensional uncertainty specifically we consider a mean variance formulation of the control objective and employ a taylor expansion with respect to the uncertain parameter field either to directly approximate the control objective or as a control variate for variance reduction the expressions for the mean and variance of the taylor approximation are known analytically although their evaluation requires efficient computation of the trace of the preconditioned hessian of the control objective we propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the hessian on a small number of random directions then the computational work does not depend on the nominal dimension of the uncertain parameter but depends only on the effective dimension i e the rank of the preconditioned hessian thus ensuring scalability to high dimensional problems moreover to increase the estimation accuracy of the mean and variance of the control objective by the taylor approximation we use it as a control variate for variance reduction which results in considerable computational savings several orders of magnitude compared to a plain monte carlo method in summary our approach amounts to solving an optimal control problem constrained by the original pde the generalized eigenvalue equations at a small number of eigenfunctions and a set of linearized pdes that arise from the computation of the gradient and hessian of the control objective with respect to the uncertain parameter we use the lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient based optimization method to solve the problem we demonstrate the accuracy efficiency and scalability of the proposed computational method for two examples with high dimensional uncertain parameters subsurface flow in a porous medium modeled as an elliptic pde and turbulent jet flow modeled by the reynolds averaged navier stokes equations coupled with a nonlinear advection diffusion equation characterizing model uncertainty in particular for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field

multifractal analyses of music sequences multifractal analysis is applied to study the fractal property of music in this paper a method is proposed to transform both the melody and rhythm of a music piece into individual sets of distributed points along a one dimensional line the structure of the musical composition is thus manifested and characterized by the local clustering pattern of these sequences of points specifically the local holder exponent and the multifractal spectrum are calculated for the transformed music sequences according to the multifractal formalism the observed fluctuations of the holder exponent along the music sequences confirm the non uniformity feature in the structures of melodic and rhythmic motions of music our present result suggests that the shape and opening width of the multifractal spectrum plot can be used to distinguish different styles of music in addition a characteristic curve is constructed by mapping the point sequences converted from the melody and rhythm of a musical work into a two dimensional graph each different pieces of music has its own unique characteristic curve this characteristic curve which also exhibits a fractal trait unveils the intrinsic structure of music

weighting the edge stabilization a modification of the edge stabilization technique is proposed to improve the behavior of the method when solving conservation equations with nonsmooth data or nonsmooth solutions the key ingredient is tempering the edge stabilization in regions of large gradients through appropriate weights the new method is shown to preserve the convergence properties of the original method on smooth solutions and numerical tests indicate that it performs better on nonsmooth solutions

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

stochastic optimal robin boundary control problems of advection dominated elliptic equations in this work we deal with a stochastic optimal robin boundary control problem constrained by an advection diffusion reaction elliptic equation with advection dominated term we assume that the uncertainty comes from the advection field and consider a stochastic robin boundary condition as control function a stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system a global error estimate in the product of physical space and stochastic space for the numerical approximation is derived illustrative numerical experiments are provided

multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations in this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by stokes equations we prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data the finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space respectively resulting in many large scale optimality systems to solve in order to reduce the unaffordable computational effort we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse grid techniques a weighted a posteriori error bound highlights the contribution stemming from each method numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient especially for solving high dimensional and large scale optimization problems

stability analysis of explicit entropy viscosity methods for non linear scalar conservation equations this material is based upon work supported by the department of homeland security under agreement 2008 dn 077 ari018 02 national science foundation grants dms 0811041 dms 0914977 dms 1015984 af office of scientific research grant fa99550 12 0358 and is partially supported by award kus c1 016 04 made by king abdullah university of science and technology kaust

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

music walk fractal geometry in music abstract in this study sequences of musical notes from various pieces of music are converted into one variable random walks here termed music walks quantitative measurements of the properties of each musical composition are then performed by applying hurst exponent and fourier spectral analyses on these music walk sequences our results show that music shares the similar fractal properties of a fractional brownian motion fbm that is music displays an anti persistent trend in its tone changes melody over decades of musical notes and music sequence exhibits generally the 1 f type spectrum fractal property with apparently two different values in two different temporal scales

fractal dimension applied to plant identification this article discusses methods to identify plants by analysing leaf complexity based on estimating their fractal dimension leaves were analyzed according to the complexity of their internal and external shapes a computational program was developed to process analyze and extract the features of leaf images thereby allowing for automatic plant identification results are presented from two experiments the first to identify plant species from the brazilian atlantic forest and brazilian cerrado scrublands using fifty leaf samples from ten different species and the second to identify four different species from genus passiflora using twenty leaf samples for each class a comparison is made of two methods to estimate fractal dimension box counting and multiscale minkowski the results are discussed to determine the best approach to analyze shape complexity based on the performance of the technique when estimating fractal dimension and identifying plants