an alternative least squares formulation of the navier stokes equations with improved mass conservation the focus of this paper is on incompressible flows in three dimensions modeled by least squares finite element methods lsfem and using a novel reformulation of the navier stokes equations lsfem are attractive because the resulting discrete equations yield symmetric positive definite systems of algebraic equations and the functional provides both a local and global error measure on the other hand it has been documented for existing reformulations that certain types of boundary conditions and high aspect ratio domains can yield very poor mass conservation it has also been documented that improved mass conservation with lsfem can be achieved by strengthening the coupling between the pressure and velocity the new reformulation presented here is demonstrated to provide both improved multigrid convergence rates because it is differentially diagonally dominant and improved mass conservation over existing methods because it increases the pressure velocity coupling along the inflow and outflow boundaries

family optimal investment strategy for a random household expenditure under the cev model abstract the family optimal investment strategy for a random household expenditure is investigated assume that the family is allowed to invest in a financial market consisting of one risk free asset and one risky asset whose price process satisfies the constant elasticity of variance cev model the target is to maximize the expected exponential utility of the family terminal wealth and obtain the optimal investment strategy employing techniques of stochastic control theory and the dual theory we derive the hamilton jacobi bellman hjb equation and obtain an approximate expression for the optimal investment strategy in the slow fluctuating regime numerical examples are presented to illustrate the effects of parameters on the optimal strategies

enhanced mass conservation in least squares methods for navier stokes equations there are many applications of the least squares finite element method for the numerical solution of partial differential equations because of a number of benefits that the least squares method has however one of the most well known drawbacks of the least squares finite element method is the lack of exact discrete mass conservation in some contexts due to the fact that the least squares method minimizes the continuity equation in l 2 norm in this paper we explore the reason for the mass loss and provide new approaches to retain the mass even in a severely underresolved grid

fosll method for the eddy current problem with three dimensional edge singularities in the case that the domain has reentrant edges the standard finite element method loses its global accuracy because of singularities on the boundary to overcome this difficulty foslla is applied in this paper foslla is a methodology for solving pdes using the dual operator here a modified foslla method is developed that employs a partially weighted functional and allows the use of a standard finite element scheme without losing global accuracy

nonadiabatic transitions for a decaying two level system geometrical and dynamical contributions we study the landau zener problem for a decaying two level system described by a non hermitian hamiltonian depending analytically on time use of a super adiabatic basis allows us to calculate the survival probability p in the slow sweep limit without specifying the hamiltonian explicitly it is found that p consists of a dynamical and a geometrical factor the former is determined by the complex adiabatic eigenvalues e t only whereas the latter solely requires the knowledge of t the ratio of the components of each of the adiabatic eigenstates both factors can be split into a universal one depending only on the complex level crossing points and a nonuniversal one involving the full time dependence of e t and t this general result is applied to the akulin schleich model where the initial upper level is damped with damping constant for analytic power law sweeps we find that stuckelberg oscillations of p exist for smaller than a critical value c and disappear for c a physical interpretation of this behaviour will be presented by use of a damped harmonic oscillator

analysis of first order system least squares fosls for elliptic problems with discontinuous coefficients part i first order system least squares fosls is a recently developed methodology for solving partial differential equations among its advantages are that the finite element spaces are not restricted by the inf sup condition imposed for example on mixed methods and that the least squares functional itself serves as an appropriate error measure this paper studies the fosls approach for scalar second order elliptic boundary value problems with discontinuous coefficients irregular boundaries and mixed boundary conditions a least squares functional is defined and ellipticity is established in a natural norm of an appropriately scaled least squares bilinear form for some geometries this ellipticity is independent of the size of the jumps in the coefficients the occurrence of singularities at interface corners cross points reentrant corners and irregular boundary points is discussed and a basis of singular functions with local support around singular points is established a companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques also independent of the size of coefficient jumps for some geometries

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

hybrid first order system least squares finite element methods with application to stokes equations this paper combines first order system least squares fosls with first order system ll fosll to create a hybrid method the fosls approach minimizes the error bf e h bf u h bf u over a finite element subspace cal v h in the operator norm min bf u h in cal v h l bf u h bf u the fosll method looks for an approximation in the range of l setting bf u h l bf w h and choosing bf w h in cal w h a standard finite element space fosll minimizes the l 2 norm of the error over l cal w h min bf w h in cal w h l bf w h bf u fosls enjoys a locally sharp globally reliable and easily computable a posteriori error estimate while fosll does not however fosll has the major advantage that it applies to problems that do not exhibit enough smoothness to enable the full advantages that the fosls approach otherwise provides the hybrid method attempts to retain the best properties of both

efficiency based adaptive local refinement for first order system least squares formulations in this paper we propose new adaptive local refinement alr strategies for first order system least squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration the goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements to accomplish this the refinement decision at each refinement level is determined based on optimizing efficiency measures that take into account both error reduction and computational cost two efficiency measures are discussed predicted error reduction and predicted computational cost these methods are first applied to a two dimensional 2d poisson problem with steep gradients and the results are compared with the threshold based methods described in w dorfler siam j numer anal 33 1996 pp 1106 1124 next these methods are applied to a 2d reduced model of the incompressible resistive magnetohydrodynamic equations these equations are used to simulate instabilities in a large aspect ratio tokamak we show that by using the new alr strategies on this system we are able to resolve the physics using only 10 percent of the computational cost used to approximate the solutions on a uniformly refined mesh within the same error tolerance

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

the cev model and its application to financial markets with volatility uncertainty abstract we survey the financial markets whose risks are caused by uncertain volatilities the financial markets focus on the assets which are effectively allocated in one risk free asset and one risky asset whose price process is governed by the constant elasticity of variance c e v for short model which contains the g brownian motion rather than the classical brownian motion such the c e v model which includes the g brownian motion utilized to financial markets is the extension of the classical c e v model applying the concept of arbitrage and the properties of g expectation we consider stock price dynamics which exclude arbitrage opportunities moreover the interval of no arbitrage price for the general european contingent claims is found in the markovian case

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

optimal investment policies for a firm with a random risk process exponential utility and minimizing the probability of ruin we consider a firm that is faced with an uncontrollable stochastic cash flow or random risk process there is one investment opportunity a risky stock and we study the optimal investment decision for such firms there is a fundamental incompleteness in the market in that the risk to the investor of going bankrupt cannot be eliminated under any investment strategy since the random risk process ensures that there is always a positive probability of ruin bankruptcy we therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin in particular we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level this policy also minimizes the probability of ruin we prove that when there is no risk free interest rate this policy is equivalent to the policy that maximizes utility from terminal wealth for a fixed terminal time when the firm has an exponential utility function this validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility when there is a positive risk free interest rate the conjecture is shown to be false we also solve for the optimal policy for the related objective of minimizing the expected discounted penalty paid upon ruin

optimal investment problem under non extensive statistical mechanics abstract the optimal investment problem in which the asset price process is modeled by the non extensive statistical mechanics is studied in this paper by the methods of deterministic control and the dynamic programming we obtain the optimal strategy with logarithmic utility function power utility function and quadratic utility function respectively moreover the numerical results indicate that the optimal investment strategy is affected by the non extensive parameter the proportion invested in the risky asset decreases as the wealth increases under quadratic utility function but it remains unchanged under power utility and logarithmic utility function

the fractional and mixed fractional cev model abstract the continuous observation of the financial markets has identified some stylized facts which challenge the conventional assumptions promoting the born of new approaches on the one hand the long range dependence has been faced replacing the traditional gauss wiener process brownian motion characterized by stationary independent increments by a fractional version on the other hand the cev model addresses the leverage effect and smile skew phenomena efficiently in this paper these two insights are merging and both the fractional and mixed fractional extensions for the cev model are developed using the fractional versions of both the ito s calculus and the fokker planck equation the transition probability density function of the asset price is obtained as the solution of a non stationary feller process with time varying coefficients getting an analytical valuation formula for a european call option besides the greeks are computed and compared with the standard case

nonlinear energy transfer in classical and quantum systems in this paper we investigate the effect of slowly varying parameters on the energy transfer in a weakly coupled system for definiteness we consider a system of two nonlinear oscillators in which the directly excited first oscillator with constant parameter is attached to the oscillator with slowly time varying frequency it is proved that the equations of the slow passage through resonance in this system are identical to the equations of nonlinear landau zener lz tunneling three types of dynamical behavior are distinguished namely quasilinear moderately nonlinear and strongly nonlinear ones quasilinear systems exhibit a gradual energy transfer from the excited to the attached oscillator while moderately nonlinear systems are characterized by an abrupt transition from the energy localization on the excited oscillator to the localization on the attached oscillator in strongly nonlinear systems the transition from the energy localization to strong energy exchange between the oscillators is revealed explicit approximate solutions describing the transient processes in moderately and strongly nonlinear systems are suggested correctness of the constructed approximations is confirmed by numerical results the results presented in this paper in addition to providing an analytical framework for understanding the transient dynamics suggest an approximate procedure for solving the nonlinear lz problem with arbitrary initial conditions over a finite time interval

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

numerical solution of an optimal investment problem with proportional transaction costs this paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite horizon optimal investment problem with proportional transaction costs the problem is initially posed in terms of an evolutive hjb equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable the proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain a characteristics method for time discretization to deal with the large gradients of the solution a newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle free boundary feature moreover piecewise linear lagrange finite elements for spatial discretization are considered numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in dai and yi 2009 6

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

first order system cl cl fosll scalar elliptic partial differential equations the l2 norm version of first order system least squares fosls attempts to reformulate a given system of partial differential equations so that applying a least squares principle yields a functional whose bilinear part is h1 elliptic this ellipticity means that the minimization process amounts to solving a weakly coupled system of poisson like scalar equations an unfortunate limitation of the l2 norm fosls approach is that this product h1 equivalence generally requires sufficient smoothness of the original problem inverse norm fosls overcomes this limitation but at a substantial loss of real efficiency the fosll approach introduced here is a promising alternative that is based on recasting the original problem as a minimization principle involving the adjoint equations this paper provides a theoretical foundation for the fosll methodology and illustrates its performance by applying it numerically to several examples results for the so called two stage approach applied to discontinuous coefficient problems show promising robustness and optimality indeed fosll appears to exhibit the generality of the inverse norm fosls approach while retaining the full efficiency of the l2 norm approach

fresnel integrals and irreversible energy transfer in an oscillatory system with time dependent parameters we demonstrate that in significant limiting cases the problem of irreversible energy transfer in an oscillatory system with time dependent parameters can be efficiently solved in terms of the fresnel integrals for definiteness we consider a system of two weakly coupled linear oscillators in which the first oscillator with constant parameters is excited by an initial impulse whereas the coupled oscillator with a slowly varying frequency is initially at rest but then acts as an energy trap we show that the evolution equations of the slow passage through resonance are identical to the equations of the landau zener tunneling problem and therefore the suggested asymptotic solution of the classical problem provides a simple analytic description of the quantum landau zener tunneling with arbitrary initial conditions over a finite time interval a correctness of approximations is confirmed by numerical simulations

bifurcation structure of the special class of nonstationary regimes emerging in the 2d inertially coupled unit cell model analytical study abstract present work is devoted to the analytical investigation of the bifurcation structure of special class of nonstationary low energy regimes emerging in the locally resonant unit cell model system under consideration comprises an outer mass with internal rotator and subject to the 2d nonlinear local potential these regimes are characterized by the slow purely rotational motion of the rotator synchronized with the periodic energy beats between the axial and the lateral vibrations of the outer element thus the angular speed of the rotator and the beating frequency of the outer element satisfy the 1 2 resonance condition in the present study these regimes are referred to as regimes of synchronous nonlinear beats rsnb using the regular muti scale analysis in the limit of low energy excitation we derive the slow flow model to showcase the evolution of rsnbs we used the special poincare map technique applied on the slow flow model results of the poincare sections unveiled some interesting local bifurcations undergone by these regimes further analysis of the slow flow model enabled us to describe the rsnbs analytically as well as exposed their entire bifurcation structure the bifurcation analysis has shown the coexistence of several branches of rsnbs corresponding to the regimes of weak and strong two dimensional recurrent energy channeling we substantiate the results of the analytical study with numerical simulations of the full model and find them to be in the very good agreement