regularized decomposition of high dimensional multistage stochastic programs with markov uncertainty we develop a quadratic regularization approach for the solution of high dimensional multistage stochastic optimization problems characterized by a potentially large number of time periods stages e g hundreds a high dimensional resource state variable and a markov information process the resulting algorithms are shown to converge to an optimal policy after a finite number of iterations under mild technical assumptions computational experiments are conducted using the setting of optimizing energy storage over a large transmission grid which motivates both the spatial and temporal dimensions of our problem our numerical results indicate that the proposed methods exhibit significantly faster convergence than their classical counterparts with greater gains observed for higher dimensional problems

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

matrix free subcell residual distribution for bernstein finite element discretizations of linear advection equations abstract in this work we introduce a new residual distribution rd framework for the design of bound preserving high resolution finite element schemes the continuous and discontinuous galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing led approximations to that end we perform mass lumping and redistribute the element residuals in a manner which guarantees the led property the hierarchical correction procedure for high order bernstein finite element discretizations involves localization to subcells and definition of bound preserving weights for subcell contributions using strong stability preserving ssp runge kutta methods for time integration we prove the validity of discrete maximum principles under cfl like time step restrictions the low order version of our method has roughly the same accuracy as the one derived from a piecewise multi linear approximation on a submesh with the same nodal points in high order extensions we use an element based flux corrected transport fct algorithm which can be interpreted as a nonlinear rd scheme the proposed led corrections are tailor made for matrix free implementations which avoid the rapidly growing cost of matrix assembly for high order bernstein elements the results for 1d 2d and 3d test problems compare favorably to those obtained with the best matrix based approaches

accelerating the regularized decomposition method for two stage stochastic linear problems practical improvements of the regularized decomposition algorithm for two stage stochastic problems are presented they are associated with the primal simplex method for solving subproblems a penalty formulation of the subproblems is used which facilitates crash and warm starts and allows more freedom when creating the model the computational results are highly encouraging

a sequential sampling procedure for stochastic programming we develop a sequential sampling procedure for a class of stochastic programs we assume that a sequence of feasible solutions with an optimal limit point is given as input to our procedure such a sequence can be generated by solving a series of sampling problems with increasing sample size or it can be found by any other viable method our procedure estimates the optimality gap of a candidate solution from this sequence if the point estimate of the optimality gap is sufficiently small according to our termination criterion then we stop otherwise we repeat with the next candidate solution from the sequence under an increased sample size we provide conditions under which this procedure i terminates with probability one and ii terminates with a solution that has a small optimality gap with a prespecified probability

fixed width sequential stopping rules for a class of stochastic programs monte carlo sampling based methods are frequently used in stochastic programming when exact solution is not possible a critical component of monte carlo sampling based methods is determining when to stop sampling to ensure the desired quality of the solutions in this paper we develop stopping rules for sequential sampling procedures that depend on the width of an optimality gap confidence interval estimator the procedures solve a sequence of sampling approximations with increasing sample size to generate solutions and stop when the width of the confidence interval on the current solution s optimality gap plus an inflation factor fall below a prespecified value epsilon we first present a method that takes the schedule of sample sizes as an input and provide guidelines on the growth of sample sizes then we present a method that increases the sample sizes according to the current estimates of the optimality gap the larger the estimates the larger the increases in sample sizes we provide conditi

a regularized decomposition method for minimizing a sum of polyhedral functions a problem of minimizing a sum of many convex piecewise linear functions is considered in view of applications to two stage linear programming where objectives are marginal values of lower level problems it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise linear induced feasibility constraints we propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component the master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts the method goes through nonbasic points in general and is finitely convergent without any nondegeneracy assumptions next we present a special technique for solving the regularized master problem that uses an active set strategy and qr factorization and exploits the structure of the master finally some numerical evidence is given

the benders decomposition algorithm a literature review the benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems this paper presents a state of the art survey of this algorithm emphasizing its use in combinatorial optimization we discuss the classical algorithm the impact of the problem formulation on its convergence and the relationship to other decomposition methods we introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm the taxonomy provides the framework to synthesize the literature and to identify shortcomings trends and potential research directions we also discuss the use of the benders decomposition to develop efficient meta heuristics describe the limitations of the classical algorithm and present extensions enabling its application to a broader range of problems

an adaptive partition based approach for solving two stage stochastic programs with fixed recourse we study an adaptive partition based approach for solving two stage stochastic programs with fixed recourse a partition based formulation is a relaxation of the original stochastic program and we study a finitely converging algorithm in which the partition is adaptively adjusted until it yields an optimal solution a solution guided refinement strategy is developed to refine the partition by exploiting the relaxation solution obtained from a partition in addition to refinement we show that in the case of stochastic linear programs it is possible to merge some components in a partition without weakening the corresponding relaxation bound thus allowing the partition size to be kept small we also show that for stochastic linear programs with simple recourse there exists a small partition that yields an optimal solution the size of this partition is independent of the number of scenarios used in the model our computational results show that the proposed adaptive partition based approach converges v

monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws abstract using the theoretical framework of algebraic flux correction and invariant domain preserving schemes we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations nonlinear scalar conservation laws and hyperbolic systems in contrast to flux corrected transport fct algorithms that apply limited antidiffusive corrections to bound preserving low order solutions our new limiting strategy exploits the fact that these solutions can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and preserves appropriate local bounds there is no free parameter and no need for limit fluxes associated with the consistent mass matrix of time derivative term separately moreover the steady state limit of the nonlinear discrete problem is well defined and independent of the pseudo time step in the case study for the euler equations the components of the bar states are constrained sequentially to satisfy local maximum principles for the density velocity and specific total energy in addition to positivity preservation for the density and pressure the results of numerical experiments for standard test problems illustrate the ability of built in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner

high order accurate schemes for euler and navier stokes equations on staggered cartesian grids abstract we present here a new class of staggered schemes for solving the compressible 1d euler equations in internal energy formulation on uniform grids the schemes are applicable to arbitrary equation of states and can be extended to high order of accuracy in both time and space on smooth flows high order accuracy in time is reached thanks to cauchy kovalevskaya procedure modifications on the initial schemes are performed to give sufficient conditions for stability on 1d wave equations results obtained for wave equations are extended to 1d euler equations and then to 2d compressible navier stokes equations using directional splitting methods results on the conservation of total energy are given proper shock capturing is observed experimentally numerical results are provided up to 4th order accuracy in 1d and 2d

a probability metrics approach for reducing the bias of optimality gap estimators in two stage stochastic linear programming monte carlo sampling based estimators of optimality gaps for stochastic programs are known to be biased when bias is a prominent factor estimates of optimality gaps tend to be large on average even for high quality solutions this diminishes our ability to recognize high quality solutions in this paper we present a method for reducing the bias of the optimality gap estimators for two stage stochastic linear programs with recourse via a probability metrics approach motivated by stability results in stochastic programming we apply this method to the averaged two replication procedure a2rp by partitioning the observations in an effort to reduce bias which can be done in polynomial time in sample size we call the resulting procedure the averaged two replication procedure with bias reduction a2rp b we provide conditions under which a2rp b produces strongly consistent point estimators and an asymptotically valid confidence interval we illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small sample behavior of a2rp b on a number of two stage stochastic linear programs from the literature our computational results indicate that the procedure effectively reduces bias we also observe variance reduction in certain circumstances

staggered grid residual distribution scheme for lagrangian hydrodynamics this paper is focused on the residual distribution rd interpretation of the dobrev kolev and rieben scheme siam j sci comput 34 2012 pp b606 b641 for the numerical solution of the euler equations in lagrangian form the first ingredient of the original scheme is the staggered grid formulation which uses continuous node based finite element approximations for the kinematic variables and cell centered discontinuous finite elements for the thermodynamic parameters the second ingredient of the dobrev et al scheme is an artificial viscosity technique applied in order to make possible the computation of strong discontinuities the aim of this paper is to provide an efficient mass matrix diagonalization method in order to avoid the inversion of the global sparse mass matrix while keeping all the accuracy properties and to construct a parameter free stabilization of the scheme to get rid of the artificial viscosity in addition we study the conservation and entropy properties of the constructed

explicit runge kutta residual distribution schemes for time dependent problems second order case in this paper we construct spatially consistent explicit second order discretizations for time dependent hyperbolic problems starting from a given residual distribution rd discrete approximation of the steady operator we review the existing knowledge on consistent rd mass matrices and highlight the relations between different definitions we then introduce our explicit approach which is based on three main ingredients first recast the rd discretization as a stabilized galerkin scheme then use a shifted time discretization in the stabilization operator and lastly apply high order mass lumping on the galerkin component of the discretization the discussion is particularly relevant for schemes of the residual distribution type 18 3 which we will use for all our numerical experiments however similar ideas can be used in the context of residual based finite volume discretizations such as the ones proposed in 14 12 the schemes are tested on a wide variety of classical problems confirming the theoretical expectations

simulation based confidence bounds for two stage stochastic programs this paper provides a rigorous asymptotic analysis and justification of upper and lower confidence bounds proposed by dantzig and infanger a probabilistic lower bound for two stage stochastic programs stanford university ca 1995 for an iterative sampling based decomposition algorithm introduced by dantzig and glynn ann oper res 22 1 21 1990 and infanger ann oper res 39 41 67 1992 for solving two stage stochastic programs the paper provides confidence bounds in the presence of both independent sampling across iterations and when common samples are used across different iterations confidence bounds for sample average approximation then follow as a special case extensions of the theory to cover use of variance reduction and the dropping of cuts are also presented an extensive empirical investigation of the performance of these bounds establishes that the bounds perform reasonably on realistic problems

assessing solution quality in stochastic programs determining whether a solution is of high quality optimal or near optimal is fundamental in optimization theory and algorithms in this paper we develop monte carlo sampling based procedures for assessing solution quality in stochastic programs quality is defined via the optimality gap and our procedures output is a confidence interval on this gap we review a multiple replications procedure that requires solution of say 30 optimization problems and then we present a result that justifies a computationally simplified single replication procedure that only requires solving one optimization problem even though the single replication procedure is computationally significantly less demanding the resulting confidence interval might have low coverage probability for small sample sizes for some problems we provide variants of this procedure that require two replications instead of one and that perform better empirically we present computational results for a newsvendor problem and for two stage stochastic linear programs from the literature we also discuss when the procedures perform well and when they fail and we propose using optimal solutions to strengthen the performance of our procedures

probe waveforms and deconvolution in the experimental determination of elastic green s functions we propose a new time domain method for the experimental determination of the impulse response of linear systems the technique centers around the use of specifically designed probe waveforms these waveforms are particular c infty approximations to the dirac delta function and the heaviside unit step function and lead to a subsequent time domain deconvolution problem which can be implemented as a cauchy initial value problem this approach allows for continuous deconvolution a powerful option in the presence of noise we orient the discussion to the context of acoustic emission and elastic green s functions and present several numerical reconstructions of sharp signals from smooth synthetic data

a general framework to construct schemes satisfying additional conservation relations application to entropy conservative and entropy dissipative schemes abstract we are interested in the approximation of a steady hyperbolic problem in some cases the solution can satisfy an additional conservation relation at least when it is smooth this is the case of an entropy in this paper we show starting from the discretization of the original pde how to construct a scheme that is consistent with the original pde and the additional conservation relation since one interesting example is given by the systems endowed by an entropy we provide one explicit solution and show that the accuracy of the new scheme is at most degraded by one order in the case of a discontinuous galerkin scheme and a residual distribution scheme we show how not to degrade the accuracy this improves the recent results obtained in 1 2 3 4 in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved we study the behavior of the method on a non linear scalar problem however the method is not restricted to scalar problems

high order staggered schemes for compressible hydrodynamics weak consistency and numerical validation abstract staggered grids schemes formulated in internal energy are commonly used for cfd applications in industrial context here we prove the consistency of a class of high order lagrange remap staggered schemes for solving the euler equations in 1d and 2d on cartesian grids the main result of the paper is that using an a posteriori internal energy corrector the lagrangian schemes are proved to be conservative in mass momentum and total energy and to be weakly consistent with the 1d lagrangian formulation of the euler equations extension in 2d is done using directional splitting methods and face staggering numerical examples in both 1d and 2d illustrate the accuracy the convergence and the robustness of the schemes

a multicut algorithm for two stage stochastic linear programs outer linearization methods such as van slyke and wets s l shaped method for stochastic linear programs generally apply a single cut on the nonlinear objective at each major iteration the structure of stochastic programs allows for several cuts to be placed at once this paper describes a multicut algorithm to carry out this procedure it presents experimental and theoretical justification for reductions in major iterations

second order invariant domain preserving ale approximation of hyperbolic systems abstract in this paper we introduce an invariant domain preserving arbitrary lagrangian eulerian method for solving hyperbolic systems the time stepping is explicit and the approximation in space is done with continuous finite elements the method is second order in space and made invariant domain preserving by using a newly introduced convex limiting technique