wait and judge scenario optimization we consider convex optimization problems with uncertain probabilistically described constraints in this context scenario optimization is a well recognized methodology where a sample of the constraints is used to describe uncertainty one says that the scenario solution generalizes well or has a high robustness level if it satisfies most of the other constraints besides those in the sample over the past 10 years the main theoretical investigations on the scenario approach have related the robustness level of the scenario solution to the number of optimization variables this paper breaks into the new paradigm that the robustness level is a posteriori evaluated after the solution is computed and the actual number of the so called support constraints is assessed wait and judge a new theory is presented which shows that a posteriori observing k support constraints in dimension d k allows one to draw conclusions close to those obtainable when the problem is from the outset in dimension k this new theory provides evaluations of the robustness that largely outperform those carried out based on the number of optimization variables

near wall treatment for the simulation of turbulent flow by the cumulant lattice boltzmann method abstract we present a new wall function implementation for the cumulant lattice boltzmann method that sets a partial slip velocity on the wall by computing a skin frictional coefficient our approach uses local information and is particularly appropriate for implementations on general purpose graphics processing units the validation of the model has been conducted by performing numerical simulations of the turbulent channel flow test case with different grid resolutions and for different reynolds numbers the results showed encouraging agreement with direct numerical simulation data for velocity profile reynolds shear and normal stresses

introducing a new kinetic model which admits an h theorem for simulating the nearly incompressible fluid flows we present a new kinetic model for simulating the nearly incompressible fluid flows the model uses constant speed particles which are free to move in all directions in the new model the maxwellian is replaced by a fractional form while the navier stokes and continuity equations are recovered it is known that the conventional lattice boltzmann model lbm with polynomial equilibrium distribution function cannot admit an h theorem wagner 1998 9 in the present work we show that the new model admits an h theorem and the numerical schemes which stem from the new model are more stable than the conventional lbm for the streaming stage two different approaches namely the lattice based and the discontinuous galerkin based schemes are introduced the former is more cost effective than the conventional lattice boltzmann models while it maintains their simplicity the latter is a higher order spectral element method using which one can employ non uniform triangular elements with high accuracy and geometrical flexibility unlike the conventional lbm the new model is derived from continuous relations hence it does not require a symmetric discrete velocity model the accuracy and stability of the model have been verified by simulating three benchmark problems the plane couette flow lid driven square cavity and flow around an impulsively started cylinder have been simulated the results of the present work are in excellent agreement with the exact solutions and with the results reported by others

a stable richardson iteration method for complex linear systems the richardson iteration method is conceptually simple as well as easy to program and parallelize this makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues we change the ordering of the relaxation parameters of a richardson iteration method proposed by eiermann niethammer and varga for the solution of such problems the new method obtained is shown to be stable and to have better convergence properties

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

on the sample size of random convex programs with structured dependence on the uncertainty the scenario approach provides an intuitive method to address chance constrained problems arising in control design for uncertain systems it addresses these problems by replacing the chance constraint with a finite number of sampled constraints scenarios the sample size critically depends on helly s dimension a quantity always upper bounded by the number of decision variables however this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability we derive improved bounds of helly s dimension for problems where the chance constraint has certain structural properties the improved bounds lower the number of scenarios required for these problems leading both to improved objective value and reduced computational complexity our results are generally applicable to randomized model predictive control of chance constrained linear systems with additive uncertainty and affine disturbance feedback

a nonlinearly preconditioned inexact newton algorithm for steady state lattice boltzmann equations most existing methods for calculating the steady state solution of the lattice boltzmann equations are based on pseudo time stepping which often requires a large number of time steps especially for high reynolds number problems to calculate the steady state solution directly without the time integration in this paper we propose and study a nonlinearly preconditioned inexact newton algorithm with a domain decomposition based linear solver for parallelization more precisely the proposed algorithmic framework involves an implicit second order discretization a two level inexact newton method and a nonlinear elimination preconditioner to accelerate the convergence of newton iteration a nonstandard pollution removing coarse space is introduced for the two level method numerical experiments are presented to demonstrate the robustness and efficiency of the algorithm especially for problems at a high reynolds number a comparison is also included to show the superiority of the proposed approach over o

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

the conjugate function method and conformal mappings in multiply connected domains the conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains in this paper the conjugate function method is generalized for

on the computational complexity and generalization properties of multi stage and stage wise coupled scenario programs abstract we discuss the computational complexity and feasibility properties of scenario sampling techniques for uncertain optimization programs we propose an alternative way of dealing with a special class of stage wise coupled programs and compare it with existing methods in the literature in terms of feasibility and computational complexity we identify trade offs between different methods depending on the problem structure and the desired probability of constraint satisfaction to illustrate our results an example from the area of approximate dynamic programming is considered

acceleration of steady state lattice boltzmann simulations on non uniform mesh using local time step method in this paper the local time step method is applied to the lattice boltzmann method lbm on non uniform grid generalized form of interpolation supplemented lbm gilbm for steady flow simulations the local time step method is known to be an effective technique to accelerate the solution to the converged steady state on non uniform grid first code validation for global time step method is performed by solving flow around a cylinder at low reynolds numbers our results show good agreement with previous studies two dimensional flow simulations around an airfoil were performed in order to validate the code using the local time step method and the results are compared with the global time step solutions the pressure distribution and the aerodynamic coefficients of global and local time step results are in good agreement they are also consistent with the previous studies at the same time cpu time required to obtain the steady state solutions using the local time step are reduced by 70 80 compared with that using global time step

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

semi lagrangian implicit bhatnagar gross krook collision model for the finite volume discrete boltzmann method in order to increase the accuracy of temporal solutions reduce the computational cost of time marching and improve the stability associated with collisions for the finite volume discrete boltzmann method an advanced implicit bhatnagar gross krook bgk collision model using a semi lagrangian approach is proposed in this paper unlike existing models in which the implicit bgk collision is resolved either by a temporal extrapolation or by a variable transformation the proposed model removes the implicitness by tracing the particle distribution functions pdfs back in time along their characteristic paths during the collision process an interpolation scheme is needed to evaluate the pdfs at the traced back locations by using the first order interpolation the resulting model allows for the straightforward replacement of f eq n 1 by f eq n no matter where it appears after comparing the proposed model with the existing models under different numerical conditions e g different flux schemes and time marching schemes and using the proposed model to successfully modify the variable transformation technique three conclusions can be drawn first the proposed model can improve the accuracy by almost an order of magnitude second it can slightly reduce the computational cost therefore the proposed scheme improves accuracy without extra cost finally the proposed model can significantly improve the t limit compared to the temporal interpolation model while having the same t limit as the variable transformation approach the proposed scheme with a second order interpolation is also developed and tested however that technique displays no advantage over the simple first order interpolation approach both numerical and theoretical analyses are also provided to explain why the developed implicit scheme with simple first order interpolation can outperform the same scheme with second order interpolation as well as the existing temporal extrapolation and variable transformation schemes

a charge simulation method for the numerical conformal mapping of interior exterior and doubly connected domains abstract a simple numerical method is described for computing the following three conformal maps a from a domain interior to a closed jordan curve onto the interior of the unit disk b from a domain exterior to a closed jordan curve onto the exterior of the unit disk and c from a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on the charge simulation method or the method of fundamental solutions applied to the dirichlet problem of laplace s equation in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials without integration typical examples show that numerical results of high accuracy can be obtained by the simple method if the domains have no concave corners

a charge simulation method for numerical conformal mapping onto circular and radial slit domains a simple numerical method is described for computing the following two conformal maps a from a domain exterior to closed jordan curves onto a circular slit domain and b from a domain exterior to closed jordan curves onto a radial slit domain they constitute a dual problem and can be computed in a dual way the numerical method is based on the charge simulation method or the method of fundamental solutions applied to the dirichlet problem of laplace s equation in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials the unknown coefficients are determined by the collocation condition imposed on the real part or the imaginary part the modulus or the argument of the approximate mapping function effectiveness of the method is demonstrated by some typical examples

numerical conformal mappings of bounded multiply connected domains by the charge simulation method we present here a simple method of numerical conformal mappings from bounded multiply connected domains onto the bounded canonical slit domains of nehari i e a disk with concentric circular slits and an annulus with concentric circular slits in this method we express the mapping functions in terms of a pair of conjugate harmonic functions and approximate them using the charge simulation method by a linear combination of complex logarithmic functions some numerical examples show the effectiveness of the method

numerical conformal mapping of multiply connected regions onto the fifth category of koebe s canonical slit regions abstract this paper presents a boundary integral method for approximating the conformal mapping from bounded multiply connected regions onto the fifth category of koebe s classical canonical slit regions the method is based on a uniquely solvable boundary integral equation with generalized neumann kernel the results of some test calculations illustrate the performance of the presented method

high order upwind compact finite difference lattice boltzmann method for viscous incompressible flows abstract in this work a high order upwind compact finite difference lattice boltzmann method ucdlbm is developed to efficiently solve viscous incompressible flow problems a fifth order upwind compact difference scheme is adopted to discretize the spatial derivatives of the lattice boltzmann equation and the third order total variation diminishing runge kutta scheme is utilized for the discretization of the temporal term compared to the existing central compact finite difference lattice boltzmann method cfdlbm the present ucdlbm can prevent non physical oscillations without filtering due to the natural dissipative property of upwind schemes three benchmark problems involving the taylor green vortex problem the doubly periodic shear layer flow problem and the lid driven square cavity flow problem are numerically solved to demonstrate the accuracy and efficiency of the present method numerical results computed are in good agreement with the analytical solution or other available numerical results and the present ucdlbm is less time consuming than the cfdlbm without degenerating the order of accuracy of the numerical solutions

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

equivalence of nystrom s method and fourier methods for the numerical solution of fredholm integral equations on demontre un resultat d equivalence de la methode de nystrom et de la methode de fourier avec des fonctions orthogonales pour la solution numerique des equations integrales de fredholm on en deduit des estimations d erreur pour la methode de nystrom on decrit exactement la regle de quadrature pour l integration numerique

high order finite elements applied to the discrete boltzmann equation a discontinuous galerkin approach for solving the discrete boltzmann equation is presented allowing to compute approximate solutions for fluid flow problems based on a two dimensional high order finite element and an explicit euler time stepping scheme the d2q9 model is discretized and the results are compared to the exact solution of the navier stokes equation four numerical examples are considered including stationary and instationary problems with curved boundaries it is demonstrated that the proposed method allows to obtain the desired highly efficient exponential convergence copyright 2006 john wiley sons ltd

a fully implicit method for lattice boltzmann equations existing approaches for solving the lattice boltzmann equations with finite difference methods are explicit and semi implicit both have certain stability constraints on the time step size in this work a fully implicit second order finite difference scheme is developed we focus on a parallel highly scalable newton krylov ras algorithm for the solution of a large sparse nonlinear system of equations arising at each time step here ras is a restricted additive schwarz preconditioner based on a first order spatial discretization we show numerically that by using the fully implicit method the time step size is no longer constrained by the cfl condition and the newton krylov ras algorithm is scalable on a supercomputer with more than ten thousand processors moreover to calculate the steady state solution we investigate an adaptive time stepping strategy the total compute time required by the implicit method with adaptive time stepping is much smaller than that of an explicit method for several t

the scenario approach for stochastic model predictive control with bounds on closed loop constraint violations many practical applications in control require that constraints on the inputs and states of the system are respected while some performance criterion is optimized in the presence of model uncertainties or disturbances it is often sufficient to satisfy the state constraints for at least a prescribed share of the time such as in building climate control or load mitigation for wind turbines for such systems this paper presents a new method of scenario based model predictive control scmpc the basic idea is to optimize the control inputs over a finite horizon subject to robust constraint satisfaction under a finite number of random scenarios of the uncertainty and or disturbances previous scmpc approaches have suffered from a substantial gap between the rate of constraint violations specified in the optimal control problem and that actually observed in closed loop operation of the controlled system this paper identifies the two theoretical explanations for this gap first accounting for the special structure of the optimal control problem leads to a substantial reduction of the problem dimension second the probabilistic constraints have to be interpreted as average in time rather than pointwise in time based on these insights a novel scmpc method can be devised for general linear systems with additive and multiplicative disturbances for which the number of scenarios is significantly reduced the presented method retains the essential advantages of the general scmpc approach namely a low computational complexity and the ability to handle arbitrary probability distributions moreover the computational complexity can be adjusted by a sample and remove strategy

on implementation of boundary conditions in the application of finite volume lattice boltzmann method a new implementation of boundary condition based on the half covolume and bounce back rule for the non equilibrium distribution function for the finite volume lbm is proposed here the numerical simulation results for the expansion channel flow and driven cavity problem indicate that this method is workable for arbitrary meshes in addition the fourth order runge kutta scheme is found to be a practical way in the lbm to accelerate the calculation speed

implementation of a high order compact finite difference lattice boltzmann method in generalized curvilinear coordinates abstract in this work the implementation of a high order compact finite difference lattice boltzmann method cfdlbm is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non uniform grids the incompressible form of the discrete boltzmann equation with the bhatnagar gross krook bgk approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates herein the spatial derivatives in the resulting lattice boltzmann lb equation in the computational plane are discretized by using the fourth order compact finite difference scheme and the temporal term is discretized with the fourth order runge kutta scheme to provide an accurate and efficient incompressible flow solver a high order spectral type low pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions flow non linearities and grid non uniformity all boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates the accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems a sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are unsteady couette flow and steady flow in a 2 d cavity with non uniform grid and steady and unsteady flows over a circular cylinder and the naca0012 hydrofoil at different flow conditions results obtained for the above test cases are in good agreement with the existing numerical and experimental results the study shows the present solution methodology based on the implementation of the high order compact finite difference lattice boltzmann method cfdlbm in the generalized curvilinear coordinates is robust efficient and accurate for solving steady and unsteady incompressible flows over practical geometries

fast conformal mapping of multiply connected regions abstract an iterative method is presented which constructs for an unbounded region g with m holes and sufficiently smooth boundary a circular region h and a conformal mapping from h to g with the usual normalization both h and are uniquely determined by g with a few modifications the method can also be applied to a bounded region g with m holes the canonical region h is then the unit disc with m circular holes the proposed method also determines the centers and radii of the boundary circles of h and requires at each iterative step the solution of a riemann hilbert rh problem which has a unique solution numerically the rh problem can be treated efficiently by the method of successive conjugation using the fast fourier transform fft the iteration for the solution of the rh problem converges linearly the conformal mapping method converges quadratically the results of some test calculations exemplify the performance of the method

chebyshev collocation spectral lattice boltzmann method for simulation of low speed flows in this study the chebyshev collocation spectral lattice boltzmann method ccslbm is developed and assessed for the computation of low speed flows both steady and unsteady flows are considered here the discrete boltzmann equation with the bhatnagar gross krook approximation based on the pressure distribution function is considered and the space discretization is performed by the chebyshev collocation spectral method to achieve a highly accurate flow solver to provide accurate unsteady solutions the time integration of the temporal term in the lattice boltzmann equation is made by the fourth order runge kutta scheme to achieve numerical stability and accuracy physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used an iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows the main advantage of using the ccslbm over other high order accurate lattice boltzmann method lbm based flow solvers is the decay of the error at exponential rather than at polynomial rates note also that the ccslbm applied does not need any numerical dissipation or filtering for the solution to be stable leading to highly accurate solutions three two dimensional 2d test cases are simulated herein that are a regularized cavity the taylor vortex problem and doubly periodic shear layers the results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement the computational efficiency of the proposed solution methodology based on the ccslbm is also examined by comparison with those of the standard streaming collision classical lbm and two finite difference lbm solvers the study indicates that the ccslbm provides more accurate and efficient solutions than these lbm solvers in terms of cpu and memory usage and an exponential convergence is achieved rather than polynomial rates the solution methodology proposed the ccslbm is also extended to three dimensions and a 3d regularized cavity is simulated the corresponding results are presented and validated indications are that the ccslbm developed and applied herein is robust efficient and accurate for computing 2d and 3d low speed flows note also that high accuracy solutions obtained by applying the ccslbm can be used as benchmark solutions for the assessment of other lbm based flow solvers

an eulerian description of the streaming process in the lattice boltzmann equation this paper presents a novel strategy for solving discrete boltzmann equation dbe for simulation of fluid flows this strategy splits the solution procedure into streaming and collision steps as in the lattice boltzmann equation lbe method the streaming step can then be carried out by solving pure linear advection equations in an eulerian framework this offers two significant advantages over previous methods first the relationship between the relaxation parameter and the discretization of the collision term developed from the lbe method is directly applicable to the dbe method the resulting dbe collision step remains local and poses no constraint on time step second decoupling of the advection step from the collision step facilitates implicit discretization of the advection equation on arbitrary meshes an implicit unstructured dbe method is constructed based on this strategy and is evaluated using several test cases of flow over a backward facing step lid driven cavity flow and flow past a circular cylinder the speedup of convergence for some cases improves by a factor of about 20

a solution adaptive lattice boltzmann method for two dimensional incompressible viscous flows a stencil adaptive lattice boltzmann method lbm is developed in this paper it incorporates the stencil adaptive algorithm developed by ding and shu 26 for the solution of navier stokes n s equations into the lbm calculation based on the uniform mesh the stencil adaptive algorithm refines the mesh by two types of 5 points symmetric stencils which are used in an alternating sequence for increased refinement levels the two types of symmetric stencils can be easily combined to form a 9 points symmetric structure using the one dimensional second order interpolation recently developed by wu and shu 27 along the straight line and the d2q9 model the adaptive lbm calculation can be effectively carried out note that the interpolation coefficients are only related to the lattice velocity and stencil size hence the simplicity of lbm is not broken down and the accuracy is maintained due to the use of adaptive technique much less mesh points are required in the simulation as compared to the standard lbm as a consequence the computational efficiency is greatly enhanced the numerical simulation of two dimensional lid driven cavity flows is carried out accurate results and improved efficiency are reached in addition the steady and unsteady flows over a circular cylinder are simulated to demonstrate the capability of proposed method for handling problems with curved boundaries the obtained results compare well with data in the literature

a hybrid wavelet based adaptive immersed boundary finite difference lattice boltzmann method for two dimensional fluid structure interaction abstract a second generation wavelet based adaptive finite difference lattice boltzmann method fd lbm is developed in this paper in this approach the adaptive wavelet collocation method awcm is firstly to the best of our knowledge incorporated into the fd lbm according to the grid refinement criterion based on the wavelet amplitudes of density distribution functions an adaptive sparse grid is generated by the omission and addition of collocation points on the sparse grid the finite differences are used to approximate the derivatives to eliminate the special treatments in using the fd based derivative approximation near boundaries the immersed boundary method ibm is also introduced into fd lbm by using the adaptive technique the adaptive code requires much less grid points as compared to the uniform mesh code as a consequence the computational efficiency can be improved to justify the proposed method a series of test cases including fixed boundary cases and moving boundary cases are invested a good agreement between the present results and the data in previous literatures is obtained which demonstrates the accuracy and effectiveness of the present awcm ib lbm

lattice boltzmann method on quadtree grids for simulating fluid flow through porous media a new automatic algorithm abstract during the past two decades the lattice boltzmann lb method has been introduced as a class of computational fluid dynamic methods for fluid flow simulations in this method instead of solving the navier stocks equation the boltzmann equation is solved to simulate the flow of a fluid this method was originally developed based on uniform grids however in order to model complex geometries such as porous media it can be very slow in comparison with other techniques such as finite differences and finite elements to eliminate this limitation a number of studies have aimed to formulate the lattice boltzmann on the unstructured grids this paper deals with simulating fluid flow through a synthetic porous medium using the lb method and on the quadtree grid structure to this end the lb method was used on nonuniform grids coupled with a technique for image reconstruction which resulted in the quadtree grids for simulation of fluid flow through porous media accuracy and efficiency of this algorithm is compared against the conventional lb method based on uniform grids while the decrease in computational time in the proposed lb method on nonuniform grids is found to be significant regarding the size of the initial and reconstructed images the same level of accuracy is obtained when compared with the conventional lb method on uniform grids

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

an immersed boundary method for fluids using the xfem and the hydrodynamic boltzmann transport equation abstract this paper presents a stabilized finite element formulation of the hydrodynamic boltzmann transport equation hbte to predict nearly incompressible fluid flow the hbte is discretized with hermite polynomials in the velocity variable and a streamline upwind petrov galerkin formulation is used to discretize the spatial variable a nonlinear stabilization scheme is presented from which a simple linear stabilization scheme is constructed in contrast to the navier stokes ns equations the hbte is a first order equation and allows for conveniently enforcing dirichlet conditions along immersed boundaries a simple and efficient formulation for enforcing dirichlet boundary conditions is presented and its accuracy is studied for immersed boundaries captured by the extended finite element method xfem numerical experiments indicate that both the linear and non linear stabilization methods are sufficiently accurate and stable but the linear formulation reduces the computational cost significantly the accuracy of enforcing boundary conditions is satisfactory and shows second order convergence as the mesh is refined augmenting the boundary condition formulation with a penalty term increases the accuracy of enforcing the boundary condition constraints but may degrade the accuracy of the global solution comparisons with results of a single relaxation time lattice boltzmann method show that the proposed finite element method features greater robustness and lesser dependence of the computational costs on the level of mesh refinement

a spectral element discontinuous galerkin lattice boltzmann method for nearly incompressible flows we present a spectral element discontinuous galerkin lattice boltzmann method for solving nearly incompressible flows decoupling the collision step from the streaming step offers numerical stability at high reynolds numbers in the streaming step we employ high order spectral element discontinuous galerkin discretizations using a tensor product basis of one dimensional lagrange interpolation polynomials based on gauss lobatto legendre grids our scheme is cost effective with a fully diagonal mass matrix advancing time integration with the fourth order runge kutta method we present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous galerkin approach we show convergence studies for couette flows and demonstrate two benchmark cases with lid driven cavity flows for re 400 5000 and flows around an impulsively started cylinder for re 550 9500 computational results are compared with those of other theoretical and computational work that used a multigrid method a vortex method and a spectral element model

finite volume versus streaming based lattice boltzmann algorithm for fluid dynamics simulations a one to one accuracy and performance study a finite volume fv discretization method for the lattice boltzmann lb equation which combines high accuracy with limited computational cost is presented in order to assess the performance of the fv method we carry out a systematic comparison focused on accuracy and computational performances with the standard streaming lattice boltzmann equation algorithm in particular we aim at clarifying whether and in which conditions the proposed algorithm and more generally any fv algorithm can be taken as the method of choice in fluid dynamics lb simulations for this reason the comparative analysis is further extended to the case of realistic flows in particular thermally driven flows in turbulent conditions we report the successful simulation of high rayleigh number convective flow performed by a lattice boltzmann fv based algorithm with wall grid refinement

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

risk and complexity in scenario optimization scenario optimization is a broad methodology to perform optimization based on empirical knowledge one collects previous cases called scenarios for the set up in which optimization is being performed and makes a decision that is optimal for the cases that have been collected for convex optimization a solid theory has been developed that provides guarantees of performance and constraint satisfaction of the scenario solution in this paper we open a new direction of investigation the risk that a performance is not achieved or that constraints are violated is studied jointly with the complexity as precisely defined in the paper of the solution it is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk this result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios in particular no extra knowledge on the distribution by which scenarios are generated is assumed so that the result is broadly applicable this deep seated result unveils a fundamental and general structure of data driven optimization and suggests practical approaches for risk assessment

on the connection between compression learning and scenario based single stage and cascading optimization problems we investigate the connections between compression learning and scenario based optimization we first show how to strengthen or relax the consistency assumption at the basis of compression learning and provide novel learnability conditions for the underlying algorithms we then consider different constrained optimization problems affected by uncertainty represented by means of scenarios we show that the compression learning perspective provides a unifying framework for scenario based optimization since the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that satisfies some consistency assumption to illustrate this we revisit the scenario approach within the developed context moreover using the compression learning machinery we provide novel results on the probability of constraint violation for the class of cascading optimization problems

parallel slits map of bounded multiply connected regions abstract in this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a parallel slit region the method is based on some uniquely solvable boundary integral equations with adjoint classical adjoint generalized and modified neumann kernels these boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region some numerical examples are presented to illustrate the efficiency of the presented method

randomized solutions to convex programs with multiple chance constraints the scenario based optimization approach scenario approach provides an intuitive way of approximating the solution to chance constrained optimization programs based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty scenarios a key merit of this approach is that it neither requires explicit knowledge of the uncertainty set as in robust optimization nor of its probability distribution as in stochastic optimization the scenario approach is also computationally efficient because it only requires the solution to a convex optimization program even if the original chance constrained problem is nonconvex recent research has obtained a rigorous foundation for the scenario approach by establishing a direct link between the number of scenarios and bounds on the constraint violation probability these bounds are tight in the general case of an uncertain optimization problem with a single chance constraint this paper shows that the bounds can be improved

a scenario approach for non convex control design randomized optimization is an established tool for control design with modulated robustness while for uncertain convex programs there exist efficient randomized approaches this is not the case for non convex problems methods based on statistical learning theory are applicable to non convex problems but they usually are conservative in achieving the desired probabilistic guarantees in this paper we derive a novel scenario approach for a wide class of random non convex programs with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program but for all feasible solutions inside a set of a priori chosen complexity we also address measure theoretic issues for uncertain convex and non convex programs among the family of non convex control design problems that can be addressed via randomization we apply our scenario approach to stochastic model predictive control for chance constrained nonlinear control affine systems

a characteristic galerkin method for discrete boltzmann equation abstract the characteristic galerkin finite element method for the discrete boltzmann equation is presented to simulate fluid flows in complex geometries the inherent geometric flexibility of the finite element method permits the easy use of simple cartesian variables on unstructured meshes and the mesh clustering near large gradients the characteristic galerkin procedure with appropriate boundary condition results in accurate solutions with little numerical diffusion several test cases are conducted including unsteady couette flows lid driven cavity flows and steady flow past a circular cylinder on unstructured meshes the numerical results are in good agreement with previous analytical if applicable numerical and experimental results

investigation of the les wale turbulence model within the lattice boltzmann framework turbulence models which can perform the transition from laminar flow to fully developed turbulent flow are of key importance in industrial applications a promising approach is the les wale model which can be used without wall functions or global damping functions the model produces an efficient and fast scheme due to its algebraic character additionally its prediction of the transition from laminar to turbulent regimes has shown promising results in this work the les wale model is investigated within the lattice boltzmann framework for validation purposes various test cases are presented first a channel flow at a reynolds number of 6876 is investigated secondly the flow around a wall mounted cube at various reynolds numbers is determined the flow regime varies from laminar to transitional to fully turbulent conditions at a reynolds number of 40 000 with respect to the cube height

numerical conformal mapping of multiply connected regions by fornberg like methods we develop a new algorithm for computing conformal maps from regions exterior to non overlapping disks to unbounded multiply connected regions exterior to non overlapping smoothly bounded jordan regions the method is an extension of fornberg s original newton like method for mapping of the disk to simply connected regions a fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case numerical examples are reported

least squares finite element scheme for the lattice boltzmann method on an unstructured mesh a numerical model of the lattice boltzmann method lbm utilizing least squares finite element method in space and the crank nicolson method in time is developed this method is able to solve fluid flow in domains that contain complex or irregular geometric boundaries by using the flexibility and numerical stability of a finite element method while employing accurate least squares optimization fourth order accuracy in space and second order accuracy in time are derived for a pure advection equation on a uniform mesh while high stability is implied from a von neumann linearized stability analysis implemented on unstructured mesh through an innovative element by element approach the proposed method requires fewer grid points and less memory compared to traditional lbm accurate numerical results are presented through two dimensional incompressible poiseuille flow couette flow and flow past a circular cylinder finally the proposed method is applied to estimate the permeability of a randomly generated porous media which further demonstrates its inherent geometric flexibility

non body fitted cartesian mesh simulation of highly turbulent flows using multi relaxation time lattice boltzmann method this paper presents a lattice boltzmann method lbm based study aimed at numerical simulation of highly turbulent and largely inclined flow around obstacles of curved geometry using non body fitted cartesian meshes the approach features 1 combining the interpolated bounce back scheme with the lbm of multi relaxation time mrt type to enable the use of simple cartesian mesh for the flow cases even with complex geometries and 2 incorporating the spalart allmaras sa turbulence model into lbm in order to represent the turbulent flow effect the numerical experiments are performed corresponding to flows around an naca0012 airfoil at re 5x10 5 and around a flat plate at re 2x10 4 respectively the agreement between all simulation results obtained from this study and the data provided by other literature demonstrates the reliability of the enhanced lbm proposed in this paper for simulating simply on cartesian meshes complex flows that may involve bodies of curved boundary high reynolds number and large angle of attack

a fast method for solving certain integral equations of the first kind with application to conforma mapping abstract we describe a fast iterative method for solving fredholm integral equations of the first kind whose kernels have a logarithmic principal part the method is a fourier galerkin method and due to the singularity of the kernel the linear system of simultaneous equations is block diagonally dominant and can be solved rapidly by an iterative method we present examples of the computation of conformal mappings of simply and multiply connected domains

three time integration methods for incompressible flows with discontinuous galerkin boltzmann method abstract this paper presents three time integration methods for incompressible flows with finite element method in solving the lattice bgk boltzmann equation the space discretization is performed using nodal discontinuous galerkin method which employs unstructured meshes with triangular elements and high order approximation degrees the time discretization is performed using three different kinds of time integration methods namely direct decoupling and splitting from the storage cost temporal accuracy numerical stability and time consumption we systematically compare three time integration methods then benchmark fluid flow simulations are performed to highlight efficient time integration methods numerical results are in good agreement with others or exact solutions

least squares finite element lattice boltzmann method a new numerical model of the lattice boltzmann method utilizing least squares finite element in space and crank nicolson method in time is presented the new method is able to solve problem domains that contain complex or irregular geometric boundaries by using finite element method s geometric flexibility and numerical stability while employing efficient and accurate least squares optimization for the pure advection equation on a uniform mesh the proposed method provides for fourth order accuracy in space and second order accuracy in time with unconditional stability in the time domain accurate numerical results are presented through two dimensional incompressible poiseuille flow and couette flow

finite volume lattice boltzmann method we present a finite volume formulation for the lattice boltzmann method fvlbm based on standard bilinear quadrilateral elements in two dimensions the accuracy of this scheme is demonstrated by comparing the velocity field with the analytical solution of the navier stokes equations for time dependent rotating couette flow and taylor vortex flow to demonstrate the flexibility of the scheme we have also simulated a modified rotating couette flow where the inner cylinder has an elliptical shape the results agree with those obtained from the traditional marker and cell method the fvlbm scheme is applicable to arbitrarily shaped two dimensional regions and thus the range of applicability of the lattice boltzmann method has been significantly extended

stochastic control with uncertain parameters via chance constrained control chance constrained methods handle problem uncertainty by formulating the constraints probabilistically and only guaranteeing their satisfaction up to a given violation level this paper extends existing chance constrained methods to handle both uncertainty in the system state as well as in the constraint parameters due to the imperfect knowledge of the system state caused by motion sensor and environment uncertainty the system constraints cannot be guaranteed to be satisfied and consequently must be considered probabilistically prior work has primarily only dealt with uncertainty in the system state or constraint parameters but not both simultaneously however in many motivating applications both frequently occur to handle this case previous approaches either approximate the distribution leading to a nonconvex optimization program or use sampling alone to represent the uncertainty which requires a large number of samples to accurately represent the distribution to address these limitations a novel hybrid method is proposed that uses both analytical functions and sampling to represent the uncertainty it is shown that under certain conditions the resulting optimization program using this hybrid representation is convex to check the convexity an efficient a priori sufficient condition is developed furthermore by using this hybrid representation this method drastically reduces the computational complexity over previous methods which is demonstrated through two examples consequently this method has the ability to enable real time stochastic control for the motivating applications

on polynomial approximation in the complex plane with application to conformal mapping we consider the selection of polynomial bases for polynomial approximation of analytic functions on bounded simply connected regions in the complex plane while a monomial basis may be very ill conditioned we show that a basis of lagrange polynomials with fejer points as nodes is well conditioned numerical examples where we compute polynomial approximations of conformal mappings conclude the paper

on the determination of boundary collocation points for solving some problems for the laplace operator abstract boundary collocation is a method for obtaining approximate solutions of boundary problems for linear partial differential equations for which complete families of particular solutions are explicitly known the method contains various decisions which are important for its performance such as choice of solution subspace choice of basis for the subspace and choice of collocation points using a model problem some particular strategies for the determination of collocation points are investigated

linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits abstract in this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a disk with circular slits the method is based on some uniquely solvable boundary integral equations with classical adjoint and generalized neumann kernels these boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region some numerical examples are presented to illustrate the efficiency of the presented method

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions