error estimates for the discretization of the velocity tracking problem in this paper we are continuing our work casas and chrysafinos siam j numer anal 50 5 2281 2306 2012 concerning a priori error estimates for the velocity tracking of two dimensional evolutionary navier stokes flows the controls are of distributed type and subject to point wise control constraints the discretization scheme of the state and adjoint equations is based on a discontinuous time stepping scheme in time combined with conforming finite elements in space for the velocity and pressure provided that the time and space discretization parameters tau and h h respectively satisfy tau le ch 2 ch2 error estimates of order mathcal o h 2 o h2 and mathcal o h frac 3 2 frac 2 p o h32 2p with p 3 p 3 depending on the regularity of the target and the initial velocity are proved for the difference between the locally optimal controls and their discrete approximations when the controls are discretized by the variational discretization approach and by using piecewise linear functions in space respectively both results are based on new duality arguments for the evolutionary navier stokes equations

a modified split radix fft with fewer arithmetic operations recent results by van buskirk have broken the record set by yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power of two discrete fourier transform dft here we present a simple recursive modification of the split radix algorithm that computes the dft with asymptotically about 6 fewer operations than yavne matching the count achieved by van buskirk s program generation framework we also discuss the application of our algorithm to real data and real symmetric discrete cosine transforms where we are again able to achieve lower arithmetic counts than previously published algorithms

ciarlet raviart mixed finite element approximation for an optimal control problem governed by the first bi harmonic equation the ciarlet raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi harmonic equation the optimality conditions consisting of the state and the co state equations is derived also the a priori error estimates are analyzed in the analysis of the a priori error estimates the improved convergent rate of the higher order than existed results is proved some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

on computing the split radix fft this paper presents an efficient fortran program that computes the duhamel hollmann split radix fft an indexing scheme is used that gives a three loop structure for the split radix fft that is very similar to the conventional cooley tukey fft both a decimation in frequency and a decimation in time program are presented an arithmetic analysis is made to compare the operation count of the cooley tukey fft fo several different radixes to that of the split radix fft the split radix fft seems to require the least total arithmetic of any power of two dft algorithm

a posteriori error estimation for a pde constrained optimization problem involving the generalized oseen equations we derive globally reliable a posteriori error estimators for a linear quadratic optimal control problem involving the generalized oseen equations as state equations control constraints are also considered the corresponding local error indicators are locally efficient the assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods when stabilized methods are considered no a priori relation between the stabilization terms for the state and adjoint equations is required if a lower bound for the inf sup constant is available a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained we illustrate the theory with numerical examples

least squares finite element methods for optimization and control problems for the stokes equations the approximate solution of optimization and control problems for systems governedby the stokes equations is considered modern computational techniques for such problems are predominantly based on the application of the lagrange multiplier rule while penalty formulations even though widely used in other settings have not enjoyed the same level of popularity for this class of problems a discussion is provided that explains why naively defined penalty methods may not be practical then practical penalty methods are defined using methodologies associated with modern least squares finite element methods the advantages with respect to efficiency of penalty leasts squares methods for optimal control problems compared to methods based on lagrange multipliers are highlighted a tracking problem for the stokes system is used for illustrative purposes

a partition of unity method for a class of fourth order elliptic variational inequalities abstract we consider a partition of unity method pum for a class of fourth order elliptic variational inequalities on convex polygonal domains that include obstacle problems of simply supported kirchhoff plates and elliptic distributed optimal control problems with pointwise state constraints as special cases by including singular functions in the local approximation spaces we are able to show that the partition of unity method converges optimally numerical results that corroborate the theoretical estimates are also presented

scaled radix 2 8 algorithm for efficient computation of length n 2 m dfts this paper presents a scaled radix 2 8 fast fourier transform fft sr28fft algorithm for computing length n 2m discrete fourier transforms dfts scaled by complex number rotating factors the idea of the sr28fft algorithm is from the modified split radix fft msrfft algorithm and its purpose is to furnish other algorithms with high efficiency but without shortcomings of the msrfft algorithm a novel radix 2 4 fft nr24fft algorithm and a novel radix 2 8 fft nr28fft algorithm are proposed these two algorithms use sr28fft to calculate their sub dfts of odd indexed terms several aspects of the two algorithms such as computational complexity computation accuracy and coefficient evaluations or accesses to the lookup table all are improved the bit reverse method can be used for their order permutation and no extra memory is required to store their extra coefficients by the two novel algorithms which contribute significantly to the performance of the fft algorithms the sr28fft algorithm can also be applied to other algorithms whose decomposition contains sub dfts of powers of two the appendix presents an algorithm named sr28fft 2 for further reducing the number of arithmetic operations and nr24fft and nr28fft algorithms based on sr28fft 2 requires fewer real operations than that required by the msrfft algorithm

an accurate error analysis model for fast fourier transform an error propagation model is proposed for the in place decimation in time version of the radix 2 fft algorithm with the model an accurate error expression and error variance for the computation of fft are derived this correspondence deals with fixed point and block floating point arithmetic simulation results agree closely with the theoretical predicted ones we find that some roundoff errors at different stages correlate with each other the density of correlations is closely associated with the round off approach used in butterfly calculations

a feasible active set method for strictly convex quadratic problems with simple bounds a primal dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of kunisch and rendl siam j optim 14 2003 pp 35 52 based on a guess of the active set a primal dual pair x alpha is computed that satisfies stationarity and the complementary condition if x is not feasible the variables connected to the infeasibilities are added to the active set and a new primal dual pair x alpha is computed this process is iterated until a primal feasible solution is generated then a new active set is determined based on the feasibility information of the dual variable alpha strict convexity of the quadratic problem is sufficient for the algorithm to stop after a finite number of steps with an optimal solution computational experience indicates that this approach also performs well in practice

a priori error estimates for space time finite element discretization of parabolic optimal control problems part i problems without control constraints in this paper we develop a priori error analysis for galerkin finite element discretizations of optimal control problems governed by linear parabolic equations the space discretization of the state variable is done using usual conforming finite elements whereas the time discretization is based on discontinuous galerkin methods for different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters the paper is divided into two parts in the first part we develop some stability and error estimates for space time discretization of the state equation and provide error estimates for optimal control problems without control constraints in the second part of the paper the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable

analysis and finite element approximation of optimal control problems for the stationary navier stokes equations with distributed and neumann controls we examine certain analytic and numerical aspects of optimal control problems for the stationary navier stokes equations the controls considered may be of either the distributed or neumann type the functionals minimized are either the viscous dissipation or the l4 distance of candidate flows to some desired flow we show the existence of optimal solutions and justify the use of lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced we study the regularity of solutions of this system then we consider the approximation by finite element methods of solutions of the optimality system and derive optimal error estimates

optimal control of the convection diffusion equation using stabilized finite element methods in this paper we analyze the discretization of optimal control problems governed by convection diffusion equations which are subject to pointwise control constraints we present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case moreover the in general different approaches optimize then discretize and discretize then optimize coincide for the proposed discretization scheme this allows for a symmetric optimality system at the discrete level and optimal order of convergence

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

primal dual active set strategy for a general class of constrained optimal control problems an algorithm for efficient solution of optimal control problems with pointwise constraints is introduced it is based on an active set strategy using primal as well as dual variables under certain assumptions a decrease of an appropriately chosen merit function and convergence of the algorithm are proved numerical examples for a parabolic optimal control problem demonstrate the efficiency of the proposed method for control constrained problems

incomplete data problems in x ray computerized tomography i singular value decomposition of the limited angle transform etude des problemes intrinseques relatifs aux donnees incompletes dans la tomographie aux rayons x assistee par ordinateur decomposition de la transformation de radon

on the fixed point accuracy analysis of fft algorithms in this paper we investigate the effect of fixed point arithmetics with limited precision for different fast fourier transform fft algorithms a matrix representation of error propagation model is proposed to analyze the rounding effect an analytic expression of overall quantization loss due to the arithmetic quantization errors is derived to compare the performance with decimation in time dit and decimation in frequency dif configurations from the simulation results the radix 2 dit fft algorithm has better accuracy in term of signal to quantization noise ratio sqnr based on the results a simple criterion of wordlength optimization is proposed to yield comparable accuracy with fewer bit budget

on stabilized mixed methods for generalized stokes problem based on the velocity pseudostress formulation a priori error estimates abstract in this paper we present an augmented mixed formulation applied to generalized stokes problem and uses it as state equation in an optimal control problem the augmented scheme is obtained adding suitable least squares terms to the corresponding velocity pseudostress formulation of the generalized stokes problem to ensure the existence and uniqueness of solution at continuous and discrete levels we prove coerciveness of the corresponding augmented bilinear form and using approximation properties of the respective discrete subspaces we deduce the optimal rate of convergence as by product and considering the associated optimal control problem we derive error estimates for the approximated control unknown finally we present several numerical examples confirming the theoretical properties of this approach

a priori error estimates for space time finite element discretization of parabolic optimal control problems in this article we summarize recent results on a priori error estimates for space time finite element discretizations of linear quadratic parabolic optimal control problems we consider the following three cases problems without inequality constraints problems with pointwise control constraints and problems with state constraints pointwise in time for all cases error estimates with respect to the temporal and to the spatial discretization parameters are derived the results are illustrated by numerical examples

accumulation of coefficient roundoff error in fast fourier transforms implemented with logarithmic number system when a fast fourier transform fft is implemented on a digital computer or with special purpose hardware quantization errors will arise due to finite word lengths in the digital system this correspondence presents an analysis of error accumulation due to coefficient rounding in the fft implemented with a logarithmic number system the theoretical result of the coefficient roundoff error analysis is verified by computer simulations the results indicate that the fft implemented with the logarithmic number system provides better signal to noise performance than that implemented with a fixed point or floating point number system

edge stabilization for the generalized stokes problem a continuous interior penalty method in this note we introduce and analyze a stabilized finite element method for the generalized stokes equation stability is obtained by adding a least squares penalization of the gradient jumps acro

numerical sensitivity analysis for the quantity of interest in pde constrained optimization in this paper we consider the efficient computation of derivatives of a functional the quantity of interest which depends on the solution of a pde constrained optimization problem with inequality constraints and which may be different from the cost functional the optimization problem is subject to perturbations in the data we derive conditions under with the quantity of interest possesses first and second order derivatives with respect to the perturbation parameters an algorithm for the efficient evaluation of these derivatives is developed with considerable savings over a direct approach especially in the case of high dimensional parameter spaces the computational cost is shown to be small compared to that of the overall optimization algorithm numerical experiments involving a parameter identification problem for navier stokes flow and an optimal control problem for a reaction diffusion system are presented which demonstrate the efficiency of the method

the ill conditioned nature of the limited angle tomography problem the instability of inverting the limited angle radon transform is studied by constructing the singular value decomposition of this operator connections are established with previously known series expansions and with the discrete prolate spheroidal wave functions mollification methods are shown to delay but not prevent the onset of instability as the angular range of projections is decreased

a note on an accurate error analysis model for fast fourier transform the expressions derived by ma see ibid vol 45 p 1641 1997 for error variance and average power expectation using the model that has been proposed are not accurate in this paper all the possible correlations have not been considered the expressions have been modified to make them exact closed form expressions for the correlation between errors of different stages have been derived the expressions have been verified by comparing the predicted values using these expressions with the simulation values given by ma

l infty estimates for approximated optimal control problems an optimal control problem for a two dimensional elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise linear functions the state and the adjoint state are discretized by linear finite elements approximation of order h in the l infty norm is proved in the main result

biorthogonal decompositions of the radon transform the concept of singular value decompositions is a valuable tool in the examination of ill posed inverse problems af g such as the inversion of the radon transform a singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators a a a a in this paper we consider a new approach which generalizes this concept by application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces although it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator with respect to the radon transform for functions with support in the unit ball we apply the system of appell polynomials which is a natural generalization of the univariate system of gegenbauer ultraspherical polynomials to the multivariate case the corresponding biorthogonal decompositions show some advantages in comparison with the known singular value decompositions vice versa by application of our decompositions we are able to prove new properties of the appell polynomials

feedback optimal control of distributed parameter systems by using finite dimensional approximation schemes optimal control for systems described by partial differential equations is investigated by proposing a methodology to design feedback controllers in approximate form the approximation stems from constraining the control law to take on a fixed structure where a finite number of free parameters can be suitably chosen the original infinite dimensional optimization problem is then reduced to a mathematical programming one of finite dimension that consists in optimizing the parameters the solution of such a problem is performed by using sequential quadratic programming linear combinations of fixed and parameterized basis functions are used as the structure for the control law thus giving rise to two different finite dimensional approximation schemes the proposed paradigm is general since it allows one to treat problems with distributed and boundary controls within the same approximation framework it can be applied to systems described by either linear or nonlinear elliptic parabolic and hyperbolic equations in arbitrary multidimensional domains simulation results obtained in two case studies show the potentials of the proposed approach as compared with dynamic programming

a family of second order methods for convex ell _1 l1 regularized optimization this paper is concerned with the minimization of an objective that is the sum of a convex function f and an ell 1 l1 regularization term our interest is in active set methods that incorporate second order information about the function f to accelerate convergence we describe a semismooth newton framework that can be used to generate a variety of second order methods including block active set methods orthant based methods and a second order iterative soft thresholding method the paper proposes a new active set method that performs multiple changes in the active manifold estimate at every iteration and employs a mechanism for correcting these estimates when needed this corrective mechanism is also evaluated in an orthant based method numerical tests comparing the performance of three active set methods are presented

semidiscretization and error estimates for distributed control of the instationary navier stokes equations optimal control problems governed by the two dimensional instationary navier stokes equations and their spatial discretizations with finite elements are investigated a concept of semi discrete solutions to the control problem is introduced which is utilized to prove existence and uniqueness of discrete controls in neighborhoods of regular continuous solutions furthermore an optimal error estimate in terms of the spatial discretization parameter is given

on the interplay between interior point approximation and parametric sensitivities in optimal control this paper is concerned with the sensitivities of function space oriented interior point approximations in parameter dependent optimization problems for an abstract setting that covers control constrained optimal control problems the convergence of interior point sensitivities to the sensitivities of the optimal solution is shown error bounds for lq norms are derived and illustrated with numerical examples ams msc 2000 90c51 90c31 49m30

a locking free fem in active vibration control of a timoshenko beam in this paper we analyze the numerical approximation of an active vibration control problem of a timoshenko beam in order to avoid locking we focus on the finite element method used to compute the beam vibration to minimize it optimal order error estimates are obtained for the control variable which is the amplitude of secondary forces modeled as dirac s delta distributions these estimates are valid with constants that do not depend on the thickness of the beam in order to assess the performance of the method numerical tests are reported

fixed point analysis and parameter selections of msr cordic with applications to fft designs mixed scaling rotation msr coordinate rotation digital computer cordic is an attractive approach to synthesizing complex rotators this paper presents the fixed point error analysis and parameter selections of msr cordic with applications to the fast fourier transform fft first the fixed point mean squared error of the msr cordic is analyzed by considering both the angle approximation error and signal round off error incurred in the finite precision arithmetic the signal to quantization noise ratio sqnr of the output of the fft synthesized using msr cordic is thereafter estimated based on these analyses two different parameter selection algorithms of msr cordic are proposed for general and dedicated msr cordic structures the proposed algorithms minimize the number of adders and word length when the sqnr of the fft output is constrained design examples show that the fft designed by the proposed method exhibits a lower hardware complexity than existing methods

optimal control of the stokes equations a priori error analysis for finite element discretization with postprocessing an optimal control problem for 2d and 3d stokes equations is investigated with pointwise control constraints this paper is concerned with the discretization of the control by piecewise constant functions the state and the adjoint state are discretized by finite element schemes in the paper a postprocessing strategy is suggested which allows for significant improvement of the accuracy

multigrid solution of a distributed optimal control problem constrained by the stokes equations in this work we construct multigrid preconditioners to accelerate the solution process of a linear quadratic optimal control problem constrained by the stokes system the first order optimality conditions of the control problem form a linear system the kkt system connecting the state adjoint and control variables our approach is to eliminate the state and adjoint variables by essentially solving two stokes systems and to construct efficient multigrid preconditioners for the schur complement of the block associated with the state and adjoint variables these multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the stokes system in particular the number of conjugate gradient iterations is shown to decrease as the resolution increases a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems

a level set approach for the solution of a state constrained optimal control problem state constrained optimal control problems for linear elliptic partial differential equations are considered the corresponding first order optimality conditions in primal dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary interface between the active and inactive sets as optimization variable the new algorithm is based on the level set methodology the speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization encouraging numerical results attained by the new algorithm are reported on

singular value decompositions and inversion methods for the exterior radon transform and a spherical transform abstract the classical radon transform r maps an integrable function in rn to its integrals over all n 1 dimensional hyperplanes and the exterior radon transform is the transform r restricted to hyperplanes that do not intersect a given disc a singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc this decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space a modification of this method is given that will exactly invert functions of compact support these results generalize theorems of r m perry and the author a singular value decomposition for the radon transform that integrates over spheres in rn containing the origin is also given this follows from the singular value decomposition for r and yields the null space and a new inversion method for this transform

a c0 interior penalty method for the dirichlet control problem governed by biharmonic operator an energy space based dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a c0 interior penalty method is proposed and analyzed an abstract a priori error estimate is derived under the minimal regularity conditions the abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular further an optimal order l2 norm error estimate is derived numerical experiments illustrate the theoretical findings

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2

semismooth newton and augmented lagrangian methods for a simplified friction problem in this paper a simplified friction problem and iterative second order algorithms for its solution are analyzed in infinite dimensional function spaces motivated from the dual formulation a primal dual active set strategy and a semismooth newton method for a regularized problem as well as an augmented lagrangian method for the original problem are presented and their close relation is analyzed local as well as global convergence results are given by means of numerical tests we discuss among others convergence properties the dependence on the mesh and the role of the regularization and illustrate the efficiency of the proposed methodologies