nonlinear multichannel impedance inversion by total variation regularization abstractthe analysis of acoustic impedance ai allows for the mapping of seismic reflection data to lithology and hence it plays an important role in the interpretation of poststack seismic data the ai is obtainable from the inversion of the earth reflectivity series efficient deconvolution methods have been developed for recovering the reflectivity series from band limited poststack data which are multiple free zero offset and migrated however calculation of the ai from the reflectivity when considering the spatial correlation of the impedance parameters demands the solution of a constrained nonlinear inverse problem two efficient algorithms are proposed for solving the nonlinear impedance problem in multichannel form with the total variation tv constraint to recover impedance maps with blocky structures the first uses the continuous earth model for reflectivity which allows linearizing the problem in the logarithm domain the second uses the discrete layered earth model for reflectivit

feasibility of approximate broadband estimation of acoustic impedance profile from first principles and band limited reflection data abstractbecause seismic reflection data are band limited acoustic impedance profiles derived from them are nonunique the conventional inversion methods counter the nonuniqueness either by stabilizing the answer with respect to an initial model or by imposing mathematical constraints such as sparsity of the reflection coefficients by making a nominal assumption of an earth model locally consisting of a stack of homogeneous and horizontal layers we have formulated a set of linear equations in which the reflection coefficients are the unknowns and the recursively integrated seismic trace constitute the data drawing only on first principles the zoeppritz equation in this case the approach makes a frontal assault on the problem of reconstructing reflection coefficients from band limited data the local layer cake assumption and the strategy of seeking a singular value decomposition solution of the linear equations counter the nonuniqueness provided that the objective is to reconstruct a smooth version

approximate computation of the acoustic impedance from seismic data the approximate computation of the acoustic impedance from seismic data is usually based on the recursive formula h 1 ck ktl k gtegrated seismic pulse multiplied with twice the impedance in the top layer for impedance values less than 0 2 in absolute value this is also equal to the acoustic impedance function minus the acoustic impedance in the top layer convolved with the seismic pulse where h is the acoustic impedance in layer number k and ck is the pressure reflection coefficient for the interface between layer k and k 1 the above formula is derived from a discrete layered earth model when we consider a continuous earth model and discretize the results we obtain the recursive formula the computation of the acoustic impedance from band limited seismic data corresponds to an exponential transformation of the integrated seismic trace on a band limited acoustic impedance section with well separated reflectors and low noise level the direction of change in the acoustic impedance can be correctly identified

simultaneous multitrace impedance inversion with transform domain sparsity promotion abstractthe impedance inversion technique plays a crucial role in seismic reservoir properties prediction however most existing impedance inversion methods often suffer from spatial discontinuities and instability because each vertical profile is processed independently in the inversion we tested a transform domain sparsity promotion simultaneous multitrace impedance inversion method to address this issue the approach was implemented through minimizing a data misfit term and a transform domain sparsity constraint term that incorporates the 2d or 3d structural information into the inversion processing a 2d synthetic data example was applied to mainly explain the roles of the transform domain sparsity constraint we determined that the transform domain sparsity constraint can help in stabilizing the inversion reducing the influence of high wavenumber noise on the inverted result and exploring spatial continuities of structures furthermore a 3d field data example was used to examine the effectiven

robust wavelet estimation and blind deconvolution of noisy surface seismics robust blind deconvolution is a challenging problem particularly if the bandwidth of the seismic wavelet is narrow to very narrow that is if the wavelet bandwidth is similar to its principal frequency the main problem is to estimate the phase of the wavelet with sufficient accuracy the mutual information rate is a general purpose criterion to measure whiteness using statistics of all orders we modified this criterion to measure robustly the amplitude and phase spectrum of the wavelet in the presence of noise no minimum phase assumptions were made after wavelet estimation we obtained an optimal deconvolution output using wiener filtering the new procedure performs well even for very band limited data and it produces frequency dependent phase estimates

seismic sparse layer reflectivity inversion using basis pursuit decomposition a basis pursuit inversion of seismic reflection data for reflection coefficients is introduced as an alternative method of incorporating a priori information in the seismic inversion process the inversion is accomplished by building a dictionary of functions representing reflectivity patterns and constituting the seismic trace as a superposition of these patterns basis pursuit decomposition finds a sparse number of reflection responses that sum to form the seismic trace when the dictionary of functions is chosen to be a wedge model of reflection coefficient pairs convolved with the seismic wavelet the resulting reflectivity inversion is a sparse layer inversion rather than a sparse spike inversion synthetic tests suggest that a sparse layer inversion using basis pursuit can better resolve thin beds than a comparable sparse spike inversion application to field data indicates that sparse layer inversion results in the potentially improved detectability and resolution of some thin layers and reveals a

a fast automatic multichannel blind seismic inversion for high resolution impedance recovery abstractthe inversion of seismic reflection data for acoustic impedance ai is a common and accepted method for the interpretation of poststack seismic data the original mathematical problem is nonlinear due to the nonlinearity of relation between the ai and the earth reflectivity series and also the insufficiency of information about the source wavelet furthermore the problem is ill posed due to the lack of low and high frequencies in the data we have developed a multichannel blind inversion and solved it for obtaining the ai model and the wavelet directly from seismic reflection data we found a solution to the overall problem by alternating between two subproblems corresponding to the ai and wavelet recovery having an estimation of the wavelet the algorithm directly inverts multichannel data for a high resolution ai model having blocky structures in the sense of total variation tv regularization while satisfying a priori low frequency information the advantages of the split bregman techniq

multichannel spatially correlated reflectivity inversion using block sparse bayesian learning abstractthe reflectivity inversion approach based on a variety of regularization terms was extensively developed and applied to image subsurface structure in recent years in addition multichannel reflectivity inversion or deconvolution considering the lateral continuity of reflection interfaces or reflectivity in adjacent channels has been developed however these processing operations seldom adaptively judge the stratal continuity or automatically alter the parameters of the corresponding algorithm to use the special correlation of the reflection information contained in the seismic data a multichannel spatially correlated reflectivity inversion using block sparse bayesian learning bsbl is introduced the method adopts a covariance matrix that describes the spatial relationship of reflectivity and simultaneously controls the temporal sparsity with an expectation maximization algorithm we can obtain the parameters of the multichannel reflectivity model including the mean i e the estimated mul

on certain definite quadrature formulae gauss and lobatto quadrature formulae related to spaces of cubic splines with double and equidistant knots are constructed such quadrature formulae are known as asymptotically optimal definite formulae of order 4 some monotonicity results concerning the associated quadrature processes are established

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

preconditioning landweber iteration in hilbert scales in this paper we investigate convergence of landweber iteration in hilbert scales for linear and nonlinear inverse problems as opposed to the usual application of hilbert scales in the framework of regularization methods we focus here on the case s 0 which for tikhonov regularization corresponds to regularization in a weaker norm in this case the hilbert scale operator l 2 s appearing in the iteration acts as a preconditioner which significantly reduces the number of iterations needed to match an appropriate stopping criterion additionally we carry out our analysis under significantly relaxed conditions i e we only require inlinemediaobject not available see fulltext instead of inlinemediaobject not available see fulltext which is the usual condition for regularization in hilbert scales the assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones

asymptotically optimal definite quadrature formulae of 4th order we construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants besides the asymptotical optimality an advantage of our quadrature formulae is the explicit form of their weights and nodes for the remainders of our quadrature formulae monotonicity properties are established when the integrand has a non negative or non positive fourth derivative and a posteriori error estimates are proven

efficient preconditioners for optimality systems arising in connection with inverse problems this paper is devoted to the numerical treatment of linear optimality systems os that arise in connection with inverse problems for partial differential equations if such inverse problems are regularized by tikhonov regularization then it follows from standard theory that the associated os is well posed provided that the regularization parameter alpha is positive and that the involved state equation satisfies suitable assumptions we explain and analyze how certain mapping properties of the operators appearing in the os can be employed to define efficient preconditioners for finite element fe approximations of such systems the key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter h used in the fe discretization and increases only moderately as alpha rightarrow0 more specifically if the stopping criterion for the iteration process is defined in terms of the associated energy norm then the number of iterations required in the severely ill posed case cannot grow faster than o ln alpha 2 our analysis is based on a careful study of the operators involved which yields the distribution of the eigenvalues of the preconditioned os finally the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by borzi kunisch and kwak siam j control optim 41 2003 pp 1477 1497 and an inverse problem arising in connection with electrocardiography nielsen cai and lysaker math biosci 210 2007 pp 523 553

on schwarz type smoothers for saddle point problems with applications to pde constrained optimization problems in this paper we consider a one shot multigrid strategy for solving the discretized optimality system kkt system of a pde constrained optimization problem in particular we discuss the construction of an additive schwarz type smoother for a certain class of optimal control problems a rigorous multigrid convergence analysis is presented numerical experiments are shown which confirm the theoretical results

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

phase retrieval through regularization for seismic problems abstractthe most advanced data processing imaging algorithms aim at optimizing a suitably defined cost function consisting of a data misfit term posed in the data domain and a regularization term for strong phase errors in the data however such algorithms fail to achieve an acceptable solution because they force us to match the incorrect phase information to remedy this deficiency we replaced the time domain misfit term by a fourier magnitude fidelity term to recover the desired signal from just the amplitude spectrum of the observations such a problem is known as phase retrieval and it is a subject of interest for those who cannot measure the phase information about the system under study the proposed phase retrieval problem is solved by a fast iterative shrinkage thresholding algorithm and it is used to tackle two common phase problems arising in seismic processing data recovery in the presence of residual static shifts and deconvolution with a missing wavelet phase under sparsity constrain

recovery of the acoustic impedance from reflection seismograms this paper examines the problem of recovering the acoustic impedance from a band limited normal incidence reflection seismogram the convolutional model for the seismogram is adopted at the outset and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible in the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace that is the band limited version of the reflectivity function into the standard recursion formula for the acoustic impedance the formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance moreover the averaging function is identical to that used in deconvolving the initial seismogram the advantage of these averages is that they are unique their disadvantage is that low frequency information which is crucial to making a g

analysis of the minimal residual method applied to ill posed optimality systems we analyze the performance of the minimal residual minres method applied to linear karush kuhn tucker systems arising in connection with inverse problems such optimality systems typically have a saddle point structure and have unique solutions for all alpha 0 where alpha is the parameter employed in the tikhonov regularization unfortunately the associated spectral condition number is very large for small values of alpha which strongly indicates that their numerical treatment is difficult our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the minres method this result is obtained by carefully analyzing the spectrum of the associated saddle point operator except for a few isolated eigenvalues the spectrum consists of three bounded intervals krylov subspace methods handle such problems very well for severely ill posed cases techniques based on chebyshev polynomials are applied to prove that the number of iterations needed by

a robust multigrid method for elliptic optimal control problems we consider the discretized optimality system of a special class of elliptic optimal control problems and propose an all at once multigrid method for solving this discretized system under standard assumptions the convergence of the multigrid method and the robustness of the convergence rates with respect to the involved parameter are shown numerical experiments are presented for illustrating the theoretical results

on the monotonicity of sequences of quadrature formulae leti f l f k 0 r 0 vk 1 a k f x k be a quadrature formula and let s n f n 1 be successive approximations of the definite integrali f 0 1 f x dx obtained by the composition ofl i e s n f l n where varphi n x frac 1 n sum nolimits k 0 n 1 f left frac k x n right we prove sufficient conditions for monotonicity of the sequence s n f n 1 as particular cases the monotonicity of well known newton cotes and gauss quadratures is shown finally a recovery theorem based on the monotonicity results is presented

multitrace impedance inversion with lateral constraints abstractwe have developed a lateral constraint to the inversion of 1d seismic impedance models to suppress the effect of data noise and improve the fidelity of formation boundaries in 2d models for situations with dips of less than 20 typical inversion frameworks rely on a 1d forward model with each 1d trace being inverted independently adjacent inversion models are combined together to form a 2d impedance model adding a lateral constraint improves the fidelity of the 2d impedance models while retaining much of the advantage of the low computational cost associated with typical 1d inversion schemes solving the 1d lateral constraint inversion 1d lci problem involves the simultaneous inversion of multiple 1d traces producing layered sections with lateral smoothed transition in addition to enforcing lateral continuity in the inversion model this algorithm allows for the inclusion of a priori knowledge from boreholes we determined the effectiveness of this algorithm on two synthetic models as well

optimal quadrature for univariate and tensor product splines abstract numerical integration is a core subroutine in many engineering applications including the finite element method fem isogeometric analysis is a fem technology that uses smooth b spline and nurbs basis functions traditionally gaussian quadrature was used for numerical integration but this yields suboptimal performance this is attributed to the fact that gaussian quadrature rules do not take inter element smoothness of the spline basis into account resulting in over integration the equations for exact quadrature are well known but prove notoriously hard to solve due to their nonlinear nature these generalized gauss rules were first introduced in hughes et al 2010 in the context of isogeometric analysis where newton iteration was utilized to study and tabulate several cases later homotopy continuation barto and calo 2015 was used as an alternative strategy for finding these rules in this paper we describe an algorithm to generalize on both of these techniques it is optimal in the sense that it will integrate a space of dimension n using no more than n 1 2 quadrature points the algorithm works on arbitrary nonuniform knot vectors of any polynomial degree and continuity and is demonstrated on polynomial orders up to 15 it extends to 2d and 3d integrals by tensor product

multichannel blind deconvolution of seismic signals a new algorithm for simultaneous wavelet estimation and deconvolution of seismic reflection signals is given to remove the inherent ambiguity in this blind deconvolution problem we introduce relevant a priori information our major assumption is sparseness of the reflectivity which corresponds to a layered earth model this allows nonminimum phase wavelets to be recovered reliably and closely spaced reflectors to be resolved to combine a priori knowledge and data we use a bayesian framework and derive a maximum a posteriori estimate computing this estimate is a difficult optimization problem solved by a suboptimal iterative procedure the procedure alternates steps of wavelet estimation and reflectivity estimation the first step only requires a simple least squares fit while the second step is solved by the iterated window maximization algorithm proposed by kaaresen this enables better efficiency and optimality than established alternatives the resulting optimization method can easily handle multichannel models with only a moderate increase of the computational load lateral continuity of the reflectors is achieved by modeling local dependencies between neighboring traces major improvements in both wavelet and reflectivity estimates are obtained by taking the wavelet to be invariant across several traces the practicality of the algorithm is demonstrated on synthetic and real seismic data an application to multivariate well log segmentation is also given

sparse multichannel blind deconvolution abstractwe developed a sparse multichannel blind deconvolution smbd method the method is a modification of the multichannel blind deconvolution technique often called euclid deconvolution in which the multichannel impulse response of the earth is estimated by solving an homogeneous system of equations classical euclid deconvolution is unstable in the presence of noise and requires the correct estimation of the length of the seismic wavelet the proposed method on the other hand can tolerate moderate levels of noise and does not require a priori knowledge of the length of the wavelet smbd solves the homogeneous system of equations arising in euclid deconvolution by imposing sparsity on the unknown multichannel impulse response trivial solutions to the aforementioned homogeneous system of equations are avoided by seeking sparse solutions on the unit sphere we tested smbd with synthetic and real data examples synthetic examples were used to judge the viability of the method in terms of noise we f

nonlinear impedance inversion for attenuating media applications of seismic impedance inversion normally assume the data are free of multiples and transmission effects requiring knowledge of the seismic pulse that is assumed to be stationary an alternative formulation for impedance inversion is based on an exact frequency domain zero offset reflectivity function for a 1d medium analytical formulas for the frechet derivatives are derived for efficient implementation of an iterative nonlinear inversion the exact zero offset reflectivity accounts for internal multiples and transmission effects in the data absorption and dispersion are also conveniently handled if a reasonable estimate for the quality q factor of the medium is available a series of convenient features are included in the inversion algorithm an automatic estimation of the amplitude spectrum of the seismic pulse an impedance transform that makes the inversion independent from the initial smooth model and a practical approach to estimate the regularization weight numerical tests usin

explicit gaussian quadrature rules for c 1 cubic splines with symmetrically stretched knot sequences we provide explicit expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences the quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal that is it requires minimal number of nodes numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented the paper provides expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences any function from the space is exactly integrated by the rule the quadrature nodes and weights are derived via explicit recursion formulae with no intervention of any numerical solver the rule is optimal i e it requires minimal number of nodes

stochastic sparse spike deconvolution sparse spike deconvolution can be viewed as an inverse problem where the locations and amplitudes of a number of spikes reflectivity are estimated from noisy data seismic traces the main objective is to find the least number of spikes that when convolved with the available band limited seismicwaveletestimate fitthedatawithinagiventolerance errormisfit thedetectionofthespikes timelagsisahighly nonlinearoptimizationproblemthatcanbesolvedusingvery fastsimulatedannealingsa amplitudesareeasilyestimated using linear least squares at each sa iteration at this stage quadratic regularization is used to stabilize the solution to reduce its nonuniqueness and to provide meaningful reflectivitysequences thusavoidingtheneedtoconstrainthe spikes time lags and or amplitudes to force valid solutions impedanceconstraintsalsocanbeincludedatthisstage providing the low frequencies required to recover the acoustic impedance oneadvantageoftheproposedmethodoverother sparse spike deconvolution techniques is that the uncertaintyoftheobtainedsolutionscanbeestimatedstochastically further errorsinthephaseofthewaveletestimatearetolerated foranoptimumconstant phaseshiftisobtainedtocalibratetheeffectivewaveletthatispresentinthedata results using synthetic data including simulated data for the marmousi2 modelandfield 3d data show that physically meaningful high resolution sparse spike sections can be derived fromband limitednoisydata evenwhentheavailablewaveletestimateisinaccurate

a domain embedding preconditioner for the lagrange multiplier systems finite element approximations for the dirichlet problem associated to a second order elliptic differential equation are studied the purpose of this paper is to discuss domain embedding preconditioners for discrete systems the essential boundary condition on the interior interface is removed by introducing lagrange multipliers the associated discrete system with a saddle point structure is preconditioned by a block diagonal preconditioner the main contribution of this paper is to propose a new operator constructed from the h div inner product for the block of the preconditioner corresponding to the multipliers

optimal quadrature rules for odd degree spline spaces and their application to tensor product based isogeometric analysis abstract we introduce optimal quadrature rules for spline spaces that are frequently used in galerkin discretizations to build mass and stiffness matrices using the homotopy continuation concept barto and calo 2016 that transforms optimal quadrature rules from source spaces to target spaces we derive optimal rules for splines defined on finite domains starting with the classical gaussian quadrature for polynomials which is an optimal rule for a discontinuous odd degree space we derive rules for target spaces of higher continuity we further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points we demonstrate it by deriving optimal rules for various odd degree spline spaces particularly with non uniform knot sequences and non uniform multiplicities we also discuss convergence of our rules to their asymptotic counterparts that is the analogues of the midpoint rule of hughes et al 2010 that are exact and optimal for infinite domains for spaces of low continuities we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values

exit criteria and monotonicity in compound quadrature of gaussian type in this short note for compound quadrature rules of gaussian type we prove stopping rules and monotonicity results based on peano kernel methods

preconditioning saddle point systems with applications in optimization saddle point systems arise in many applications areas in fact in any situation where an extremum principle arises with constraints the stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from pdes and in the area of optimization such problems are ubiquitous in this paper we present a framework into which many well known methods for solving saddle point systems fit based on this description we show how new approaches for the solution of saddle point systems arising in optimization can be derived from the bramble pasciak conjugate gradient approach widely used in pdes and more recent generalizations thereof in particular we derive a class of new solution methods based on the use of preconditioned conjugate gradients in nonstandard inner products and demonstrate how these can be understood through more standard machinery we show connections to constraint preconditioning and give the results of numerical computations on a number of standard optimization test examples

generalized linear inversion of reflection seismic data generalized linear inversion sometimes known as model perturbation nonlinear regression or inverse modeling is applied to synthetic and real seismic data sets with the objective of obtaining an impedance profile as a function of time the impedances solved for are parameterized in a manner that describes the unknown earth using fewer variables than previous seismic generalized linear inversion techniques in this application only single traces of common midpoint cmp processed data will be inverted the method of generalized linear inversion gli presented here is designed to improve on the shortcomings of recursive inversion with respect to relative and absolute scale of the impedance results resolution of impedance boundaries and distortion from residual wavelet effects in obtaining these goals other advantageous aspects of gli were discovered for example it is insensitive to noise in many cases and it will allow an interpreter to fix the impedance of any number of known lithologies in an in

autoregressive recovery of the acoustic impedance this paper presents a method of recovering the acoustic impedance from reflection seismograms using autoregressive ar modeling an approach originally applied to deconvolution by lines and clayton 1977 the algorithm which we describe is novel both in the manner in which the missing low and high frequency information is predicted and in the fact that the prediction may be constrained if acoustic impedance information is available the prediction of the low frequencies treats the missing data as a gap which extends from the low frequency cut off in the negative frequency band to the corresponding frequency in the positive frequency band the conjugate symmetry which governs the behavior of the spectrum in the band is taken into account in the prediction the missing high frequencies are predicted using a modified minimum entropy norm in the frequency domain both synthetic and field examples are presented and illustrate the robustness of the new ar algorithm under a variety of conditions the field e

multigrid algorithms for inverse problems with linear parabolic pde constraints we present a multigrid algorithm for the solution of source identification inverse problems constrained by variable coefficient linear parabolic partial differential equations we consider problems in which the inversion variable is a function of space only we consider the case of l 2 tikhonov regularization the convergence rate of our algorithm is mesh independent even in the case of no regularization this feature makes the method algorithmically robust to the value of the regularization parameter and thus useful for the cases in which we seek high fidelity reconstructions the inverse problem is formulated as a pde constrained optimization we use a reduced space approach in which we eliminate the state and adjoint variables and we iterate in the inversion parameter space using conjugate gradients we precondition the hessian with a v cycle multigrid scheme the multigrid smoother is a two step stationary iterative solver that inexactly inverts an approximate hessian by iterating exclusively in the high frequency subspace using a high pass filter we analyze the performance of the scheme for the constant coefficient case with full observations we analytically calculate the spectrum of the reduced hessian and the smoothing factor for the multigrid scheme the forward and adjoint problems are discretized using a backward euler finite difference scheme the overall complexity of our inversion algorithm is mathcal o n tn n log 2n where n is the number of grid points in space and n t is the number of time steps we provide numerical experiments that demonstrate the effectiveness of the method for different diffusion coefficients and values of the regularization parameter we also provide heuristics and we conduct numerical experiments for the case with variable coefficients and partial observations we observe the same complexity as in the constant coefficient case finally we examine the effectiveness of using the reduced space solver as a preconditioner for a full space solver

optimal solvers for pde constrained optimization optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering particularly in problems of design the solution of such pde constrained optimization problems is usually a major computational task here we consider simple problems of this type distributed control problems in which the 2 and 3 dimensional poisson problem is the pde the large dimensional linear systems which result from discretization and which need to be solved are of saddle point type we introduce two optimal preconditioners for these systems which lead to convergence of symmetric krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem these preconditioners are block structured and involve standard multigrid cycles the optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases the theoretical proof indicates that these approaches may have much broader applicability for other pdes

gaussian quadrature for splines via homotopy continuation we introduce a new concept for generating optimal quadrature rules for splines to generate an optimal quadrature rule in a given target spline space we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one while preserving the number of quadrature points and therefore optimality the quadrature nodes and weights are considered as a higher dimensional point a zero of a particular system of polynomial equations as the space is continuously deformed by changing the source knot vector the quadrature rule gets updated using polynomial homotopy continuation for example starting with c 1 cubic splines with uniform knot sequences we demonstrate the methodology by deriving the optimal rules for uniform c 2 cubic spline spaces where the rule was only conjectured to date we validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen