anisotropic mesh adaptation for crack detection in brittle materials the quasi static brittle fracture model proposed by g francfort and j j marigo can be gamma approximated at each time evolution step by the ambrosio tortorelli functional in this paper we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field second we build on this variational model an adapted discretization to numerically compute the time evolving minimizing solution we present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure the main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack but only far away from the crack tip while they show a highly isotropic behavior in a neighborhood of the crack tip instead as a consequence of these properties the resulting discretizations follow very closely the propagation of the fracture

on generalized fractional integral inequalities for twice differentiable convex functions abstract in this article some new generalized fractional integral inequalities of midpoint and trapezoid type for twice differentiable convex functions are obtained in view of this we obtain new integral inequalities of midpoint and trapezoid type for twice differentiable convex functions in a form classical integral and riemann liouville fractional integrals finally we apply our new inequalities to construct inequalities involving moments of a continuous random variable

isogeometric boundary element analysis using unstructured t splines we couple collocated isogeometric boundary element methods and unstructured analysis suitable t spline surfaces for linear elastostatic problems we extend the definition of analysis suitable t splines to encompass unstructured control grids unstructured meshes and develop basis functions which are smooth rational polynomials defined in terms of the bezier extraction framework and which pass standard patch tests we then develop a collocation procedure which correctly accounts for sharp edges and corners extraordinary points and t junctions this approach is applied to several three dimensional problems including a real world t spline model of a propeller we believe this work clearly illustrates the power of combining new analysis suitable computer aided design technologies with established analysis methodologies in this case the boundary element method

stability estimates and structural spectral properties of saddle point problems for a general class of saddle point problems sharp estimates for babuska s inf sup stability constants are derived in terms of the constants in brezzi s theory in the finite dimensional hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented which are helpful for the convergence analysis of common krylov subspace methods the theoretical results are applied to two model problems from optimal control with time periodic state equations numerical experiments with the preconditioned minimal residual method are reported

implementation of an adaptive finite element approximation of the mumford shah functional we present and detail a method for the numerical solving of the mumford shah problem based on a finite element method and on adaptive meshes we start with the formulation introduced in 13 detail its numerical implementation and then propose a variant which is proved to converge to the mumford shah problem a few experiments are illustrated

isogeometric analysis and hierarchical refinement for higher order phase field models abstract while the interest in higher order models in physics and mechanics grows their numerical simulation still poses a challenge especially for arbitrary shaped three dimensional domains this contribution presents the mathematical framework as well as the application to different problems in the field of material science fracture mechanics and diffusion problems all models under consideration require at least c 1 continuity which prevents the application of standard finite element analysis and local mesh refinements introducing isogeometric analysis iga for the discretization in a finite element framework enables us to deal with these requirements moreover a general hierarchical refinement scheme based on a subdivision projection is presented here for one two and three dimensional simulations this technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization using this mathematical framework the improved convergence of a kuramoto sivashinsky model a mesh adapted thermal diffusion simulation and computations of a priori unknown crack propagation in different fracture modes underline the versatility of the presented hierarchical refinement scheme

on the stability of the reduced basis method for stokes equations in parametrized domains we present an application of reduced basis method for stokes equations in domains with affine parametric dependence the essential components of the method are i the rapid convergence of global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n selected points in parameter space ii the off line on line computational procedures decoupling the generation and projection stages of the approximation process the operation count for the on line stage in which given a new parameter value we calculate an output of interest depends only on n typically very small and the parametric complexity of the problem the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation design optimization and real time control particular attention is given i to the pressure treatment of incompressible stokes problem ii to find an equivalent inf sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem iii to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions iv to reduce computational costs in order to allow real time solution of parametrized problem

reduced basis approximation of parametrized optimal flow control problems for the stokes equations this paper extends the reduced basis method for the solution of parametrized optimal control problems presented in negri et al 2013 to the case of noncoercive elliptic equations such as the stokes equations we discuss both the theoretical properties with particular emphasis on the stability of the resulting double nested saddle point problems and on aggregated error estimates and the computational aspects of the method then we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three dimensional flow through boundary control demonstrating the effectivity of the methodology

thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field in this paper we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids develop incremental variational principles and consider their numerical implementations by multi field finite element methods we start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that converges for vanishing length scale parameter to a sharp crack topology functional this functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase field here we propose alternative rate independent and viscous over force models that ensure the local growth of the phase field next we define an energy storage function whose positive tensile part degrades with increasing phase field with these constitutive functionals at hand we derive the coupled balances of quasi static stress equilibrium and gradient type phase field evolution in the solid from the argument of virtual power here we consider a canonical two field setting for rate independent response and a time regularized three field formulation with viscous over force response it is then shown that these balances follow as the euler equations of incremental variational principles that govern the multi field problems these principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation including features such as the symmetry of the monolithic tangent matrices we demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples copyright 2010 john wiley sons ltd

an energy space finite element approach for elliptic dirichlet boundary control problems in this paper we present a finite element analysis for a dirichlet boundary control problem where the dirichlet control is considered in a convex closed subspace of the energy space h 1 2 gamma h 1 2 as an equivalent norm in h 1 2 gamma h 1 2 we use the energy norm induced by the so called steklov poincare operator which realizes the dirichlet to neumann map and which can be implemented by using standard finite element methods the presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems and their finite element approximations some numerical results are given which confirm on one hand the theoretical estimates but on the other hand indicate the differences when modelling the control in l 2 gamma l 2

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints in this paper we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations we consider the standard linear quadratic problem setting with distributed control and unilateral control constraints for this problem class we propose two different reduced basis approximations and associated error estimation procedures in our first approach we directly consider the resulting optimality system introduce suitable reduced basis approximations for the state adjoint control and lagrange multipliers and use a projection approach to bound the error in the reduced optimal control for our second approach we first reformulate the optimal control problem using a slack variable then develop a reduced basis approximation for the slack problem by suitably restricting the solution space and finally derive error bounds for the slack based optimal control we discuss benefits and drawbacks of b

reduced basis isogeometric methods rb iga for the real time simulation of potential flows about parametrized naca airfoils abstract we present a reduced basis rb method based on isogeometric analysis iga for the rapid and reliable evaluation of pde systems characterized by complex geometrical features at the current state of the art this is the first case of coupling between rb and iga methods the construction of the rb method relies on an isogeometric boundary element method iga bem as the high fidelity technique allowing a direct interface with computer aided design cad tools a suitable empirical interpolation method eim ensures an efficient offline online decomposition between the construction and the evaluation of the rb method we consider the real time simulation of potential flows past airfoils parametrized with respect to the angle of attack and the naca number identifying their shape and we provide a validation of our methodology with respect to experimental data and reference numerical codes showing in both cases a very good agreement

an adaptive finite element approximation of a variational model of brittle fracture the energy of the francfort marigo model of brittle fracture can be approximated in the sense of gamma convergence by the ambrosio tortorelli functional in this work we formulate and analyze two adaptive finite element algorithms for the computation of its local minimizers for each algorithm we combine a newton type method with residual driven adaptive mesh refinement we present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the ambrosio tortorelli functional

error analysis for a finite element approximation of elliptic dirichlet boundary control problems we consider the galerkin finite element approximation of an elliptic dirichlet boundary control model problem governed by the laplacian operator the analytical setting of this problem uses l 2 controls and a very weak formulation of the state equation however the corresponding finite element approximation uses standard continuous trial and test functions for this approximation we derive a priori error estimates of optimal order which are confirmed by numerical experiments the proofs employ duality arguments and known results from the l p error analysis for the finite element dirichlet projection

a higher order phase field model for brittle fracture formulation and analysis within the isogeometric analysis framework phase field models based on the variational formulation for brittle fracture have recently been gaining popularity these models have proven capable of accurately and robustly predicting complex crack behavior in both two and three dimensions in this work we propose a fourth order model for the phase field approximation of the variational formulation for brittle fracture we derive the thermodynamically consistent governing equations for the fourth order phase field model by way of a variational principle based on energy balance assumptions the resulting model leads to higher regularity in the exact phase field solution which can be exploited by the smooth spline function spaces utilized in isogeometric analysis this increased regularity improves the convergence rate of the numerical solution and opens the door to higher order convergence rates for fracture problems we present an analysis of our proposed theory and numerical examples that support this claim we also demonstrate the robustness of the model in capturing complex three dimensional crack behavior

certified reduced basis method for the electric field integral equation in b fares et al j comput phys 230 2011 pp 5532 5555 a reduced basis method rbm for the electric field integral equation efie using the boundary element method bem is developed based on a simplified a posteriori error estimator for the greedy based snapshot selection in this paper we extend this work and propose a certified rbm for the efie based on a mathematically rigorous a posteriori estimator a central difficulty of the certified method is that the intrinsic solution space of the efie is boldsymbol h 1 2 rm div gamma inducing a relatively complicated norm since the measured error consists of the difference between the reduced basis solution and the boundary element solution which is a member of the discrete boundary element space we clarify that the intrinsic norm can be replaced by an alternative norm and in this work use the boldsymbol h rm div norm which is computable and demonstrated to not degrade the quality of the error estimator a success

stabilized reduced basis methods for parametrized steady stokes and navier stokes equations abstract it is well known in the reduced basis approximation of saddle point problems that the galerkin projection on the reduced space does not guarantee the inf sup approximation stability even if a stable high fidelity method was used to generate snapshots for problems in computational fluid dynamics the lack of inf sup stability is reflected by the inability to accurately approximate the pressure field in this context inf sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions the main goal of this work is to propose an alternative approach which relies on the residual based stabilization techniques customarily employed in the finite element literature such as brezzi pitkaranta franca hughes streamline upwind petrov galerkin galerkin least square in the spirit of offline online reduced basis computational splitting two such options are proposed namely offline only stabilization and offline online stabilization these approaches are then compared to and combined with the state of the art supremizer enrichment approach numerical results are discussed highlighting that the proposed methodology allows to obtain smaller reduced basis spaces i e neglecting supremizer enrichment for which a modified inf sup stability is still preserved at the reduced order level

nonsingular isogeometric boundary element method for stokes flows in 3d abstract isogeometric analysis iga is emerging as a technology bridging computer aided geometric design cagd most commonly based on non uniform rational b splines nurbs surfaces and engineering analysis in finite element and boundary element isogeometric methods fe iga and iga bem the nurbs basis functions that describe the geometry define also the approximation spaces in the fe iga approach the surfaces generated by the cagd tools need to be extended to volumetric descriptions a major open problem in 3d this additional passage can be avoided in principle when the partial differential equations to be solved admit a formulation in terms of boundary integral equations leading to boundary element isogeometric analysis iga bem the main advantages of such an approach are given by the dimensionality reduction of the problem from volumetric based to surface based by the fact that the interface with cagd tools is direct and by the possibility to treat exterior problems where the computational domain is infinite by contrast these methods produce system matrices which are full and require the integration of singular kernels in this paper we address the second point and propose a nonsingular formulation of iga bem for 3d stokes flows whose convergence is carefully tested numerically standard gaussian quadrature rules suffice to integrate the boundary integral equations and carefully chosen known exact solutions of the interior stokes problem are used to correct the resulting matrices extending the work by klaseboer et al 2012 27 to iga bem

new bounds for moments of continuous random variables in this paper some gruss and ostrowski gruss type inequalities are applied to estimate the moments of continuous random variables whose probability density function is in l p space some results of kumar are generalized p kumar inequalities involving moments of a continuous random variable defined over a finite interval comput math appl 48 2004 257 273 p kumar the ostrowski type moment integral inequalities and moment bounds for continuous random variables comput math appl 49 2005 1929 1940

approximation of a brittle fracture energy with a constraint of non interpenetration linear fracture mechanics or at least the initiation part of that theory can be framed in a variational context as a minimization problem over a sbd type space the corresponding functional can in turn be approximated in the sense of convergence by a sequence of functionals involving a phase field as well as the displacement field we show that a similar approximation persists if additionally imposing a non interpenetration constraint in the minimization namely that only nonnegative normal jumps should be permissible 2010 mathematics subject classification 26a45

existence uniqueness and approximation for generalized saddle point problems extensions of the well known results of brezzi on saddle point problems are presented the class of problems is generalized to include the unsymmetric case and the known stability and approximation results are strengthened and applied to the generalized problem an application to the solution of certain exterior interface problems is indicated

the ostrowski type moment integral inequalities and moment bounds for continuous random variables we establish ostrowski type integral inequalities involving moments of a continuous random variable defined on a finite interval we also derive bounds for moments from these inequalities further we discuss applications of these bounds to the euler s beta mappings and illustrate their behaviour

numerical approximation of optimal flow control problems by a penalty method error estimates and numerical results the purpose of this paper is to present numerically convenient approaches to solve optimal dirichlet control problems governed by the steady navier stokes equations we will examine a penalized neumann control approach for solving dirichlet control problems from numerical and computational points of view the control is affected by the suction or injection of fluid through the boundary or by boundary surface movements in the tangential direction the control objective is to minimize the vorticity in the flow or to drive the velocity field to a desired one we develop sequential quadratic programming methods to solve these optimal control problems the effectiveness of the optimal control techniques in flow controls and the feasibility of the proposed penalized neumann control approaches for flow control problems are demonstrated by numerical experiments for a viscous incompressible fluid flow in a two dimensional channel and in a cavity geometry

inequalities involving moments of a continuous random variable defined over a finite interval some results based on the korkine s identity and integral inequalities of holder and gruss are obtained for the moments of a continuous random variable whose probability distribution is a convex function on the interval of real numbers applications of these results are considered in deriving the inequalities involving higher moments and special means and also in evaluating moments of a beta random variable

γ convergence for high order phase field fracture continuum and isogeometric formulations abstract we consider high order phase field functionals introduced in borden et al 2014 and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy we take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in sobolev spaces and in spaces of tensor product b splines in the latter convergence holds when the mesh size vanishes faster than the internal length of the phase field model on the theoretical level this condition is natural since the size of the phase field layer around the crack itself scales like the internal length on the numerical level it should be satisfied by local h refinement technically convergence holds in the sense of convergence with respect to the strong topology of l 1 while the sharp crack energy is defined in g s b d 2 the constraint on the phase field to take values in 0 1 is taken into account both in the sobolev setting and in the iso geometric setting in the latter it requires a special treatment since the projection operator on the space of tensor product b splines is not lagrangian i e interpolatory

reduced basis method for parametrized elliptic optimal control problems we propose a suitable model reduction paradigm the certied reduced basis method rb for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations pdes in particular we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable firstly we recast the optimal control problem in the framework of saddle point problems in order to take advantage of the already developed rb theory for stokes type problems then the usual ingredients of the rb methodology are called into play a galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure an efficient and rigorous a posteriori error estimate on the state control and adjoint variables as well as on the cost functional finally we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique

certified reduced basis methods for parametrized saddle point problems we present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems first we develop new a posteriori error estimates that unlike earlier approaches provide upper bounds for the errors in the approximations of the primal variable and the lagrange multiplier separately the proposed method is an application of brezzi s theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods second based on an analysis of brezzi s theory we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost finally we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and compared to earlier approaches computationally much more efficient the method is applied to a stokes flow problem in a two dimensional channel with a parametrized rectangular obstacle

a natural norm successive constraint method for inf sup lower bounds we present a new approach for the construction of lower bounds for the inf sup stability constants required in a posteriori error analysis of reduced basis approximations to anely parametrized partial dierential equations we combine the linearized inf sup statement of the natural norm approach with the approximation procedure of the successive constraint method scm the former natural norm provides an economical parameter expansion and local concavity in parameter a small er optimization problem which enjoys intrinsic lower bound properties the latter scm provides a systematic optimization framework a linear program lp relaxation which readily incorporates continuity and stability constraints the natural norm scm requires a parameter domain decomposition we propose a greedy algorithm for selection of the scm control points as well as adaptive construction of the optimal subdomains the ecacy of the natural norm scm is illustrated through numerical results for two types of non coercive problems the helmholtz equation for acoustics elasticity and electromagnetics and the convection diusi on equation

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

certified reduced basis vms smagorinsky model for natural convection flow in a cavity with variable height abstract in this work we present a reduced basis vms smagorinsky boussinesq model applied to natural convection problems in a variable height cavity in which the buoyancy forces are involved we take into account in this problem both physical and geometrical parametrizations considering the rayleigh number as a parameter so as the height of the cavity we perform an empirical interpolation method to approximate the sub grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters we construct an a posteriori error estimator based upon the brezzi rappaz raviart theory used in the greedy algorithm for the selection of the basis functions finally we present several numerical tests for different parameter configuration

taylor approximation and variance reduction for pde constrained optimal control under uncertainty abstract in this work we develop a scalable computational framework for the solution of pde constrained optimal control problems under high dimensional uncertainty specifically we consider a mean variance formulation of the control objective and employ a taylor expansion with respect to the uncertain parameter field either to directly approximate the control objective or as a control variate for variance reduction the expressions for the mean and variance of the taylor approximation are known analytically although their evaluation requires efficient computation of the trace of the preconditioned hessian of the control objective we propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the hessian on a small number of random directions then the computational work does not depend on the nominal dimension of the uncertain parameter but depends only on the effective dimension i e the rank of the preconditioned hessian thus ensuring scalability to high dimensional problems moreover to increase the estimation accuracy of the mean and variance of the control objective by the taylor approximation we use it as a control variate for variance reduction which results in considerable computational savings several orders of magnitude compared to a plain monte carlo method in summary our approach amounts to solving an optimal control problem constrained by the original pde the generalized eigenvalue equations at a small number of eigenfunctions and a set of linearized pdes that arise from the computation of the gradient and hessian of the control objective with respect to the uncertain parameter we use the lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient based optimization method to solve the problem we demonstrate the accuracy efficiency and scalability of the proposed computational method for two examples with high dimensional uncertain parameters subsurface flow in a porous medium modeled as an elliptic pde and turbulent jet flow modeled by the reynolds averaged navier stokes equations coupled with a nonlinear advection diffusion equation characterizing model uncertainty in particular for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

optimal control of two and three dimensional incompressible navier stokes flows the focus of this work is on the development of large scale numerical optimization methods for optimal control of steady incompressible navier stokes flows the control is affected by the suction or injection of fluid on portions of the boundary and the objective function represents the rate at which energy is dissipated in the fluid we develop reduced hessian sequential quadratic programming methods that avoid converging the flow equations at each iteration both quasi newton and newton variants are developed and compared to the approach of eliminating the flow equations and variables which is effectively the generalized reduced gradient method optimal control problems are solved for two dimensional flow around a cylinder and three dimensional flow around a sphere the examples demonstrate at least an order of magnitude reduction in time taken allowing the optimal solution of flow control problems in as little as half an hour on a desktop workstation

reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries roles of the inf sup stability constants in this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady stokes flows in affinely parametrized geometries focusing on the role played by the brezzi s and babuska s stability constants the crucial ingredients of the methodology are a galerkin projection onto a low dimensional space of basis functions properly selected an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables the combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction ideally suited for real time simulation and many query contexts e g optimization control or parameter identification in particular in this work we focus on i the stability of the reduced basis approximation based on the brezzi s saddle point theory and the introduction of a supremizer operator on the pressure terms ii a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the babuska s inf sup constant including residuals calculations iii the computation of a lower bound of the stability constant and iv different options for the reduced basis spaces construction we present some illustrative results for both interior and external steady stokes flows in parametrized geometries representing two parametrized classical poiseuille and couette flows a channel contraction and a simple flow control problem around a curved obstacle

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations in this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by stokes equations we prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data the finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space respectively resulting in many large scale optimality systems to solve in order to reduce the unaffordable computational effort we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse grid techniques a weighted a posteriori error bound highlights the contribution stemming from each method numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient especially for solving high dimensional and large scale optimization problems

stabilized reduced basis method for parametrized advection diffusion pdes the aim of this master thesis is to study and develop a stabilized reduced basis method suitable for the approximation of the solution of parametrized advection diffusion pdes with high peclet number that is roughly the ratio between the advection coefficient and the diffusion one advection diffusion equations are very important in many engineering applications because they are used to model for example heat transfer phenomena or the diffusion of pollutants in the atmosphere in such applications we often need very fast evaluations of the approximated solution depending on some input parameters this happens for example in the case of real time simulation moreover we need rapid evaluations also if we have to perform repeated approximation of the solution for different input parameters an important case of this many query situation is represented by some optimization problems in which the objective function to optimize depends on the parameters through the solution of a pde the reduced basis rb method meets our need for rapidity and it is also able to guarantee the reliability of the solution thanks to sharp a posteriori error bounds we can find in literature many works about the application of the rb method to advection diffusion problems but they mainly deal with equations in which the peclet number is low the need for stabilization arise from the fact that the finite element fe approximated solution that the rb method aims to recover shows strong instability problems that have to be fixed in this work we want to go further in the study of the stabilization of the rb method for advection dominated problem in both steady and unsteady case as regards the steady case we first compare two possible stabilization strategies by testing them on some test problems in order to design an efficient stabilized reduced basis method we will then test this method using the piecewise quadratic fe approximation as reference solution instead of the usual piecewise linear one we extend the method designed for the steady case to the time dependent case and we will carry out some numerical tests

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

bezier extraction and adaptive refinement of truncated hierarchical nurbs abstract this contribution presents bezier extraction of truncated hierarchical b splines and the application of the approach to adaptive isogeometric analysis the developed procedures allow for the implementation of hierarchical b splines and nurbs without the need for an explicit truncation of the basis moreover standard procedures of adaptive finite element analysis for error estimation and marking of elements are directly applicable due to the strict use of an element viewpoint starting from a multi level nested mesh that results from uniform h refinement standard bezier extraction is applied to active elements that contribute to the hierarchical approximation this results in a multi level system of equations without communication between individual hierarchy levels a hierarchical subdivision operator is developed to recover this communication by transforming the multi level system of equations into a hierarchical system of equations it is demonstrated that this approach implicitly defines the truncated hierarchical basis in terms of a simple matrix multiplication in this way the implementation effort is reduced to a minimum as shape function routines and bezier extraction procedures remain unchanged compared to standard isogeometric analysis the convergence and the computational efficiency of the approach are examined in three different demonstration problems of heat conduction linear elasticity and the phase field modelling of brittle fracture

a reduced basis for a local high definition wind model abstract in this paper we present an application of the reduced basis method to a local high definition adjusted wind model the model provides a precise description of the wind in 3d and takes into account topography and thermal gradients on the surface by solving only 2d linear equations the buoyancy forces slope effects and mass conservation are also considered the wind field is adjusted to the point measurements through an optimal control problem in which the wind flux acts as a control on the boundary in order to use a reduced basis method we consider an affine decomposition in terms of the parameter related to the friction coefficient and the wind measures at some given observation points we also design an a posteriori error estimator that is needed to conduct our reduced basis process finally two numerical examples are presented a test problem and a real data scenario we corroborate the correct behavior of the method in both cases

reduced basis method and a posteriori error estimation for parametrized linear quadratic optimal control problems we propose the reduced basis method for the solution of parametrized optimal control problems described by parabolic partial differential equations in the unconstrained case the method which is based on an off line on line decomposition procedure allows at the on line step large computational cost savings with respect to the truth approximation used for defining the reduced basis an a posteriori error estimate is provided by means of the goal oriented analysis thus associating an error bound to each optimal solution of the parametrized optimal control problem and answering to the demand for a reliable method an adaptive procedure led by the a posteriori error estimate is considered for the generation of the reduced basis space which is set according to the optimal primal and dual solutions of the optimal control problem at hand

multi space reduced basis preconditioners for parametrized stokes equations abstract we introduce a two level preconditioner for the efficient solution of large scale saddle point linear systems arising from the finite element fe discretization of parametrized stokes equations this preconditioner extends the multi space reduced basis msrb preconditioning method proposed in dal santo et al 2018 it combines an approximated block fine grid preconditioner with a reduced basis rb solver which plays the role of coarse component a sequence of rb spaces constructed either with an enriched velocity formulation or a petrov galerkin projection is built each rb coarse component is defined to perform a single iteration of the iterative method at hand the flexible gmres fgmres algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time numerical test cases for stokes flows in three dimensional parameter dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings