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Geometric optimization of the evaluation of finite element matrices
 SIAM J. Sci. Comput
"... Abstract. Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for ..."
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Cited by 16 (13 self)
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Abstract. Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for the Laplace operator and for the trilinear form for NavierStokes. For the Laplace operator in two space dimensions, we have developed a heuristic graph algorithm that searches for such redundancies and generates code for computing the local stiffness matrices. Up to cubics, we are able to build the stiffness matrix on any triangle in less than one multiplyadd pair per entry. Up to sixth degree, we can do it in less than about two. Preliminary lowdegree results for Poisson and NavierStokes operators in three dimensions are also promising.
A framework for the adaptive finite element solution of large inverse problems. I. Basic techniques
, 2004
"... Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of a ..."
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Cited by 14 (7 self)
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Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of adaptive finite element schemes, solvers for the large linear systems arising from discretization, and methods to treat additional information in the form of inequality constraints on the parameter to be recovered. The methods to be developed will be based on an allatonce approach, in which the inverse problem is solved through a Lagrangian formulation. The main feature of the paper is the use of a continuous (function space) setting to formulate algorithms, in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed. This entails that steps such as the description of a Newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions. On the other hand, this approach avoids the dependence of finite dimensional norms on the mesh size, making individual steps of the algorithm comparable even if they used differently refined meshes. Numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10,000 parameters. Key words. Adaptive finite elements, inverse problems, Newton method on function spaces. AMS subject classifications. 65N21,65K10,35R30,49M15,65N50 1. Introduction. Parameter
A Finite Element Approach to the Immersed Boundary Method
, 2004
"... The immersed boundary method was introduced by Peskin in [31] to study the blood flow in the heart and further applied to many situations where a fluid interacts with an elastic structure. The basic idea is to consider the structure as a part of the fluid where additional forces are applied and addi ..."
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Cited by 13 (7 self)
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The immersed boundary method was introduced by Peskin in [31] to study the blood flow in the heart and further applied to many situations where a fluid interacts with an elastic structure. The basic idea is to consider the structure as a part of the fluid where additional forces are applied and additional mass is localized. The forces exerted by the structure on the fluid are taken into account as a source term in the NavierStokes equations and are mathematically described as a Dirac delta function lying along the immersed structure. In this paper we first review on various ways of modeling the elastic forces in different physical situations. Then we focus on the discretization of the immersed boundary method by means of finite elements which can handle the Dirac delta function variationally avoiding the introduction of its regularization. Practical computational aspects are described and some preliminary numerical experiment in two dimensions are reported.
AUTOMATING THE FINITE ELEMENT METHOD
, 2006
"... The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations. However, the generality of the framework provided by the finit ..."
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Cited by 13 (2 self)
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The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations. However, the generality of the framework provided by the finite element method is seldom reflected in implementations (realizations), which are often specialized and can handle only a small set of variational problems and finite elements (but are typically parametrized over the choice of mesh). This paper reviews ongoing research in the direction of a complete automation of the finite element method. In particular, this work discusses algorithms for the efficient and automatic computation of a system of discrete equations from a given variational problem, finite element and mesh. It is demonstrated that by automatically generating and compiling efficient lowlevel code, it is possible to parametrize a finite element code over variational problem and finite element in addition to the mesh.
Efficient compilation of a class of variational forms
 ACM Transactions on Mathematical Software
, 2007
"... We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry ..."
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Cited by 10 (5 self)
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We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry and variable coefficients. Based on this representation theorem, we design an algorithm for efficient pretabulation of the reference tensor. The new algorithm has been implemented in the FEniCS Form Compiler (FFC) and improves on a previous loopbased implementation by several orders of magnitude, thus shortening compiletimes and development cycles for users of FFC. Categories and Subject Descriptors: G.4 [Mathematical Software]—Algorithm design and analysis,
Reconstructions in Ultrasound Modulated Optical Tomography
"... Abstract. We describe a mathematical model for ultrasound modulated optical tomography and present a simple reconstruction algorithm and numerical simulations for this model. The computational results show that stable reconstruction of sharp features of the absorption coefficient is possible. A form ..."
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Cited by 3 (1 self)
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Abstract. We describe a mathematical model for ultrasound modulated optical tomography and present a simple reconstruction algorithm and numerical simulations for this model. The computational results show that stable reconstruction of sharp features of the absorption coefficient is possible. A formal linearization of the model leads to an equation with a Fredholm operator, which explains the stability observed in our numerical experiments. Submitted to: Inverse Problems 1.
hpGEM – A software framework for discontinuous Galerkin finite element methods. submitted to
 ACM Trans. Math. Sofw
, 2007
"... discontinuous Galerkin finite element methods ..."
On the stability of the finite element immersed boundary method
 In preparation
, 2006
"... The immersed boundary (IB) method is a mathematical formulation for fluidstructure interaction problems, where immersed incompressible viscoelastic bodies or boundaries interact with an incompressible fluid. The original numerical scheme associated to the IB method requires a smoothed approximatio ..."
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Cited by 1 (1 self)
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The immersed boundary (IB) method is a mathematical formulation for fluidstructure interaction problems, where immersed incompressible viscoelastic bodies or boundaries interact with an incompressible fluid. The original numerical scheme associated to the IB method requires a smoothed approximation of the Dirac delta distribution to link the moving Lagrangian domain with the fixed Eulerian one. We present a stability analysis of the finite element immersed boundary method, where the Dirac delta distribution is treated variationally, in a generalized viscoelastic framework and for two different time stepping schemes. Key words: immersed boundary method, finite element method, numerical stability. 1
Numerical Validation of a Class of Mixed Discontinuous Galerkin Methods for Darcy Flow ∗
, 2006
"... In this paper we present a thorough set of numerical experiments to confirm the robustness of several Discontinuous Galerkin (DG) methods proposed for the approximation of Darcy law. We perform fairly two and threedimensional tests involving both low and high order approximations. Convergence test ..."
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Cited by 1 (0 self)
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In this paper we present a thorough set of numerical experiments to confirm the robustness of several Discontinuous Galerkin (DG) methods proposed for the approximation of Darcy law. We perform fairly two and threedimensional tests involving both low and high order approximations. Convergence tests carried out on Cartesian grids as well as on structured and unstructured triangular grids confirm the theoretical error estimates in the pressure energy norm and in the L 2 norm of the velocity for both the symmetric and the nonsymmetric DG schemes considered. Moreover, an optimal convergence rate in the L 2 norm of the pressure error is observed in the cases where adjoint consistency does not hold, provided s = min{p + 1, ℓ} is odd, p and ℓ being the polynomial degrees of the velocity and the pressure approximations, respectively. Test of accuracy and robustness involving a discontinuous permeability coefficient as well as a bounded smooth permeability function are also presented. 1