Results 1  10
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140
CHARMS: A Simple Framework for Adaptive Simulation
 ACM Transactions on Graphics
, 2002
"... Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, bui ..."
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Cited by 130 (9 self)
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Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order Bsplines, Loop subdivision, etc.). The (un)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thinshell animations.
Data Oscillation and Convergence of Adaptive FEM
 SIAM J. Numer. Anal
, 1999
"... Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscillation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient ..."
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Cited by 93 (11 self)
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Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscillation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient adaptive FEM for elliptic PDE with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in 2d and 3d yield quasioptimal meshes along with a competitive performance. Key words. A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, performance, quasioptimal meshes 1991 AMS subject classification. 65N12, 65N15, 65N30, 65N50, 65Y20 1 Introduction and Main Results Adaptive procedures for the numerical solution of partial differential equations (PDE) started in the late 70's and are now standard tools...
Wavelets on Manifolds I: Construction and Domain Decomposition
 SIAM J. Math. Anal
, 1998
"... The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features ar ..."
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Cited by 84 (20 self)
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The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...
Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints
 THE JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
, 2004
"... Regularization of illposed linear inverse problems via ℓ1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ℓ1 penalized functional is via an iterative softthresholding algorithm. We propose an alternative implem ..."
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Cited by 68 (10 self)
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Regularization of illposed linear inverse problems via ℓ1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ℓ1 penalized functional is via an iterative softthresholding algorithm. We propose an alternative implementation to ℓ1constraints, using a gradient method, with projection on ℓ1balls. The corresponding algorithm uses again iterative softthresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.
Adaptive Wavelet Methods II  Beyond the Elliptic Case
 FOUND. COMPUT. MATH
, 2000
"... This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmet ..."
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Cited by 54 (14 self)
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This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [DKS]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now wellposed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a signicant modication of the ideas from [CDD]. The main departure from [CDD] is to develop an iterative scheme that directly applies to the innite dimensional problem rather than nite subproblems derived from the infinite problem. This rests on an adaptive application of the innite dimensional operator to finite vectors representing elements from finite dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding waveletbest Nterm approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces like the LBB condition no longer arise.
Fully adaptive multiresolution finite volume schemes for conservation laws
 Math. Comp
, 2003
"... Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at ..."
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Cited by 41 (13 self)
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Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity. 1.
ElementByElement Construction Of Wavelets Satisfying Stability And Moment Conditions
 SIAM J. NUMER. ANAL
, 1998
"... In this paper, we construct a class of locally supported wavelet bases for C 0 Lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H s for jsj ! 3 2 (jsj 1 on Lipschitz' manifolds), and ..."
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Cited by 34 (11 self)
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In this paper, we construct a class of locally supported wavelet bases for C 0 Lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H s for jsj ! 3 2 (jsj 1 on Lipschitz' manifolds), and the wavelets can, in principal, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used e.g. for constructing an optimal solver of discretized H s elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet ba...
Adaptive Solution Of Operator Equations Using Wavelet Frames
 SIAM J. Numer. Anal
, 2002
"... In "Adaptive Wavelet Methods II ... ..."
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