1,408 research outputs found

    A Multiscale Finite Element Method for an Elliptic Distributed Optimal Control Problem with Rough Coefficients and Control Constraints

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    We construct and analyze a multiscale finite element method for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for smooth problems and present corroborating numerical results.Comment: 26 page

    Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

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    We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (LL^\infty) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution HH) minimizing the L2L^2 norm of the source terms; its (pre-)computation involves minimizing O(Hd)\mathcal{O}(H^{-d}) quadratic (cell) problems on (super-)localized sub-domains of size O(Hln(1/H))\mathcal{O}(H \ln (1/ H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d3d\leq 3, and polyharmonic for d4d\geq 4, for the operator -\diiv(a\nabla \cdot) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (O(H)\mathcal{O}(H) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincar\'{e} inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue (2013

    Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT

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    We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients σ(x)\sigma(x) in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions s(x)s(x) over the domain Ω\Omega. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of Ω\Omega when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we obtain an optimally robust homogenization algorithm for arbitrary rough coefficients. Next, we consider inverse homogenization and show how to decompose it into a linear ill-posed problem and a well-posed non-linear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT). It is known that the EIT problem admits at most one isotropic solution. If an isotropic solution exists, we show how to compute it from any conductivity having the same boundary Dirichlet-to-Neumann map. It is known that the EIT problem admits a unique (stable with respect to GG-convergence) solution in the space of divergence-free matrices. As such we suggest that the space of convex functions is the natural space in which to parameterize solutions of the EIT problem

    A multiscale method for heterogeneous bulk-surface coupling

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    In this paper, we construct and analyze a multiscale (finite element) method for parabolic problems with heterogeneous dynamic boundary conditions. As origin, we consider a reformulation of the system in order to decouple the discretization of bulk and surface dynamics. This allows us to combine multiscale methods on the boundary with standard Lagrangian schemes in the interior. We prove convergence and quantify explicit rates for low-regularity solutions, independent of the oscillatory behavior of the heterogeneities. As a result, coarse discretization parameters, which do not resolve the fine scales, can be considered. The theoretical findings are justified by a number of numerical experiments including dynamic boundary conditions with random diffusion coefficients
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