1,020 research outputs found

    Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations

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    We study linear parabolic initial-value problems in a space-time variational formulation based on fractional calculus. This formulation uses "time derivatives of order one half" on the bi-infinite time axis. We show that for linear, parabolic initial-boundary value problems on (0,)(0,\infty), the corresponding bilinear form admits an inf-sup condition with sparse tensor product trial and test function spaces. We deduce optimality of compressive, space-time Galerkin discretizations, where stability of Galerkin approximations is implied by the well-posedness of the parabolic operator equation. The variational setting adopted here admits more general Riesz bases than previous work; in particular, no stability in negative order Sobolev spaces on the spatial or temporal domains is required of the Riesz bases accommodated by the present formulation. The trial and test spaces are based on Sobolev spaces of equal order 1/21/2 with respect to the temporal variable. Sparse tensor products of multi-level decompositions of the spatial and temporal spaces in Galerkin discretizations lead to large, non-symmetric linear systems of equations. We prove that their condition numbers are uniformly bounded with respect to the discretization level. In terms of the total number of degrees of freedom, the convergence orders equal, up to logarithmic terms, those of best NN-term approximations of solutions of the corresponding elliptic problems.Comment: 26 page

    Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

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    This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L2L^2 approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering

    Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

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    This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on σj\sigma_j with jNj\in\N, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters σj\sigma_j. We establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence σ=(σj)j1\sigma = (\sigma_j)_{j\ge 1} of the random inputs, and prove convergence rates of best NN-term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best NN-term truncations can practically be computed, by greedy-type algorithms as in [SG, Gi1], or by multilevel Monte-Carlo methods as in [KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]

    One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems

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    We consider the analysis of a one-parameter family of hphp--version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u) on a bounded open set \om\in\re^d, subject to mixed Dirichlet and Neumann boundary conditions on \pr\om. It is assumed that aa is a real--valued function which is Lipschitz-continuous and uniformly monotonic in its last argument, and ff is a real-valued function which is locally Lipschitz-continuous and satisfies a suitable growth condition in its last argument; both functions are measurable in the first and second arguments. For quasi--uniform hphp--meshes, if u\in \H^1(0,T;\H^k(\om))\cap\L^\infty(0,T;\H^1(\om)) with k312k\geq 3\frac{1}{2}, for discontinuous piecewise polynomials of degree not less than 1, the approximation error, measured in the broken H1H^1 norm, is proved to be the same as in the linear case: O(hs1/pk3/2)\mathscr{O}(h^{s-1}/p^{k-3/2}) with 1smin{p+1,k}1\leq s\leq\min\set{p+1,k}

    Optimal Control of Convective FitzHugh-Nagumo Equation

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    We investigate smooth and sparse optimal control problems for convective FitzHugh-Nagumo equation with travelling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the travelling waves. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian

    Adaptive Algorithms

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    Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations
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