107 research outputs found

    (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods

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    We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method

    An ultraweak space-Time variational formulation for the wave equation: Analysis and efficient numerical solution

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    We introduce an ultraweak space-time variational formulation for the wave equation, prove its well-posedness (even in the case of minimal regularity) and optimal inf-sup stability. Then, we introduce a tensor product-style space-time Petrov–Galerkin discretization with optimal discrete inf-sup stability, obtained by a non-standard definition of the trial space. As a consequence, the numerical approximation error is equal to the residual, which is particularly useful for a posteriori error estimation. For the arising discrete linear systems in space and time, we introduce efficient numerical solvers that appropriately exploit the equation structure, either at the preconditioning level or in the approximation phase by using a tailored Galerkin projection. This Galerkin method shows competitive behavior concerning wall-clock time, accuracy and memory as compared with a standard time-stepping method in particular in low regularity cases. Numerical experiments with a 3D (in space) wave equation illustrate our findings

    Is the Helmholtz equation really sign-indefinite?

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    The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts

    The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces

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    © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019. We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue Lp, 1 < p < ∞. The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in L p {L^{p}}, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection

    Research in the general area of non-linear dynamical systems Final report, 8 Jun. 1965 - 8 Jun. 1967

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    Nonlinear dynamical systems research on systems stability, invariance principles, Liapunov functions, and Volterra and functional integral equation

    Nonlinear Analysis and Optimization with Applications

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    Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

    Techniques for Reconstructing a Riemannian Metric Via the Boundary Control Method

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    In this dissertation, we consider some new techniques related to the solution of the inverse boundary value problem for the wave equation with partial boundary data. Most results are formulated in a geometric setting, where waves propagate in the interior of a smooth manifold with smooth boundary M, and the wave speed is modelled by an unknown Riemannian metric g. For data, we focus mostly on using the Neumann-to-Dirichlet (N-to-D) map with sources and receivers restricted to a measurement set Γ ⊂ ∂M. The goal of the inverse problem, in this setting, is to use these wave boundary measurements to recover the geometry of (M, g) near the measurement set. We note that this geometric perspective accomodates, as special cases, both the scalar acoustic wave equation and elliptically anisotropic wave speeds. We consider three problems. In the first problem, we provide a technique to use the N-to-D map to construct the travel times between interior points with known semi-geodesic coordinates and boundary points belonging to Γ. Such travel times can be used to reconstruct the metric in semi-geodesic coordinates using one of several existing techniques, so this procedure can be viewed as providing a data processing step for a metric reconstruction procedure. In the second problem, we consider a redatuming procedure, where we use data on the boundary and known near-boundary geometry to synthesize wave measurements in this known near-boundary region. This allows us to construct a map which plays a similar role to the N-to-D map, but for interior sources and interior measurements. Our motivation for this procedure is that it can serve as a data propagation step for a layer stripping reconstruction method, in which one first reconstructs the metric near the boundary and then propagates data into this region to serve as data for an interior reconstruction step. In the third problem, we restrict attention to the case where M is a domain in Rn, and consider two related procedures to use the N-to-D map or Dirichlet-to-Neumann (D-to-N) map to directly reconstruct the metric. In the anisotropic case, we construct the metric in semi-geodesic coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case, we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. In addition to providing constructive procedures, we analyze the stability of some steps from these procedures. In particular we consider the stability of the redatuming procedure and the stability of the metric reconstruction procedure from internal data (for the third problem). Moreover, we provide computational experiments to demonstrate our three main procedures

    Homotopy Based Reconstruction from Acoustic Images

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    Mutational Analysis

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    This monograph extends the classical concept of ordinary differential equations in the Euclidean space to nonempty sets which are just supplied with a family of "continuous" distance functions. In particular, these sets are not supposed to have any linear structure or to be metric spaces. The main goal is a joint framework for continuous dynamical systems beyond the traditional border of vector spaces so that examples of completely different origins can be coupled in systems. It is motivated by the mutational equations introduced by Jean-Pierre Aubin in the 1990s. Some of the examples discussed here are: nonlocal set evolutions, semilinear evolution equations, nonlinear transport equations for finite Radon measures, functional stochastic differential equations, parabolic differential equations in noncylindrical domains. This monograph is my revised Habilitationsschrift (i.e. thesis for a postdoctoral lecture qualification in Germany) submitted to the Faculty of Mathematics and Computer Science at Heidelberg University in January 2009
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