Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

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Nonlinear Model Reduction via Discrete Empirical Interpolation

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

/pdf/a-survey-of-projection-based-model-reduction-methods-for-3yjqbxk305.pdf

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

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Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

/pdf/reliable-real-time-solution-of-parametrized-partial-1wnffkep9s.pdf

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

We propose a new reduced-basis output bound method for the symmetric eigenvalue problem. The numerical procedure consists of two stages: the pre-processing stage, in which the reduced basis and associated functions are computed—“off-line”—at a prescribed set of points in parameter space; and the real-time stage, in which the approximate output of interest and corresponding rigorous error bounds are computed—“on-line”—for any new parameter value of interest. The real time calculation is very inexpensive as it requires only the solution or evaluation of very small systems. We introduce the procedure; prove the asymptotic bounding properties and optimal convergence rate of the error estimator; discuss computational considerations; and, finally, present corroborating numerical results.

Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations -- Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation -- relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures -- methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage -- in which, given a new parameter value, we calculate the output of interest and associated error bound -- depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

https://hal.archives-ouvertes.fr/hal-00798352/document

Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations

In this paper, we extend reduced-basis output bound methods developed earlier for elliptic problems, to problems described by ‘parameterized parabolic’ partial differential equations. The essential new ingredient and the novelty of this paper consist in the presence of time in the formulation and solution of the problem. First, without assuming a time discretization, a reduced-basis procedure is presented to ‘efficiently’ compute accurate approximations to the solution of the parabolic problem and ‘relevant’ outputs of interest. In addition, we develop an error estimation procedure to ‘a posteriori validate’ the accuracy of our output predictions. Second, using the discontinuous Galerkin method for the temporal discretization, the reduced-basis method and the output bound procedure are analysed for the semi-discrete case. In both cases the reduced-basis is constructed by taking ‘snapshots’ of the solution both in time and in the parameters: in that sense the method is close to Proper Orthogonal Decomposition (POD).

/pdf/reduced-basis-output-bound-methods-for-parabolic-problems-nkz5h4mcmp.pdf

Reduced-basis output bound methods for parabolic problems

We present a “flux-free” nodal Neumann subproblem approach for the inexpensive computation of lower and upper bounds for output functionals of partial differential equations. The method resembles earlier nodal Dirichlet subproblem a posteriori approaches, except that the underlying partition of unity now appears explicitly in a modified residual; the latter ensures solvability of the corresponding Neumann problems, which, in turn, provides the requisite constant-free bounds. The new approach is considerably simpler to analyze and, more importantly, to implement, than previous hybrid-flux elemental Neumann subproblem techniques.

Luc Machiels

Papers

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations

Reduced-basis output bound methods for parabolic problems

A flux-free nodal Neumann subproblem approach to output bounds for partial differential equations