100 research outputs found
Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method
This paper discusses the computation of derivatives for optimization problems
governed by linear hyperbolic systems of partial differential equations (PDEs)
that are discretized by the discontinuous Galerkin (dG) method. An efficient
and accurate computation of these derivatives is important, for instance, in
inverse problems and optimal control problems. This computation is usually
based on an adjoint PDE system, and the question addressed in this paper is how
the discretization of this adjoint system should relate to the dG
discretization of the hyperbolic state equation. Adjoint-based derivatives can
either be computed before or after discretization; these two options are often
referred to as the optimize-then-discretize and discretize-then-optimize
approaches. We discuss the relation between these two options for dG
discretizations in space and Runge-Kutta time integration. Discretely exact
discretizations for several hyperbolic optimization problems are derived,
including the advection equation, Maxwell's equations and the coupled
elastic-acoustic wave equation. We find that the discrete adjoint equation
inherits a natural dG discretization from the discretization of the state
equation and that the expressions for the discretely exact gradient often have
to take into account contributions from element faces. For the coupled
elastic-acoustic wave equation, the correctness and accuracy of our derivative
expressions are illustrated by comparisons with finite difference gradients.
The results show that a straightforward discretization of the continuous
gradient differs from the discretely exact gradient, and thus is not consistent
with the discretized objective. This inconsistency may cause difficulties in
the convergence of gradient based algorithms for solving optimization problems
Recommended from our members
High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs
Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplinesā boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching
Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation
We consider one-dimensional distributed optimal control problems with the state equa-tion being given by the viscous Burgers equation. We discretize using a space-time dis-continuous Galerkin approach. We use upwind ļ¬ux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our ļ¬ndings by numerical results
An energy stable approach for discretizing hyperbolic equations with nonconforming discontinuous Galerkin methods
When nonconforming, discontinuous Galerkin methods are implemented for hyperbolic equations using quadrature, exponential energy growth can result even when the underlying scheme with exact integration does not support such growth. Using linear elasticity as a model problem, we proposes a skew-symmetric formulation that has the same energy stability properties for both exact and inexact, quadrature-based integration. These stability properties are maintained even when the material properties are variable and discontinuous, and the elements non-affine (e.g., curved). The analytic stability results are confirmed through numerical experiments demonstrating the stability as well as the accuracy of the method.National Science Foundation (NSF)Office of Naval Research (ONR)EAR-1547596 (NSF)N0001416WX02190 (ONR
GPU-accelerated discontinuous Galerkin methods on hybrid meshes: applications in seismic imaging
Seismic imaging is a geophysical technique assisting in the understanding of subsurface structure on a regional and global scale. With the development of computer technology, computationally intensive seismic algorithms have begun to gain attention in both academia and industry. These algorithms typically produce high-quality subsurface images or models, but require intensive computations for solving wave equations.
Achieving high-fidelity wave simulations is challenging: first, numerical wave solutions may suffer from dispersion and dissipation errors in long-distance propagations; second, the efficiency of wave simulators is crucial for many seismic applications. High-order methods have advantages of decreasing numerical errors efficiently and hence are ideal for wave modelings in seismic problems.
Various high order wave solvers have been studied for seismic imaging. One of the most popular solvers is the finite difference time domain (FDTD) methods. The strengths of finite difference methods are the computational efficiency and ease of implementation, but the drawback of FDTD is the lack of geometric flexibility. It has been shown that standard finite difference methods suffer from first order numerical errors at sharp media interfaces.
In contrast to finite difference methods, discontinuous Galerkin (DG) methods, a class of high-order numerical methods built on unstructured meshes, enjoy geometric flexibility and smaller interface errors. Additionally, DG methods are highly parallelizable and have explicit semi-discrete form, which makes DG suitable for large-scale wave simulations. In this dissertation, the discontinuous Galerkin methods on hybrid meshes are developed and applied to two seismic algorithms---reverse time migration (RTM) and full waveform inversion (FWI).
This thesis describes in depth the steps taken to develop a forward DG solver for the framework that efficiently exploits the element specific structure of hexahedral, tetrahedral, prismatic and pyramidal elements. In particular, we describe how to exploit the tensor-product property of hexahedral elements, and propose the use of hex-dominant meshes to speed up the computation.
The computational efficiency is further realized through a combination of graphics processing unit (GPU) acceleration and multi-rate time stepping. As DG methods are highly parallelizable, we build the DG solver on multiple GPUs with element-specific kernels. Implementation details of memory loading, workload assignment and latency hiding are discussed in the thesis. In addition, we employ a multi-rate time stepping scheme which allows different elements to take different time steps.
This thesis applies DG schemes to RTM and FWI to highlight the strengths of the DG methods. For DG-RTM, we adopt the boundary value saving strategy to avoid data movement on GPUs and utilize the memory load in the temporal updating procedure to produce images of higher qualities without a significant extra cost. For DG-FWI, a derivation of the DG-specific adjoint-state method is presented for the fully discretized DG system. Finally, sharp media interfaces are inverted by specifying perturbations of element faces, edges and vertices
A Hybrid Radial Basis Function - Pseudospectral Method for Thermal Convection in a 3-D Spherical Shell
A novel hybrid spectral method that combines radial basis function (RBF) and Chebyshev pseudospectral (PS) methods in a ā2+1ā approach is presented for numerically simulating thermal convection in a 3-D spherical shell. This is the first study to apply RBFs to a full 3D physical model in spherical geometry. In addition to being spectrally accurate, RBFs are not defined in terms of any surface based coordinate system such as spherical coordinates. As a result, when used in the lateral directions, as in this study, they completely circumvent the pole issue with the further advantage that nodes can be āscatteredā over the surface of a sphere. In the radial direction, Chebyshev polynomials are used, which are also spectrally accurate and provide the necessary clustering near the boundaries to resolve boundary layers. Applications of this new hybrid methodology are given to the problem of convection in the Earthās mantle,which is modeled by a Boussinesq fluid at infinite Prandtl number. To see whether this numerical technique warrants further investigation, the study limits itself to an isoviscous mantle.Benchmark comparisons are presented with other currently used mantle convection codes for Rayleigh number 7 Ā· 103 and 105. The algorithmic simplicity of the code (mostly due to RBFs)allows it to be written in less than 400 lines of Matlab and run on a single workstation. We find that our method is very competitive with those currently used in the literature
Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations
High-order Hadamard-form entropy stable multidimensional summation-by-parts
discretizations of the Euler and compressible Navier-Stokes equations are
considerably more expensive than the standard divergence-form discretization.
In search of a more efficient entropy stable scheme, we extend the
entropy-split method for implementation on unstructured grids and investigate
its properties. The main ingredients of the scheme are Harten's entropy
functions, diagonal- summation-by-parts operators with diagonal
norm matrix, and entropy conservative simultaneous approximation terms (SATs).
We show that the scheme is high-order accurate and entropy conservative on
periodic curvilinear unstructured grids for the Euler equations. An entropy
stable matrix-type interface dissipation operator is constructed, which can be
added to the SATs to obtain an entropy stable semi-discretization.
Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta
method. Entropy stable viscous SATs, applicable to both the Hadamard-form and
entropy-split schemes, are developed for the compressible Navier-Stokes
equations. In the absence of heat fluxes, the entropy-split scheme is entropy
stable for the compressible Navier-Stokes equations. Local conservation in the
vicinity of discontinuities is enforced using an entropy stable hybrid scheme.
Several numerical problems involving both smooth and discontinuous solutions
are investigated to support the theoretical results. Computational cost
comparison studies suggest that the entropy-split scheme offers substantial
efficiency benefits relative to Hadamard-form multidimensional SBP-SAT
discretizations.Comment: 34 pages, 8 figure
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