8,262 research outputs found
Space-time adaptive solution of inverse problems with the discrete adjoint method
Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method.
This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms.
In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, -refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem
A fully discrete framework for the adaptive solution of inverse problems
We investigate and contrast the differences between the discretize-then-differentiate and differentiate-then-discretize approaches to the numerical solution of parameter estimation problems. The former approach is attractive in practice due to the use of automatic differentiation for the generation of the dual and optimality equations in the first-order KKT system. The latter strategy is more versatile, in that it allows one to formulate efficient mesh-independent algorithms over suitably chosen function spaces. However, it is significantly more difficult to implement, since automatic code generation is no longer an option. The starting point is a classical elliptic inverse problem. An a priori error analysis for the discrete optimality equation shows consistency and stability are not inherited automatically from the primal discretization. Similar to the concept of dual consistency, We introduce the concept of optimality consistency. However, the convergence properties can be restored through suitable consistent modifications of the target functional. Numerical tests confirm the theoretical convergence order for the optimal solution. We then derive a posteriori error estimates for the infinite dimensional optimal solution error, through a suitably chosen error functional. This estimates are constructed using second order derivative information for the target functional. For computational efficiency, the Hessian is replaced by a low order BFGS approximation. The efficiency of the error estimator is confirmed by a numerical experiment with multigrid optimization
Domain decomposition finite element/finite difference method for the conductivity reconstruction in a hyperbolic equation
We present domain decomposition finite element/finite difference method for
the solution of hyperbolic equation. The domain decomposition is performed such
that finite elements and finite differences are used in different subdomains of
the computational domain: finite difference method is used on the structured
part of the computational domain and finite elements on the unstructured part
of the domain. The main goal of this method is to combine flexibility of finite
element method and efficiency of a finite difference method.
An explicit discretization schemes for both methods are constructed such that
finite element and finite difference schemes coincide on the common structured
overlapping layer between computational subdomains. Then the resulting scheme
can be considered as a pure finite element scheme which allows avoid
instabilities at the interfaces.
We illustrate efficiency of the domain decomposition method on the
reconstruction of the conductivity function in the hyperbolic equation in three
dimensions
Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide
We consider an inverse problem of reconstructing the conductivity function in
a hyperbolic equation using single space-time domain noisy observations of the
solution on the backscattering boundary of the computational domain. We
formulate our inverse problem as an optimization problem and use Lagrangian
approach to minimize the corresponding Tikhonov functional. We present a
theorem of a local strong convexity of our functional and derive error
estimates between computed and regularized as well as exact solutions of this
functional, correspondingly. In numerical simulations we apply domain
decomposition finite element-finite difference method for minimization of the
Lagrangian. Our computational study shows efficiency of the proposed method in
the reconstruction of the conductivity function in three dimensions
An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations
We propose an adaptive finite element method for the solution of a
coefficient inverse problem of simultaneous reconstruction of the dielectric
permittivity and magnetic permeability functions in the Maxwell's system using
limited boundary observations of the electric field in 3D. We derive a
posteriori error estimates in the Tikhonov functional to be minimized and in
the regularized solution of this functional, as well as formulate corresponding
adaptive algorithm. Our numerical experiments justify the efficiency of our a
posteriori estimates and show significant improvement of the reconstructions
obtained on locally adaptively refined meshes.Comment: Corrected typo
Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations
We consider the inverse problem of the simultaneous reconstruction of the
dielectric permittivity and magnetic permeability functions of the Maxwell's
system in 3D with limited boundary observations of the electric field. The
theoretical stability for the problem is provided by the Carleman estimates.
For the numerical computations the problem is formulated as an optimization
problem and hybrid finite element/difference method is used to solve the
parameter identification problem.Comment: in Inverse Problems and Imaging Volume: 9, Number: 1 February 2015.
arXiv admin note: text overlap with arXiv:1510.0752
- …