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Hierarchical matrix approximations of Hessians arising in inverse problems governed by PDEs. (English) Zbl 1453.65289
The authors analyze the role of Hessians in deterministic or Bayesian inverse problems governed by partial differential equations. When discretizing over a $$d$$-dimensional domain $$\Omega$$, an inverse problem governed by a partial differential equation with distributed parameter fields to be inferred from data, an optimization problem of the form $$minimize_{m}$$ $$J(m)=F(u(m))+\alpha R(m)$$ is obtained, where the state variables $$u(m)\in \mathbb{R}^{N}$$ depend on the model parameters $$m\in\mathbb{R}^{n}$$ via the solution of the discretized partial differential equations $$g(m,u)=K(m)u-f=0$$, with coefficient matrix $$K\in \mathbb{R}^{N\times N}$$ and source $$f\in \mathbb{R}^{N}$$. The data misfit term $$F(u(m))$$ measures the difference between recorded observations and the response of the model at receivers placed on the boundary or the interior of $$\Omega$$. The authors introduce the partial derivative $$G=\partial _{m}g=\partial _{m}K\times _{2}u$$, where $$\partial _{m}K$$ is a third-order tensor of size $$N\times N\times n$$ and $$\times _{2}$$ is a 2-mode tensor vector product operation. They observe that, unlike the local sparse matrices $$K$$ and $$G$$, the inverse operator $$K^{-1}$$ is a discretized, nonlocal solution operator and is in general formally dense. They compute the gradient of the data misfit term of the objective function as $$\nabla _{m}F=(\partial _{m}K\times _{2}u)^{T}p=-G^{T}K^{-T}\partial _{u}F$$, where $$p\in \mathbb{R}^{N}$$ is an adjoint variable defined via the adjoint equations $$l(m,p)=K^{T}(m)p+ \partial _{u}F=0$$. The authors then compute the Hessian $$\nabla_{mm}^{2}F=G^{T}K^{-T}\partial_{uu}^{2}FK^{-1}G-G^{T}K^{-T}L-L^{T}K^{-1}G+(\partial _{mm}^{2}K\times _{2}u)\times _{1}p$$, with $$L=\partial _{m}l=\partial _{m}K^{T}\times _{2}p$$. They start proving that Hessian matrices can be approximated with low-rank hierarchical matrices. They show how to draw fast operations on hierarchical matrices and how to construct the hierarchical Hessians ab initio. The main part of the paper presents four situations: density inversion in a one-dimensional time-dependent diffusion equation, source inversion in stationary advection diffusion, frequency-domain wave equation inversion, and transient controlled-source electromagnetic inversion. In each case, the authors write the functional to be minimized, the associated partial differential equation, the misfit functional to be considered, and they draw the associated computations. In each case, the authors present the results of numerical simulations.

##### MSC:
 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs 35Q93 PDEs in connection with control and optimization 49M25 Discrete approximations in optimal control 49N45 Inverse problems in optimal control 65F10 Iterative numerical methods for linear systems 65F99 Numerical linear algebra 65Y05 Parallel numerical computation 65Y10 Numerical algorithms for specific classes of architectures 78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
##### Software:
FEniCS; GitHub; hIPPYlib; mctoolbox; MFEM; PCBDDC; PETSc
Full Text:
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