Dynamic Data-Driven Inversion for Terascale Simulations: Real-Time Identification of Airborne Contaminants

Abstract

In contrast to traditional terascale simulations that have known, fixed data inputs, dynamic data-driven (DDD) applications are characterized by unknown data and informed by dynamic observations. DDD simulations give rise to inverse problems of determining unknown data from sparse observations. The main difficulty is that the optimality system is a boundary value problem in 4D space-time, even though the forward simulation is an initial value problem. We construct special-purpose parallel multigrid algorithms that exploit the spectral structure of the inverse operator. Experiments on problems of localizing airborne contaminant release from sparse observations in a regional atmospheric transport model demonstrate that 17-million-parameter inversion can be effected at a cost of just 18 forward simulations with high parallel efficiency. On 1024 Alphaserver EV68 processors, the turnaround time is just 29 minutes. Moreover, inverse problems with 135 million parameters-corresponding to 139 billion total space-time unknowns-are solved in less than 5 hours on the same number of processors. These results suggest that ultra-high resolution data-driven inversion can be carried out sufficiently rapidly for simulation-based "real-time" hazard assessment.

Model Variational Inverse Problems Governed by Partial Differential Equations

“…We consider a time-dependent advection-diffusion equation, in which we invert for an unknown initial condition. The problem can be interpreted as finding the initial distribution of a contaminant from measurements taken after the contaminant has been subjected to diffusive transport [1]. Let Ω ⊂ R n be open and bounded (we choose n = 2 in the sequel) and consider measurements on a part Γ m ⊂ ∂Ω of the boundary over the time horizon [T 1 , T ], with 0 < T 1 < T .…”

“…Nevertheless, some knowledge of the structure underlying these packages is required since the optimality systems arising in inverse problems with PDEs often cannot be solved using generic PDE solvers, which do not exploit the optimization structure of the problems. For illustration purposes, this report includes implementations of the model problems in COMSOL Multiphysics (linked together with MATLAB) 1 . Since our implementations use little finite element functionality that is specific to COMSOL Multiphysics, only few code pieces have to be changed in order to have these implementations available in other finite element packages.…”

Optimal order multilevel preconditioners for regularized ill-posed problems

“…The modulus of the analytic function f : B α (1) → C defined by f (z) = ln z/(1 − z) attains its extreme values on the boundary ∂B α (1), as f has no zeros in B α (1). On the circle C α (1) = {ζ : |1 − ζ| = α} we have |f (z)| = α −1 |ln z|; hence the extreme values of |f (z)| are attained at the same points as the extremes of |ln z| 2 .…”

“…Therefore unpreconditioned Krylov-type methods are expected to be efficient in inverting H h β . For example, we can show that, if K is the solution operator for the heat equation mapping the initial value onto the final-time state, then the number of conjugate gradient (CG) iterations required to solve the regularized inverse problem down to machine precision is mesh-independent; moreover, the needed number of iterations grows only logarithmically as β → 0 (see Chapter 7 in [16], and also [12,1]). However, the number of iterations needed for convergence, even though independent of resolution, may be too large for practical use in the case of large-scale problems, where the application of K h (i.e., the direct problem) requires, for example, solving a time-dependent three-dimensional partial differential equation.…”

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

“…Inverse problems arise frequently in the context of earth sciences, such as hydraulic tomography [1][2][3][4], crosswell seismic traveltime tomography [5][6][7], electrical resistivity tomography [8,9], contaminant source identification [10][11][12][13], etc. A common feature amongst inverse problems is that the parameters we are interested in estimating are hard to measure directly and a crucial component of inverse modeling is using sparse data to evaluate model parameters, i.e., the solution to the inverse problems.…”