Volkan Akcelik

A Newton-CG method for large-scale three-dimensional elastic full-waveform seismic inversion

We present a nonlinear optimization method for large-scale 3D elastic full-waveform seismic inversion. The method combines outer Gauss–Newton nonlinear iterations with inner conjugate gradient linear iterations, globalized by an Armijo backtracking line search, solved on a sequence of finer grids and higher frequencies to remain in the vicinity of the global optimum, inexactly terminated to prevent oversolving, preconditioned by L-BFGS/Frankel, regularized by a total variation operator to capture sharp interfaces, finely discretized by finite elements in the Lame parameter space to provide flexibility and avoid bias, implemented in matrix-free fashion with adjoint-based computation of reduced gradient and reduced Hessian-vector products, checkpointed to avoid full spacetime waveform storage, and partitioned spatially across processors to parallelize the solutions of the forward and adjoint wave equations and the evaluation of gradient-like information. Several numerical examples demonstrate the grid independence of linear and nonlinear iterations, the effectiveness of the preconditioner, the ability to solve inverse problems with up to 17 million inversion parameters on up to 2048 processors, the effectiveness of multiscale continuation in keeping iterates in the basin of attraction of the global minimum, and the ability to fit the observational data while reconstructing the model with reasonable resolution and capturing sharp interfaces.

Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations

We consider the problem of estimating the uncertainty in large-scale linear statistical inverse problems with high-dimensional parameter spaces within the framework of Bayesian inference. When the noise and prior probability densities are Gaussian, the solution to the inverse problem is also Gaussian and is thus characterized by the mean and covariance matrix of the posterior probability density. Unfortunately, explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters and is thus prohibitive when the forward problem is expensive and the parameter dimension is large. However, for many ill-posed inverse problems, the Hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero. We present a fast method for computation of an approximation to the posterior covariance that exploits the low-rank structure of the preconditioned (by the prior covariance) Hessian of the data misfit. Analysis of an infinite-dimensional model convection-diffusion problem, and numerical experiments on large-scale three-dimensional convection-diffusion inverse problems with up to 1.5 million parameters, demonstrate that the number of forward PDE solves required for an accurate low-rank approximation is independent of the problem dimension. This permits scalable estimation of the uncertainty in large-scale ill-posed linear inverse problems at a small multiple (independent of the problem dimension) of the cost of solving the forward problem.

Full Waveform Inversion for Seismic Velocity and Anelastic Losses in Heterogeneous Structures

We present a least-squares optimization method for solving the nonlinear full waveform inverse problem of determining the crustal velocity and intrinsic at- tenuation properties of sedimentary valleys in earthquake-prone regions. Given a known earthquake source and a set of seismograms generated by the source, the in- verse problem is to reconstruct the anelastic properties of a heterogeneous medium with possibly discontinuous wave velocities. The inverse problem is formulated as a constrained optimization problem, where the constraints are the partial and ordinary differential equations governing the anelastic wave propagation from the source to the receivers in the time domain. This leads to a variational formulation in terms of the material model plus the state variables and their adjoints. We employ a wave propaga- tion model in which the intrinsic energy-dissipating nature of the soil medium is mod- eled by a set of standard linear solids. The least-squares optimization approach to inverse wave propagation presents the well-known difficulties of ill posedness and multiple minima. To overcome ill posedness, we include a total variation regulariza- tion functional in the objective function, which annihilates highly oscillatory material property components while preserving discontinuities in the medium. To treat multi- ple minima, we use a multilevel algorithm that solves a sequence of subproblems on increasingly finer grids with increasingly higher frequency source components to re- main within the basin of attraction of the global minimum. We illustrate the metho- dology with high-resolution inversions for two-dimensional sedimentary models of the San Fernando Valley, under SH-wave excitation. We perform inversions for both the seismic velocity and the intrinsic attenuation using synthetic waveforms at the observer locations as pseudoobserved data.

Shape determination for deformed electromagnetic cavities

The measured physical parameters of a superconducting cavity differ from those of the designed ideal cavity. This is due to shape deviations caused by both loose machine tolerances during fabrication and by the tuning process for the accelerating mode. We present a shape determination algorithm to solve for the unknown deviations from the ideal cavity using experimentally measured cavity data. The objective is to match the results of the deformed cavity model to experimental data through least-squares minimization. The inversion variables are unknown shape deformation parameters that describe perturbations of the ideal cavity. The constraint is the Maxwell eigenvalue problem. We solve the nonlinear optimization problem using a line-search based reduced space Gauss-Newton method where we compute shape sensitivities with a discrete adjoint approach. We present two shape determination examples, one from synthetic and the other from experimental data. The results demonstrate that the proposed algorithm is very effective in determining the deformed cavity shape.

Inversion of airborne contaminants in a regional model

We are interested in a DDDAS problem of localization of airborne contaminant releases in regional atmospheric transport models from sparse observations. Given measurements of the contaminant over an observation window at a small number of points in space, and a velocity field as predicted for example by a mesoscopic weather model, we seek an estimate of the state of the contaminant at the begining of the observation interval that minimizes the least squares misfit between measured and predicted contaminant field, subject to the convection-diffusion equation for the contaminant. Once the “initial” conditions are estimated by solution of the inverse problem, we issue predictions of the evolution of the contaminant, the observation window is advanced in time, and the process repeated to issue a new prediction, in the style of 4D-Var. We design an appropriate numerical strategy that exploits the spectral structure of the inverse operator, and leads to efficient and accurate resolution of the inverse problem. Numerical experiments verify that high resolution inversion can be carried out rapidly for a well-resolved terrain model of the greater Los Angeles area.

Computational science research in support of petascale electromagnetic modeling

Computational science research components were vital parts of the SciDAC-1 accelerator project and are continuing to play a critical role in newly-funded SciDAC-2 accelerator project, the Community Petascale Project for Accelerator Science and Simulation (ComPASS). Recent advances and achievements in the area of computational science research in support of petascale electromagnetic modeling for accelerator design analysis are presented, which include shape determination of superconducting RF cavities, mesh-based multilevel preconditioner in solving highly-indefinite linear systems, moving window using h- or p- refinement for time-domain short-range wakefield calculations, and improved scalable application I/O.

Large scale shape optimization for accelerator cavities

We present a shape optimization method for designing accelerator cavities with large scale computations. The objective is to find the best accelerator cavity shape with the desired spectral response, such as with the specified frequencies of resonant modes, field profiles, and external Q values. The forward problem is the large scale Maxwell equation in the frequency domain. The design parameters are the CAD parameters defining the cavity shape. We develop scalable algorithms with a discrete adjoint approach and use the quasi-Newton method to solve the nonlinear optimization problem. Two realistic accelerator cavity design examples are presented.