arXiv subjects

Kamal Aghazade

Publications and source records attributed to Kamal Aghazade.

10 recordsLinked to original sources

arXiv2026

Dual-space posterior sampling for Bayesian inference in constrained inverse problems

Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy, incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion (FWI), which estimates Earth's subsurface properties by fitting seismic measurements subject to the wave equation, where ill-conditioning stems from noisy, band-limited, finite-aperture measurements and complex geological structures. A Bayesian framework describes the solution more comprehensively: instead of a single estimate, a posterior distribution of plausible solutions characterizes the non-uniqueness and can be sampled to quantify uncertainty. However, no clear procedure exists for translating hard physical constraints, such as the wave equation, into priors amenable to existing sampling techniques. We address this by sampling the posterior in the dual space via an augmented Lagrangian formulation, which converts hard constraints into penalties suited to sampling algorithms while enforcing them progressively through multiplier updates, so they are satisfied in the limit. We integrate the alternating direction method of multipliers (ADMM) with Stein variational gradient descent (SVGD), a particle-based sampler: the constraint is relaxed at each iteration and the multiplier updates progressively enforce its satisfaction. This enables posterior sampling under hard constraints while inheriting the favorable conditioning of dual-space solvers, where partial constraint relaxation permits productive updates even when the current model is far from the true solution. We validate the method on a stylized Rosenbrock conditional inference problem and on frequency-domain FWI for a Gaussian anomaly model and the Marmousi II benchmark, demonstrating physically consistent uncertainty estimates and posterior contraction with increasing data coverage.

physics.geo-ph↗

arXiv2026

Scalable Bayesian full waveform inversion via dual augmented Lagrangian SVGD

Full waveform inversion is an ill-posed inverse problem whose solution non-uniqueness -- i.e., arising from band-limited, finite-aperture, noisy data -- calls for uncertainty quantification to avoid overconfident geological interpretations. Bayesian inference addresses this need by characterizing the solution as a posterior distribution rather than a single point estimate. Sampling from this distribution, however, remains computationally challenging: Markov chain Monte Carlo and non-amortized variational inference require repeated wave equation solves, while amortized variational inference approaches that avoid repeated solves rely on training data that are inherently scarce in geoscience and face unresolved generalization challenges in high dimensions. To address these limitations, we integrate Stein variational gradient descent with the alternating direction method of multipliers under a dual augmented Lagrangian formulation. By fixing the wave operator at a background model that is updated between frequency batches, it need only be factorized once per particle per frequency, eliminating per-iteration refactorization and reducing the total cost to that of a handful of deterministic inversions while inheriting the favorable conditioning of extended-space formulations. Applied to the Marmousi~II model, the proposed method provides well-calibrated uncertainty estimates and achieves inversion quality comparable to that of the standard augmented Lagrangian SVGD at a fraction of the computational cost.

physics.geo-ph↗

arXiv2025

Automatic Penalty Parameter Selection by Residual Whiteness Principle (RWP) and GCV for Full Waveform Inversion

Full-waveform inversion (FWI) is a powerful seismic imaging technique used to estimate high-resolution physical properties of subsurface structures by minimizing the misfit between observed and modeled seismic data. FWI is inherently a highly non-linear and ill-posed inverse problem. Extended-source approaches, such as the augmented Lagrangian (AL) method, are employed to improve solution convexity and robustness. A key component of this formulation is the penalty parameter, which controls the trade-off between data fitting and satisfaction of the wave-equation constraint, strongly influencing convergence in the presence of noise. The main challenge lies in selecting the penalty parameter. Traditional strategies such as the Discrepancy Principle (DP) require an accurate estimate of the noise level, which is often unknown or poorly characterized. Moreover, trial-and-error tuning requires repeatedly solving the inverse problem, making it computationally expensive. To overcome these limitations and develop a parameter-free, computationally efficient extended-source FWI algorithm, we integrate two data-driven parameter-selection strategies--the Residual Whiteness Principle (RWP) and a stable variant of Generalized Cross-Validation (RGCV)--within a multiplier-oriented AL framework. Specifically, we adopt a dual-space AL formulation, which allows the background wave-equation operator to remain fixed and requires only a single LU factorization per frequency, significantly improving efficiency. This design enables dynamic adjustment of the parameter at negligible cost during iterations, making the algorithm scalable for large-scale applications. Numerical experiments on acoustic and elastic FWI with white and colored noise show that, combined with the dual-space formulation, RWP provides strong noise robustness, resulting in a reliable automated solution for large-scale seismic inversion.

physics.geo-ph↗

arXiv2025

Robust acoustic and elastic full waveform inversion by adaptive Tikhonov-TV regularization

Full Waveform Inversion (FWI) is a powerful wave-based imaging technique, but its inherent ill-posedness and non-convexity lead to local minima and poor convergence. Regularization methods stabilize FWI by incorporating prior information and enforcing structural constraints like smooth variations or piecewise-constant behavior. Among them, Tikhonov regularization promotes smoothness, while total variation (TV) regularization preserves sharp boundaries. However, in the context of FWI, we highlight two key shortcomings of these regularization methods. First, subsurface model parameters (P- and S-wave velocities, density) often exhibit complex geological formations with sharp discontinuities separating distinct layers, while parameters within each layer vary smoothly. Neither Tikhonov nor TV regularization alone can effectively constrain such piecewise-smooth structures. Second, and more critically, when the initial model is far from the true model, these regularization assumptions can lead to a local minimum. To address these issues, we propose adaptive Tikhonov-TV (TT) regularization, which decomposes the model into smooth and blocky components, enabling robust recovery of piecewise-smooth structures. Implemented within the ADMM framework, TT regularization incorporates an automated balancing strategy based on robust statistical analysis. Numerical experiments on acoustic and elastic FWI using benchmark geological models demonstrate that TT regularization significantly improves convergence and reconstruction accuracy compared to Tikhonov and TV regularization when applied separately. We show that for complex models and remote initial models, both Tikhonov and TV regularization tend to converge to local minima, whereas TT regularization effectively mitigates cycle skipping through its adaptive combination of the two regularization strategies.

physics.geo-ph↗

arXiv2025

Weighted Lagrange Multiplier Method for Robust Source-Independent Waveform Inversion

The Lagrange multiplier method has proven highly effective for mitigating the ill-conditioning of full waveform inversion (FWI), enabling robust and computationally efficient algorithms that converge to accurate velocity models even from poor initial estimates. Classical multiplier-based FWI methods optimize an augmented Lagrangian (AL) functional with a scalar penalty parameter that uniformly weights wave-equation constraint violations. While this balances data fit and wave-equation satisfaction, it applies uniform relaxation across the model, disregarding source locations and the natural decay of seismic energy. We propose a weighted proximal-point Lagrangian formulation that introduces spatially varying regularization, applying weaker enforcement near sources and progressively stronger enforcement with increasing distance. This compensates for the energy decay, promotes balanced wave-equation enforcement, and improves the convexity of the optimization landscape. The method also eliminates the need for explicit source signature estimation and relaxes the requirement for sources to lie on finite-difference grid points, increasing practical applicability. Enhanced computational efficiency is achieved through our dual-space ADMM implementation, which avoids repeated LU factorizations of the forward operator. Only a few LU factorizations are required, with all subsequent iterations solved via efficient forward-backward substitution, making the approach scalable to large-scale 2D and 3D problems. Numerical experiments on challenging synthetic benchmarks demonstrate that the proposed method broadens the basin of attraction of the AL objective, improves robustness to poor initial models and strong noise, and achieves faster, more stable convergence compared with standard multiplier-based methods.

physics.geo-ph↗

arXiv2024

Fast and Automatic Full Waveform Inversion by Dual Augmented Lagrangian

Full Waveform Inversion (FWI) stands as a nonlinear, high-resolution technology for subsurface imaging via surface-recorded data. This paper introduces an augmented Lagrangian dual formulation for FWI, rooted in the viewpoint that Lagrange multipliers serve as fundamental unknowns for the accurate linearization of the FWI problem. Once these multipliers are estimated, the determination of model parameters becomes simple. Therefore, unlike traditional primal algorithms, the proposed dual method circumvents direct engagement with model parameters or wavefields, instead tackling the estimation of Lagrange multipliers through a gradient ascent iteration. This approach yields two significant advantages: i) the background model remains fixed, requiring only one LU matrix factorization for each frequency inversion. ii) Convergence of the algorithm can be improved by leveraging techniques like quasi-Newton l-BFGS methods and Anderson acceleration. Numerical examples from elastic and acoustic FWI utilizing different benchmark models are provided, showing that the dual algorithm converges quickly and requires fewer computations than the standard primal algorithm.

physics.geo-ph↗

arXiv2024

Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields

Elastic full-waveform inversion (EFWI) is a process used to estimate subsurface properties by fitting seismic data while satisfying wave propagation physics. The problem is formulated as a least-squares data fitting minimization problem with two sets of constraints: Partial-differential equation (PDE) constraints governing elastic wave propagation and physical model constraints implementing prior information. The alternating direction method of multipliers is used to solve the problem, resulting in an iterative algorithm with well-conditioned subproblems. Although wavefield reconstruction is the most challenging part of the iteration, sparse linear algebra techniques can be used for moderate-sized problems and frequency domain formulations. The Hessian matrix is blocky with diagonal blocks, making model updates fast. Gradient ascent is used to update Lagrange multipliers by summing PDE violations. Various numerical examples are used to investigate algorithmic components, including model parameterizations, physical model constraints, the role of the Hessian matrix in suppressing interparameter cross-talk, computational efficiency with the source sketching method, and the effect of noise and near-surface effects.

math.NA↗

arXiv2022

Large-scale highly-accurate extended full waveform inversion using convergent Born series

Full-waveform inversion (FWI) is a seismic imaging method that provides quantitative inference about subsurface properties with a wavelength-scale resolution. Its frequency-domain formulation is computationally efficient when processing only a few discrete frequencies. However, classical FWI, which is formulated on the reduced-parameter space, requires starting the inversion with a sufficiently-accurate initial model and low frequency to prevent being stuck in local minima due to cycle skipping. FWI with extended search space has been proposed to mitigate this issue. It contains two main steps: first, data-assimilated (DA) wavefields are computed by allowing for wave-equation errors to match the data at receivers closely. Then, subsurface parameters are estimated from these wavefields by minimizing the wave-equation errors. The DA wavefields are the least-squares solution of an overdetermined system gathering the wave and observation equations. The numerical bandwidth of the resulting normal-equation system is two times that of the wave-equation system, which can be a limiting factor for 3D large-scale applications. Therefore, computing highly accurate DA wavefields at a reasonable computational cost is an issue in extended FWI. This issue is addressed here by rewriting the normal system such that its solution can be computed by solving the time-harmonic wave equation several times in sequence. Moreover, the computational burden of multi-right-hand side (RHS) simulations is mitigated with a sketching method. Finally, we solve the time-harmonic wave equation with the convergent Born series method, which conciliates accuracy and computational efficiency. Application of the new extended FWI algorithm on the salt benchmark shows that it reconstructs at a reasonable cost subsurface models that are similar to those obtained with the classical extended FWI.

math.OC↗

arXiv2021

Randomized source sketching for full waveform inversion

Partial differential equation (PDE) constrained optimization problems such as seismic full waveform inversion (FWI) frequently arise in the geoscience and related fields. For such problems, many observations are usually gathered by multiple sources, which form the right-hand-sides of the PDE constraint. Solving the inverse problem with such massive data sets is computationally demanding, in particular when dealing with large number of model parameters. This paper proposes a novel randomized source sketching method for the efficient resolution of multisource PDE constrained optimization problems. We first formulate the different source-encoding strategies used in seismic imaging into a unified framework based on a randomized sketching. To this end, the source dimension of the problem is projected in a smaller domain by a suitably defined projection matrix that gathers the physical sources in super-sources through a weighted summation. This reduction in the number of physical sources decreases significantly the number of PDE solves while suitable sparsity-promoting regularization can efficiently mitigate the footprint of the cross-talk noise to maintain the convergence speed of the algorithm. We implement the randomized sketching method in an extended search-space formulation of frequency-domain FWI, which relies on the alternating direction method of multipliers (ADMM). Numerical examples carried out with a series of well documented 2D benchmarks demonstrate that the randomized sketching algorithm reduces the cost of large-scale problems by at least one order of magnitude compared to the original deterministic algorithm.

physics.geo-ph↗

arXiv2021

Anderson accelerated augmented Lagrangian for extended waveform inversion

The augmented Lagrangian (AL) method provides a flexible and efficient framework for solving extended-space full-waveform inversion (FWI), a constrained nonlinear optimization problem whereby we seek model parameters and wavefields that minimize the data residuals and satisfy the wave equation constraint. The AL-based wavefield reconstruction inversion, also known as iteratively refined wavefield reconstruction inversion, extends the search space of FWI in the source dimension and decreases sensitivity of the inversion to the initial model accuracy. Furthermore, it benefits from the advantages of the alternating direction method of multipliers (ADMM), such as generality and decomposability for dealing with non-differentiable regularizers, e.g., total variation regularization, and large scale problems, respectively. In practice any extension of the method aiming at improving its convergence and decreasing the number of wave-equation solves would have a great importance. To achieve this goal, we recast the method as a general fixed-point iteration problem, which enables us to apply sophisticated acceleration strategies like Anderson acceleration. The accelerated algorithm stores a predefined number of previous iterates and uses their linear combination together with the current iteration to predict the next iteration. We investigate the performance of the proposed accelerated algorithm on a simple checkerboard model and the benchmark Marmousi II and 2004 BP salt models through numerical examples. These numerical results confirm the effectiveness of the proposed algorithm in terms of convergence rate and the quality of the final estimated model.

physics.geo-ph↗