Schedule for: 22w5118 - New Ideas in Computational Inverse Problems

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

TBD

Non-linear inverse problems turn up in many applications, including wave-based imaging. In the latter case, the inverse problem is usually formulated as a PDE-constrained optimisation problem with a wave-equation whose coefficients are unknown. The oscillatory nature of the solutions to the PDE give rise to auxiliary (near) stationary points. Not only does this pose challenges for the optimisation, it also makes subsequent uncertainty quantification difficult. In part I of this talk, I will discuss some research directions that aim at reducing the non-linearity of the resulting optimisation problem. In part II, I will discuss some recent approaches to UQ for such problems.

Having access to informative data is crucial for accurate inference of unknown parameters arising in inverse problems. In situations where data acquisition is costly or time-consuming, optimal experimental design provides a mathematical framework to answer the naturally arising question: how can we choose experimental conditions for data collection to optimally infer parameters of interest? For Bayesian inverse problems, the task of choosing designs or experimental conditions for 'optimal' inference of unknown parameters requires optimizing an expected utility function that assesses the effectiveness of any design. In contrast to inverse problems governed by linear parameter-to-observable maps, a closed-form expression for the expected utility is typically not available for nonlinear inverse problems and sample-based approximation techniques are often used. We propose a transport-map based sample approximation approach that enables fast sampling from the (approximate) posterior distribution for any realization of data and design.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We will discuss inversion methodologies in general including the aspects of direct vs optimization techniques and science-based vs data driven models. This will be exemplified in the setting of modern seismic inversion.

Recent advances in freeform optics have led to PDE formulations of optical inverse problems which can be interpreted as Optimal Transport problems on the sphere with “exotic” cost functions, some convex and some concave. These problems involve constructing an optical setup for prescribed source and target intensities. We will examine systems I) a point to far-field system with a single reflector (also known as the reflector antenna problem) II) a point to far-field system with two lenses III) a point to point system with two reflectors, and IV) a point to near-field parallel screen system with two reflectors. We demonstrate how these PDE computations on the sphere can be done with a provably convergent finite-difference discretization. To this end, we introduce how the existing PDE regularity results for system I can be extended to systems II and IV and how one can build such results for system III. Having a proper theoretical footing, we can then show the success of computations of the provably convergent finite-difference scheme. We also justify and show the speed and success of a new volumetric method for performing such computations on the sphere.

Standard data-driven techniques for learning dynamical systems struggle when observational data has been sampled slowly and derivatives cannot be accurately estimated. To address this challenge, we assume that the available measurements reliably describe the asymptotic statistics of the dynamical process in question, and we instead treat invariant measures as inference data. We reformulate the inversion as a PDE-constrained optimization problem by viewing invariant measures as stationary distributional solutions to the Fokker-Planck equation, which is discretized via an upwind finite volume scheme. The velocity is parameterized by fully-connected neural networks, and we use the adjoint-state method along with backpropagation to efficiently perform model identification. Numerical results for the Van der Pol Oscillator and Lorenz-63 system, as well as real-world applications to temperature prediction, are presented to demonstrate the effectiveness of the proposed approach.

Non-line-of-sight imaging aims at recovering obscured objects from multiple-scattered light. It has recently received widespread attention due to its potential applications, such as autonomous driving, rescue operations, and remote sensing. However, in cases with high measurement noise, obtaining high-quality reconstructions remains a challenging task. In this work, we establish a unified regularization framework, which can be tailored for different scenarios, including indoor and outdoor scenes with substantial background noise under both confocal and non-confocal settings. The proposed regularization framework incorporates sparseness and non-local self-similarity of the hidden objects as well as smoothness of the measured signals. We show that the estimated signals, albedo, and surface normal of the hidden objects can be reconstructed robustly even with high measurement noise under the proposed framework. Reconstruction results on synthetic and experimental data show that our approach recovers the hidden objects faithfully and outperforms state-of-the-art reconstruction algorithms in terms of both quantitative criteria and visual quality.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In previous work, low frequency ultrasound transmission was shown to penetrate the lung, which lends promise for its use as a non-ionizing tomographic technique for pulmonary monitoring. Here, we present a method for regularized full waveform inversion to reconstruct a numerical thorax with a large pleural effusion in one lung from simulated USCT data. A novel structural similarity index method (SSIM)-based regularization term is introduced that compares the structural similarity of the current sound speed iterate to a prior reconstruction computed by electrical impedance tomography (EIT). Tomographic reconstructions of sound speed from numerically simulated low frequency ultrasound data with 0.1% additive Gaussian noise on a belt of 64 transducers are computed and the results are compared using Tikhonov regularization, total variation regularization, as well as both terms combined with the novel EIT-SSIM regularization.

In this work we study a few basic questions for PDE learning from observed solution data. Using various types of PDEs as examples, we show 1) how large the data space spanned by all snapshots along a solution trajectory is, 2) if one can construct an arbitrary solution by superposition of snapshots of a single solution, and 3) identifiability of a differential operator from a single solution data on local patches.

TBA

We consider different inverse problems formulated on graphs, where the unknowns are edge or node based quantities that are to be found from internal functionals. One example is that of a network of resistors with unknown resistances. The internal functional is the power dissipated by each of the resistors under different voltage conditions at the boundary nodes. We prove local uniqueness for this problem with a method inspired by Bal's use of overdetermined systems of partial differential equations to prove uniqueness in the continuum for the linearization of inverse problems with internal functionals. We show that this method applies to other discrete inverse problems with internal functional, with varying success.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

To understand the movement inside the Earth, we aim to reconstruct the shape of tectonic plates and infer material properties such as density or viscosity from surface observations. Modeling lithospheric flow using incompressible instantaneous Stokes equations, the problem is formulated as an infinite-dimensional Bayesian inverse problem. Subsurface structures are described as level sets of a smooth auxiliary function, allowing for flexible topological changes. Since inverting for subsurface structures from surface observations is inherently challenging, we propose a method that incorporates prior knowledge about plate geometries stemming from seismic images into the prior probability distribution. The posterior probability distribution is explored using a Markov chain Monte Carlo method. We apply the method to a realistic model problem describing a subduction zone. The aim is to infer the geometry of the subducting plate and its density as well as the viscosity in the hinge zone. Furthermore, we investigate the effect of different data types on the inversion, comparing measurements of plate velocities, normal stresses, and their combination. We discuss the benefits and limitations of our method, show trade-offs between different parameters, and provide physical interpretations.

Bayesian inference characterizes the probability distribution of model parameters given data. The complexity of many computational inference methods, however, typically scales poorly with the growing dimensions of parameters and data. A recent approach to deal with high or possibly even infinite-dimensional parameters is to reduce their dimensions so that the Bayesian inverse problems can be approximately reformulated in low-to-moderate dimensions. In this presentation, I will introduce an information-theoretic analysis to bound the error from reducing the dimensions of both parameters and data. The bound reveals appropriate dimensions for the reduced variables and exploits gradient evaluations of the log-likelihood function to identify relevant low-dimensional subspaces. I will demonstrate the benefit of the proposed dimension reduction technique for several inference algorithms on applications ranging from mechanics to image processing.

Mirror descent is a popular method for optimization on a constrained set. Mirror Langevin is an extension of mirror descent, from an optimization context to the task of sampling constrained (unnormalized) probability distributions. More precisely, its continuous time version, Mirror Langevin Dynamics (MLD), can be viewed as a special case of Riemmanian Langevin Dynamics, which modifies Euclidean geometry to Hessian metric to enable rich applications. Its discretizations, on the other hand, correspond to interesting sampling algorithms (i.e. samplers). Using a general tool for analyzing samplers based on SDE discretizations, known as mean-square analysis for sampling [Li et al. 19, Li et al. 21], we established quantitative error bounds and analyzed their dimension dependence. Joint work with Andre Wibisono, Ruilin Li, and Santosh Vempala.

In this talk, we discuss a new regularized version of the Factorization Method. The Factorization Method uses Picard’s Criteria to define an indicator function to image an unknown region. In most applications, the data operator is compact which gives that the singular values can tend to zero rapidly which can cause numerical instabilities. The regularization of the Factorization Method presented here seeks to avoid the numerical instabilities in applying Picard’s Criteria. This method allows one to image the interior structure of an object with little a priori information in a computationally simple and analytically rigorous way. Here we will focus on an application of this method to diffuse optical tomography which will prove that this method can be used to recover an unknown subregion from the Dirichlet-to-Neumann mapping.

tba

We consider solving ill-posed imaging inverse problems under a generic forward model. Because of the ill-posedness present in such problems, prior models that encourage certain image-based structure are required to reduce the space of possible images when finding a solution. Common approaches utilize hand-crafted prior models with parameters tuned through trial and error, which can be time-intensive and prone to human bias. Other approaches based on machine learning try to learn the underlying image generation model given samples from the data distribution of interest, and use this to solve a constrained inverse problem; however, in many applications ground-truth images may be unavailable. In contrast, we propose to either select or learn an image generation model from the noisy measurements alone, without incorporating prior constraints on image structure. We first show how, given a collection of candidate models, the Evidence Lower Bound (ELBO) of a variational distribution can be used to select an appropriate prior. Then we showcase how, in the absence of available priors, one can directly learn the underlying model from a set of noisy measurements using the ELBO. We assume crucially that the ground-truth images share common structure by being drawn from the same underlying distribution. The learned model leverages this structure in its architecture, which consists of a shared generator with a compressed latent space where each measurement posterior is learned variationally. This allows the model to learn global properties of the data distribution from noisy observations without overfitting. We illustrate our framework on a variety of inverse problems, ranging from denoising to compressed sensing problems inspired by black-hole imaging.

Some new ideas and challenges from two applications involved decaying systems will be the subject of this talk. One is the T2 Magnetic Resonance Relaxometry (T2-MRR) and the other is the time-domain FWI of the poroelastic wave equations. The T2-MRR problem is closely related to the inverse Laplace transform of noisy data. A multi-regularization method that demonstrates numerically better stability against the data noise will be presented. This part is based on a recent paper [1]. We will also present an RKHS approach for solving this T2-MRR problem. The idea is to identify an intrinsic kernel of this problem and used it as the norm in the regularization term. Both numerical results and some analytical explanations will be presented for this RKHS approach. This part is based on the manuscript [2]. For the time-domain FWI poroelastic wave problem, the numerical scheme as well as the theoretical challenges will be presented. This is based on [3]. References: [1] "Span of regularization for solution of inverse problems with application to magnetic resonance relaxometry of the brain" by Chuan Bi and M. Yvonne Ou and Mustapha Bouhrara and Richard G. Spencer. Scientific Reports, 2022 [2] "A weighted deconvolution from noisy data" by Quanjun Lang, Fei Lu and Yvonne Ou, Preprint, 2022 [3] "On the time-domain full waveform inversion for time-dissipative and dispersive poroelastic media", by Ou, Miao-jung Yvonne, Xie, Jiangming and Plechac, Petr. Applicable Analysis, special issue on the occasion of Robert P. Gilbert's 90-th birthday, 2022.

We present a non-iterative algorithm to reconstruct an isotropic acoustic speed from the near-field data represented by the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only stable computation. Numerical validation is provided with both full and partial data.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

We survey some of the results obtained during the last 40 years on Calderon's inverse problem: Can one determine the electrical conductivity in the interior of a medium by making voltage and current measurements at the boundary? In dimension three or higher this is equivalent to a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by making appropriate measurements of the boundary values of harmonic functions? We will mention several open problems.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe a data driven approach for estimating a wave at inaccessible points inside a medium, and then show how to use it for solving the inverse scattering problem for the wave equation. The data are gathered by an array of collocated sources and receivers that probe the medium sequentially with pulses and measure the resulting waves. The inverse scattering problem seeks to estimate the wave speed from these data.

We are interested in the inverse problem of estimating unknown parameters in a mathematical model from observed data. We follow the Bayesian approach, in which the solution to the inverse problem is the distribution of the unknown parameters conditioned on the observed data, the so-called posterior distribution. We are particularly interested in the case where the mathematical model is non-linear and expensive to simulate, for example given by a partial differential equation. A major challenge in the application of sampling methods such as Markov chain Monte Carlo is then the large computational cost associated with simulating the model for a given parameter value. To overcome this issue, we consider using Gaussian process regression to approximate the likelihood of the data. This results in an approximate posterior distribution, to which sampling methods can be applied with feasible computational cost. In this talk, we will show how the uncertainty estimate provided by Gaussian process regression can be incorporated into the approximate Bayesian posterior distribution to avoid overconfident predictions, and present efficient Markov chain Monte Carlo methods in this context. This is joint work with Tianming Bai and Konstantinos Zygalakis (University of Edinburgh).

Inverse problems are ubiquitous. People probe the media with sources and measure the outputs. At the scale of quantum, classical, statistical and fluid, these are inverse Schroedinger, inverse Newton’s second law, inverse Boltzmann problem, and inverse diffusion, respectively. The universe, however, should have a universal mathematical description, as Hilbert proposed in 1900. In this talk, we initiate some discussion that aims at connecting inverse Schroedinger, inverse Newton, inverse Boltzmann, and finally inverse diffusion. We will argue these are the same problem merely represented at different scales.

In this talk, I will discuss our recent work on computational methods for mean-field games (MFG) and their inverse problems. I will start from a low-dimensional setting using conventional discretization methods and discuss an algorithm for solving the corresponding inverse problem based on a bi-level optimization method. Some theoretical aspects of the proposed method will also be discussed. After that, I will extend to high-dimensional problems by bridging the trajectory representation of MFG with normalizing flows, a special type of deep generative model. This connection does not only help solve high-dimensional MFGs, but also provides a way to improve the robustness of normalizing flows.

Inverse problem based on radiative transport equation (RTE) is at the core of optical tomography. Due to the computational complexity in the RTE, a diffusion approximation is often adopted when the scattering effect is dominant. But this would introduce instability in the inverse setting. Additionally, the co-existence of different regimes present us with a multiscale problem. In this talk, I will review the results regarding the stability degradation along the diffusion approximation. I will also share some thoughts on how to numerically deal with the multiscale problem using Newton type method.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will overview some recent works on fast algorithms for computational wave modeling and inversion, including Hadamard-Babich integrators, butterfly algorithms, ray-motivated deep-learning hybrid methods for high frequency waves, singularity-removing travel time tomography methods, and related inversion algorithms.

In this talk, I will present a weak adversarial network approach to solve a class of inverse problems. Using the weak formulation of PDE, we rewrite the inverse problem as a minimax problem. Leveraged with deep neural networks, the solution of inverse problem, including the solution of PDE and the unknown media, can be solved simultaneously by finding the network parameters for the saddle point. While the parameters are updated, the networks gradually approximate the solution of the inverse problem. Theoretical justifications are provided on the convergence of the proposed algorithm. The proposed method is mesh-free without any spatial discretization, and is suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test problems demonstrate the promising accuracy and efficiency of this approach. This presentation is based on the joint work with Gang Bao (Zhejiang), Xiaojing Ye (Georgia State Univ.) and Yaohua Zang (Zhejiang).

The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. This paper considers the case in which the training data is distributed in a linear subspace of $\mathbb R^d$. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network's depth and noise in the codimension of the data manifold. We also present additional side effects in training due to the presence of noise.

In recent years, there is great interest in using deep learning for geophysical/medical data inversion. However, the direct application of end-to-end data-driven approaches to inversion has quickly shown limitations in practical implementation. Due to the lack of prior knowledge of the objects of interest, the trained deep-learning neural networks very often have limited generalization. In this talk, we introduce a new methodology of coupling model-based inverse algorithms with deep learning for inversion problems. Here, we particularly focus on one application in full waveform inversion (FWI). In this project, we propose an offline-online computational strategy for coupling classical least-squares-based computational inversion with modern deep learning-based approaches for FWI to achieve advantages that can not be achieved with only one of the components. In a nutshell, we develop an offline learning strategy to construct a robust approximation to the inverse operator and utilize it to design a new objective function for the online inversion with new datasets. We demonstrate through numerical simulations that our coupling strategy improves the computational efficiency of FWI with reliable offline training on moderate computational resources (in terms of both the size of the training dataset and the computational cost needed).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

For most wave-based inverse problems the resolution of the reconstruction is usually limited by the so-called diffraction limit, i.e., the smallest features to be reconstructed cannot be smaller than the smallest wavelength of available data. If one properly restricts the class of features to, for example, point-scatterers, the seminal work of Donoho in the early 90’s demonstrates that the recovery of these sub-wavelength features is tractable. However, algorithms to recover a more general class of structured scatterers containing features below the diffraction limit in the presence of noise remains an open question. In this talk we aim to surpass the diffraction limit using deep learning techniques coupled with computational harmonic analysis tools. In particular, I will introduce a novel neural network architecture for inverting wide-band data to recover acoustic scatterers at resolutions finer than the classical limit. The architecture incorporates insights from the butterfly factorization and the Cooley-Tukey algorithm to explicitly account for the physics of wave propagation. The dimensions of the network seamlessly adapt to the desired image resolution, resulting in a number of trainable weights that scale quasi linearly with the image resolution and the data bandwidth. In addition, the data is optimally assimilated across frequencies thus enhancing the stability of the training stage. I will provide the rationale for such construction and showcase its properties for several classes of scatterers with sub-Nyquist features embedded in a known background media.

A novel approach to full waveform inversion (FWI), based on a data driven reduced order model (ROM) of the wave equation operator is introduced. The unknown medium is probed with pulses and the time domain pressure waveform data is recorded on an active array of sensors. The ROM, a projection of the wave equation operator is constructed from the data via a nonlinear process and is used for efficient velocity estimation. While the conventional FWI via nonlinear least-squares data fitting is challenging without low frequency information, and prone to getting stuck in local minima (cycle skipping), minimization of ROM misfit is behaved much better, even for a poor initial guess. For low-dimensional parametrizations of the unknown velocity the ROM misfit function is close to convex. The proposed approach consistently outperforms conventional FWI in standard synthetic tests.

We propose a deterministic-statistical approach for inverse problems with partial data. Certain deterministic method is first used to obtain useful (qualitative) information for the unknowns. Then the inverse problem is recasted as a statistical inference problem and the Bayesian inversion is employed to obtain more (quantitative) information of the unknowns. Several examples are presented for demonstration. Furthermore, we introduce new statistical estimators to characterize the non-unique solutions of several inverse problems.

Variable projection methods are among the classical and efficient methods to solve separable nonlinear least squares problems. In this talk, I will present the original variable projection method, its use to solve large-scale blind deconvolution problems, and some new variants that preserve the edges in the solution.

We consider identifying differential equation using numerical techniques (IDENT) from one set of noisy observation. We assume that the governing PDE can be expressed as a linear combination of different linear and nonlinear differential terms. In this talk, extending from IDENT and robust IDENT, we will discuss using weak form for differential equation identification. We consider both ODE and PDE models. Numerical results show robustness against higher level of noise and higher order derivative in underlying equation.

There are many inverse problems in machine learning, such as regression, classification and distribution estimation. In these problems, many data are in a high-dimensional space but exhibit low-dimensional structures. In mathematics, these data can be modeled as random samples on a low-dimensional manifold. In this talk, I will present mathematical and statistical theories of deep neural networks for solving inverse problems in machine learning, where data are randomly sampled on a low-dimensional manifold. The sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data.

This talk is concerned with the inverse elastic scattering problem for a random potential. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic Gaussian random field whose covariance operator is a classical pseudo-differential operator. Given the potential, the direct scattering problem is shown to be well-posed in the sense of distributions by studying the equivalent Lippmann-Schwinger integral equation. For the inverse scattering problem, we demonstrate that the microlocal strength of the random potential can be uniquely determined with probability one by a single realization of the high frequency limit of the averaged scattered wave. The analysis employs the integral operator theory, the Born approximation in the high frequency regime, the microlocal analysis for the Fourier integral operators, and the ergodicity of the wave field.

We will present a nonlinear upscaling method for solving multiscale problems. The method is based on solving local nonlinear problems to obtain effective parameters. We will also present the use of machine learning techniques to learn the effective parameters. Some examples will be presented.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

For an ill-posed inverse problem, particularly with incomplete and limited measurement data, regularization is an essential tool for stabilizing the inverse problem. Among various forms of regularization, the $\ell_p$ penalty term provides a suite of regularization of various characteristics depending on the value of $p$. When there are no explicit features to determine $p$, a spatially varying inhomogeneous $p$ can be incorporated to apply different regularization characteristics that change over the domain. This approach has been investigated and used for denoising problems where the first or the second derivatives of the true signal provide information to design the spatially varying exponent $p$ distribution. This study proposes a strategy to design the exponent distribution when the first and second derivatives of the true signal are not available, such as in the case of indirect and limited measurement data. The proposed method extracts statistical and patch-wise information using multiple reconstructions from a single measurement, which assists in classifying each patch to predefined features with corresponding $p$ values. We validate the robustness and effectiveness of the proposed approach through a suite of numerical tests in 1D and 2D, including a sea ice image recovery from partial Fourier measurement data. Numerical tests show that the exponent distribution is insensitive to the choice of multiple reconstructions.

We consider imaging absorbing as well as non-absorbing objects using intensity only measurements. Objects with high absorption contrast can be imaged effectively using multiple illuminations and/or masks as in ghost imaging. On the other hand, transparent objects with low absorption contrast are more challenging to be imaged when only intensities are measured, even when they significantly change the phase of the waves as they go through them. We present a computational imaging approach that allows quantitative imaging of both absorbing and transparent objects. This problem arises in various fields such as X-ray crystallography, electron microscopy, coherent diffractive imaging and astronomy. The proposed algorithm guarantees exact recovery if the image is sparse with respect to a given basis, and it can be used, without any modification, when the illumination is partially coherent. This is important for, for example, phase-contrast X-ray imaging because fully coherent sources of X-rays are very hard to be obtained.

Modeled and recorded seismic data is contaminated by noise due to a variety of factors including geophones recording ambient noise unrelated to seismic exploration, equipment malfunction and limitations, and restrictive modeling assumptions. Being able to estimate the noise level in the data is a valuable tool for assessing the quality of mechanical Earth parameter estimates, especially resulting from inversion. Commonly-used gradient-based local optimization techniques for estimating subsurface parameters such as wave velocity are well known to stall in geologically uninformative models if the starting guess for the optimization is not close enough to the desired global optimum. Extension-based methods relax physical constraints on model parameters to enlarge the search space of acceptable solutions, helping to reduce the impact of a poor initial guess and potentially convexifying the objective function. These extended inversion methods involve a penalty term that is added to the least squares misfit function. Then the challenge of performing the optimization shifts to adjusting the penalty weight to balance the reduction of data misfit with driving the penalty term towards physically-meaningful solutions. The source-extended objective function minimized using the discrepancy algorithm requires an estimate of the noise level in the data to proceed. We illustrate an automated algorithm for simultaneously updating this noise estimate so that we bypass the cycle-skipping problem and converge to a geologically meaningful velocity estimate.

Inverse scattering problems arise in many real life applications such as non-destructive evaluation, medical imaging, geophysical exploration. Roughly speaking, inverse problems aim to determine information about an object (scatterer) from measurements of waves scattered by that object. In recent decades, Sampling Methods have been known as fast and efficient qualitative methods to solve such inverse problems. In this talk, we will discuss the so-called Differential Sampling Method, which is the development of Sampling Methods, to determine local defects in an unknown periodic medium. By exploiting the periodicity of the medium, this method allows to identify defects without knowing the periodic structure. This is a joint work with Houssem Haddar and Fioralba Cakoni.