Browsing by Author "de Hoop, Maarten V."
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Item A Distributed-Memory Randomized Structured Multifrontal Method for Sparse Direct Solutions(Society for Industrial and Applied Mathematics, 2017) Xin, Zixing; Xia, Jianlin; de Hoop, Maarten V.; Cauley, Stephen; Balakrishnan, VenkataramananWe design a distributed-memory randomized structured multifrontal solver for large sparse matrices. Two layers of hierarchical tree parallelism are used. A sequence of innovative parallel methods are developed for randomized structured frontal matrix operations, structured update matrix computation, skinny extend-add operation, selected entry extraction from structured matrices, etc. Several strategies are proposed to reuse computations and reduce communications. Unlike an earlier parallel structured multifrontal method that still involves large dense intermediate matrices, our parallel solver performs the major operations in terms of skinny matrices and fully structured forms. It thus significantly enhances the efficiency and scalability. Systematic communication cost analysis shows that the numbers of words are reduced by factors of about $O(\sqrt{n}/r)$ in two dimensions and about $O(n^{2/3}/r)$ in three dimensions, where $n$ is the matrix size and $r$ is an off-diagonal numerical rank bound of the intermediate frontal matrices. The efficiency and parallel performance are demonstrated with the solution of some large discretized PDEs in two and three dimensions. Nice scalability and significant savings in the cost and memory can be observed from the weak and strong scaling tests, especially for some 3D problems discretized on unstructured meshes.Item An exact redatuming procedure for the inverse boundary value problem for the wave equation(Society for Industrial and Applied Mathematics, 2018) de Hoop, Maarten V.; Kepley, Paul; Oksanen, LauriRedatuming is a data processing technique to transform measurements recorded in one acquisition geometry to an analogous data set corresponding to another acquisition geometry, for which there are no recorded measurements. We consider a redatuming problem for a wave equation on a bounded domain, or on a manifold with boundary, and model data acquisition by a restriction of the associated Neumann-to-Dirichlet map. This map models measurements with sources and receivers on an open subset $\Gamma$ contained in the boundary of the manifold. We model the wavespeed by a Riemannian metric and suppose that the metric is known in some coordinates in a neighborhood of $\Gamma$. Our goal is to move sources and receivers into this known near boundary region. We formulate redatuming as a collection of unique continuation problems and provide a two-step procedure to solve the redatuming problem. We investigate the stability of the first step in this procedure, showing that it enjoys conditional Hölder stability under suitable geometric hypotheses. In addition, we provide computational experiments that demonstrate our redatuming procedure.Item Compositional heterogeneity near the base of the mantle transition zone beneath Hawaii(Springer Nature, 2018) Yu, Chunquan; Day, Elizabeth A.; de Hoop, Maarten V.; Campillo, Michel; Goes, Saskia; Blythe, Rachel A.; van der Hilst, Robert D.Global seismic discontinuities near 410 and 660 km depth in Earth’s mantle are expressions of solid-state phase transitions. These transitions modulate thermal and material fluxes across the mantle and variations in their depth are often attributed to temperature anomalies. Here we use novel seismic array analysis of SSwaves reflecting off the 410 and 660 below the Hawaiian hotspot. We find amplitude–distance trends in reflectivity that imply lateral variations in wavespeed and density contrasts across 660 for which thermodynamic modeling precludes a thermal origin. No such variations are found along the 410. The inferred 660 contrasts can be explained by mantle composition varying from average (pyrolitic) mantle beneath Hawaii to a mixture with more melt-depleted harzburgite southeast of the hotspot. Such compositional segregation was predicted, from petrological and numerical convection studies, to occur near hot deep mantle upwellings like the one often invoked to cause volcanic activity on Hawaii.Item Inverse Boundary Value Problem For The Helmholtz Equation: Quantitative Conditional Lipschitz Stability Estimates(SIAM, 2016) Beretta, Elena; de Hoop, Maarten V.; Faucher, Florian; Scherzer, OtmarWe study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of wavespeeds that are a linear combination of piecewise constant functions (following a domain partition) and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partition increases. We establish an order optimal upper bound for the stability constant. We eventually realize computational experiments to demonstrate the stability constant evolution for three-dimensional wavespeed reconstruction.Item Kinematics of Shot-Geophone Migration(2005-04) Stolk, Christiaan C.; de Hoop, Maarten V.; Symes, William W.Prestack migration methods based on data binning produce {\em kinematic artifacts}, i.e. coherent events not corresponding to actual reflectors, in the prestack image volume. Shot-geophone migration, on the other hand, generally does not produce such artifacts when events to be migrated arrive in the data along non-turning rays. This condition is required for successful implementation via wavefield depth extrapolation (``survey sinking''). In contrast to prestack migration methods based on data binning, common image gathers produced by shot-geophone migration exhibit the appropriate semblance property in either offset domain (focussing at zero offset) or angle domain (focussing at zero slope), when the migration velocity is kinematically correct. Thus shot-geophone migration may be a particularly appropriate tool for migration velocity analysis of data exhibiting structural complexity.Item Mapping Mantle Transition Zone Discontinuities Beneath the Central Pacific With Array Processing ofﾠSSﾠPrecursors(Wiley, 2017) Yu, Chunquan; Day, Elizabeth A.; de Hoop, Maarten V.; Campillo, Michel; van der Hilst, Robert D.We image mantle transition zone (MTZ) discontinuities beneath the Central Pacific using ~120,000 broadband SS waveforms. With a wave packet‐based array processing technique (curvelet transform), we improve the signal‐to‐noise ratio of SS precursors and remove interfering phases, so that precursors can be identified and measured over a larger distance range. Removal of interfering phases reveals possible phase shifts in the underside reflection at the 660, that is, S660S, which if ignored could lead to biased discontinuity depth estimates. The combination of data quantity and improved quality allows improved imaging and uncertainty estimation. Time to depth conversions after corrections for bathymetry, crustal thickness, and tomographically inferred mantle heterogeneity show that the mean depths of 410 and 660 beneath the Central Pacific are 420 ± 3 km and 659 ± 4 km, respectively. The mean MTZ thickness (239 ± 2 km) is close to global estimates and suggests an adiabatic mantle temperature of ~1,400°C for the Central Pacific. Depth variations of the 410 and 660 appear to be relatively small, with peak‐to‐peak amplitudes of the order of 10–15 km. The 410 and 660 are weakly anticorrelated, and MTZ is thinner beneath Hawaii and to the north and east of the hotspot and thicker southwest of it. The relatively small discontinuity topography argues against the presence of large‐scale (more than 5° wide) thermal anomalies with excess temperatures over 200 K across the transition zone. The data used cannot exclude stronger thermal anomalies that are of more limited lateral extent or that are not continuous across the MTZ.Item Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation(2024-01-18) Lara Benitez, Antonio; de Hoop, Maarten V.Deep learning has emerged as an incredibly successful and versatile approach within the field of machine learning, finding applications across a diverse range of domains. Originally devised for tasks such as classification and natural language processing, deep learning has made significant inroads into scientific computing. Architectures like Deeponet and Neural Operators have showcased their potential in approximating operators defined by partial differential equations (PDEs). While these architectures have shown practical success, there remains a compelling need to delve deeper into their theoretical foundations. This thesis aims to contribute to the theoretical understanding of deep learning by applying statistical learning theory to the neural operator family. Our primary focus will be on the generalization properties of this family while addressing the challenges posed by the high-frequency Helmholtz equation. To achieve this, we propose a subfamily of neural operators, known as sequential neural operators, which not only preserves all the approximation guarantees of neural operators but also exhibits enhanced generalization properties. This design draws inspiration from the self-attention mechanism found in the ubiquitous transformer architecture. To analyze both neural operators and sequential neural operators we establish upper bounds on Rademacher complexity. These bounds are instrumental in deriving the corresponding generalization error bounds. Furthermore, we leverage Gaussian-Banach spaces to shed light on the out-of-risk bounds of traditional neural operators and sequential neural operators.Item Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem(Taylor & Francis, 2023) de Hoop, Maarten V.; Lassas, Matti; Lu, Jinpeng; Oksanen, LauriWe obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular, our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface.Item Reciprocity-gap misfit functional for distributed acoustic sensing, combining data from passive and active sources(Society of Exploration Geophysicists, 2021) Faucher, Florian; de Hoop, Maarten V.; Scherzer, OtmarQuantitative imaging of subsurface earth properties in elastic media is performed from distributed acoustic sensing data. A new misfit functional based upon the reciprocity gap is designed, taking crosscorrelations of displacement and strain, and these products further associate an observation with a simulation. In comparison with other misfit functionals, this functional has the advantage of only requiring little a priori information on the exciting sources. In particular, the misfit criterion enables the use of data from regional earthquakes (teleseismic events can be included as well), followed by exploration data to perform a multiresolution reconstruction. The data from regional earthquakes contain the low-frequency content that is missing in the exploration data, allowing for the recovery of the long spatial wavelength, even with very few sources. These data are used to build prior models for the subsequent reconstruction from the higher frequency exploration data. This results in the elastic full reciprocity-gap waveform inversion method, and we illustrate its performance with a pilot experiment for elastic isotropic reconstruction.Item Reconstruction of Lamé Moduli and Density at the Boundary Enabling Directional Elastic Wavefield Decomposition(SIAM, 2017) de Hoop, Maarten V.; Nakamura, Gen; Zhai, JianWe consider the inverse boundary value problem for the system of equations describing elastic waves in isotropic media on a bounded domain in $\mathbb{R}^3$ via a finite-time Laplace transform. The data are the dynamical Dirichlet-to-Neumann map. More precisely, using the full symbol of the transformed Dirichlet-to-Neumann map viewed as a semiclassical pseudodifferential operator, we give an explicit reconstruction of both Lamé parameters and the density, as well as their derivatives, at the boundary. We also show how this boundary reconstruction leads to a decomposition of incoming and outgoing waves.Item Stable Recovery of Coefficients in an Inverse Fault Friction Problem(Springer Nature, 2024) de Hoop, Maarten V.; Lassas, Matti; Lu, Jinpeng; Oksanen, LauriWe consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.