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Maarten V. de Hoop

Maarten V. de Hoop appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
math.APmath.NAMachine Learningmath.DGmath.FAmath-phmath.MPmath.OCArtificial Intelligenceeess.SPInformation Theorymath.CAmath.PRmath.STNumerical Analysisphysics.comp-phphysics.geo-ph

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Published work

29 published item(s)

preprint2026arXiv

Dimension-Uniform Discretization Analysis of Preconditioned Annealed Langevin Dynamics for Multimodal Gaussian Mixtures

Obtaining stable diffusion-based samplers in high- and infinite-dimensional settings is challenging because errors can accumulate across high-frequency coordinates and make the dynamics unstable under refinement of the finite-dimensional approximation of the underlying function-space problem. Discretization is a typical source of such errors, and preconditioning with a suitable spectral decay is one way to control their accumulation. In this paper, we study this problem for preconditioned annealed Langevin dynamics (ALD) applied to Gaussian mixtures. We first show that Euler-Maruyama (EM) discretization, by treating the stiff linear part of the annealed score with a forward Euler step, imposes a stability constraint coupling the preconditioner with the annealed covariance scale. Together with the conditions ensuring dimension-uniform control of the annealed dynamics, this constraint forces the initial smoothed law to remain uniformly close to the target across dimensions. We then consider an exponential-integrator scheme that integrates the stiff linear part of the annealed score exactly. Under explicit spectral summability conditions coupling the smoothing covariance, the component covariance spectra, and the preconditioner, we prove a dimension-uniform Kullback-Leibler (KL) bound for this scheme. This bound can be made arbitrarily small, uniformly in dimension, by allowing enough time for annealing and then refining the time mesh accordingly. Importantly, these conditions allow regimes in which the KL divergence between the target and the initial smoothed law diverges with dimension, showing that the restrictions imposed by EM are scheme-dependent rather than intrinsic to ALD.

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preprint2026arXiv

Function graph transformers universally approximate operators between function spaces

We study the approximation of nonlinear operators between function spaces by transformers. Our approach is to lift functions to measures supported on their graphs and leverage a recently introduced measure-theoretic view of transformers. A function $h$ is represented by its graph measure $γ_h$, with finite tokens $\{(x_j,h(x_j))\}_{j=1}^N$ being its empirical approximations. We show that this framework elegantly models discretization refinement via convergence of measures and provides a natural setting for operator learning. Within this framework, we introduce function graph transformers, a graph-preserving subclass of measure-theoretic transformers that maps graph measures to graph measures, which is to say that outputs remain single-valued functions. Crucially, this additional structure does not reduce generality: we prove that the resulting graph-preserving maps can be approximated by finite compositions of standard softmax self-attention layers and pointwise MLPs, yielding universal approximation results for broad classes of nonlinear operators. Unlike existing theoretical approaches to operator learning with transformers, the measure-theoretic framework also accommodates regularized negative-order Sobolev inputs for which discretization invariance is particularly challenging, as well as query points on different output domains. Overall, function graph transformers provide a continuum viewpoint and mathematical toolkit for transformer-based operator learning, clarifying the roles of positional encodings, graph structure, regularization, and ensuring consistency across discretizations.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

On Observation Time for Recovering Latent Hawkes Networks

Dynamics of interacting systems in engineering, society, and nature often evolve over latent networks that govern which entities can interact. We study the problem of inferring these networks from event-based observations, which arise naturally in finance, seismology, and neuroscience. While there is substantial algorithmic work addressing this important problem, theoretical results are scarce. In this paper we ask the following fundamental question: what is the minimum time that one must observe the dynamics in order to exactly recover the underlying network, as a function of the number $d$ of interacting entities? For a class of stationary Hawkes processes with sparse, weak interactions, we prove that an observation time of order $\log d$ is sufficient and necessary. For the upper bound we construct a two-stage estimator that uses clipped and binned event data for screening, followed by a least-squares refinement, and apply concentration bounds derived from the Poisson cluster representation. For the lower bound we combine Fano's inequality with Jacod's Girsanov formula for point processes on a suitable subclass of networks.

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preprint2026arXiv

Training Infinitely Deep and Wide Transformers

Transformers have become the dominant architecture in modern machine learning, yet the theoretical understanding of their training dynamics remains limited. This paper develops a rigorous mathematical framework for analyzing gradient-based training of transformers in the mean-field regime, where both the depth (number of layers) and width (number of attention heads) tend to infinity. While ResNet training can be understood as controlling a neural ODE, transformer training corresponds to controlling a neural PDE, due to the coupling of multiple token distributions through the attention mechanism. Our mean-field model features two types of measure representations: token distributions evolving through layers and attention parameters at each layer. We establish well-posedness of the forward pass through infinitely deep transformers, characterizing token evolution via flow maps that satisfy ODEs in function spaces. Using adjoint sensitivity analysis, we derive an explicit formula for the conditional Wasserstein gradient of the training risk, involving adjoint variables governed by backward ODEs. We prove the existence and uniqueness of gradient flow curves in the conditional Wasserstein metric space, establishing a rigorous foundation for gradient-based transformer training. A key technical contribution is providing necessary and sufficient conditions for injectivity of the Neural Tangent Kernel (NTK) for attention mechanisms: we show that NTK injectivity is equivalent to linear independence of log-sum-exp functions modulo affine functions, a condition satisfied by diverse token distributions, including discrete distributions, uniform distributions, and Gaussian mixtures. Under this NTK injectivity assumption, we prove that gradient flow converges to global minima when the initial loss is sufficiently small, eliminating spurious local minima from the optimization landscape.

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preprint2022arXiv

Deep learning architectures for nonlinear operator functions and nonlinear inverse problems

We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks, for approximating nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution algorithms for inverse boundary value problems. Traditional neural networks treat input data as vectors, and thus they do not effectively capture the multiplicative structure associated with the linear operators that correspond to the data in such inverse problems. We therefore introduce a new family that resembles a standard neural network architecture, but where the input data acts multiplicatively on vectors. Motivated by compact operators appearing in boundary control and the analysis of inverse boundary value problems for the wave equation, we promote structure and sparsity in selected weight matrices in the network. After describing this architecture, we study its representation properties as well as its approximation properties. We furthermore show that an explicit regularization can be introduced that can be derived from the mathematical analysis of the mentioned inverse problems, and which leads to certain guarantees on the generalization properties. We observe that the sparsity of the weight matrices improves the generalization estimates. Lastly, we discuss how operator recurrent networks can be viewed as a deep learning analogue to deterministic algorithms such as boundary control for reconstructing the unknown wavespeed in the acoustic wave equation from boundary measurements.

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preprint2022arXiv

Recovery of piecewise smooth density and Lamé parameters from high-frequency exterior Cauchy data

We consider an isotropic elastic medium occupying a bounded domain D whose density and Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside the domain D and only for initial data supported outside D, and we study the recovery of the density and Lamé parameters. For known density, results have recently been obtained using the scattering control method to recover wave speeds. Here, we extend this result to include the recovery of the density in addition to the Lamé parameters under certain geometric conditions using techniques from microlocal analysis and a connection to local tensor tomography.

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preprint2022arXiv

Recovery of wave speeds and density of mass across a heterogeneous smooth interface from acoustic and elastic wave reflection operators

We revisit the problem of recovering wave speeds and density across a curved interface from reflected wave amplitudes. Such amplitudes have been exploited for decades in (exploration) seismology in this context. However, the analysis in seismology has been based on linearization and mostly flat interfaces. Here, we present a nonlinear analysis allowing curved interfaces, establish uniqueness and provide a reconstruction, while making the notion of amplitude precise through a procedure rooted in microlocal analysis.

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preprint2022arXiv

The Cost-Accuracy Trade-Off In Operator Learning With Neural Networks

The term `surrogate modeling' in computational science and engineering refers to the development of computationally efficient approximations for expensive simulations, such as those arising from numerical solution of partial differential equations (PDEs). Surrogate modeling is an enabling methodology for many-query computations in science and engineering, which include iterative methods in optimization and sampling methods in uncertainty quantification. Over the last few years, several approaches to surrogate modeling for PDEs using neural networks have emerged, motivated by successes in using neural networks to approximate nonlinear maps in other areas. In principle, the relative merits of these different approaches can be evaluated by understanding, for each one, the cost required to achieve a given level of accuracy. However, the absence of a complete theory of approximation error for these approaches makes it difficult to assess this cost-accuracy trade-off. The purpose of the paper is to provide a careful numerical study of this issue, comparing a variety of different neural network architectures for operator approximation across a range of problems arising from PDE models in continuum mechanics.

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preprint2021arXiv

A foliated and reversible Finsler manifold is determined by its broken scattering relation

The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the Earth can be in principle reconstructed from single scattering measurements at the surface.

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preprint2020arXiv

Full Reciprocity-Gap Waveform Inversion in the frequency domain, enabling sparse-source acquisition

The quantitative reconstruction of sub-surface Earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data usually consist of measurements of the pressure field but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and defines the Full Reciprocity-gap Waveform Inversion (FRgWI ) method. This misfit functional provides additional features compared to the more traditional least-squares approaches with, in particular, that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be arbitrarily chosen. Based on three-dimensional experiments, FRgWI is shown to behave better than Full Waveform Inversion (FWI) in the same context. Then, it allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to compensate. On the other hand, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance with multiple-point sources, hence reducing the numerical cost. FRgWI displays accurate reconstructions in both situations and appears more robust with respect to cross-talk than the least-squares shot-stacking.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Generic uniqueness and stability for the mixed ray transform

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 and 2+2 tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of qS-polarized elastic waves provide partial information about the mixed ray transform of 2+2 tensors fields. If in addition we include the measurement of the shear wave amplitude, the complete mixed ray transform can be recovered. We also show how one can obtain the mixed ray transform from an anisotropic perturbation of the Dirichlet-to-Neumann map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space.

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preprint2019arXiv

A weight-adjusted discontinuous Galerkin method for the poroelastic wave equation: penalty fluxes and micro-heterogeneities

We introduce a high-order weight-adjusted discontinuous Galerkin (WADG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in anisotropic porous media. We use a coupled first-order symmetric stress-velocity formulation. Careful attention is directed at (a) the derivation of an energy-stable penalty-based numerical flux, which offers high-order accuracy in presence of material discontinuities, and (b) proper treatment of micro-heterogeneities (sub-element variations) in the numerical scheme. The use of a penalty-based numerical flux avoids the diagonalization of Jacobian matrices into polarized wave constituents necessary when solving element-wise Riemann problems. Micro-heterogeneities are accurately and stably incorporated in the numerical scheme using easily-invertible weight-adjusted mass matrices. The convergence of the proposed numerical scheme is proven and verified by using convergence studies against analytical plane wave solutions. The proposed method is also compared against an existing implementation using the spectral element method to solve the poroelastic wave equation.

0 citations0 reviewsNo code linkedOpen access
preprint2018arXiv

Inverting the local geodesic ray transform of higher rank tensors

Consider a Riemannian manifold in dimension $n\geq 3$ with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic ray transform on tensor fields of rank four near a boundary point. This problem is closely related with elastic \textit{qP}-wave tomography. Under the condition that the manifold can be foliated with a continuous family of strictly convex hypersurfaces, the local invertibility implies a global result. One can straightforwardedly adapt the proof to show similar results for tensor fields of arbitrary rank.

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preprint2015arXiv

Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities

We consider the electrostatic inverse boundary value problem also known as electrical impedance tomography (EIT) for the case where the conductivity is a piecewise linear function on a domain $Ω\subset\mathbb{R}^n$ and we show that a Lipschitz stability estimate for the conductivity in terms of the local Dirichlet-to-Neumann map holds true.

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preprint2015arXiv

Multiscale reverse-time-migration-type imaging using the dyadic parabolic decomposition of phase space

We develop a representation of reverse-time migration in terms of Fourier integral operators the canonical relations of which are graphs. Through the dyadic parabolic decomposition of phase space, we obtain the solution of the wave equation with a boundary source and homogeneous initial conditions using wave packets. On this basis, we develop a numerical procedure for the reverse time continuation from the boundary of scattering data and for RTM migration. The algorithms are derived from those we recently developed for the discrete approximate evaluation of the action of Fourier integral operators and inherit from their conceptual and numerical properties.

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preprint2015arXiv

Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

In this paper, we prove optimal convergence rates results for regularisation methods for solving linear ill-posed operator equations in Hilbert spaces. The result generalises existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.

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preprint2015arXiv

Uniqueness for a seismic inverse source problem modeling a subsonic rupture

We consider an inverse problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. We assume that the sources occur along a path with subsonic velocity, and that data are collected over time on some detection surface. We explore the question of uniqueness for these problems, show how to recover the times and locations of sources microlocally, and then reconstruct the smooth part of the source assuming that it is the same at each source location.

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preprint2014arXiv

Generalized Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting. We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer, and generalize existing convergence rates results. The key tool in our analysis is a novel source condition, which we put into relation to the existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces.

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preprint2014arXiv

Inverse boundary value problem for the Helmholtz equation: Multi-level approach and iterative reconstruction

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequency as the data. We develop an explicit reconstruction of the wavespeed using a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeeds containing discontinuities. A conditional Lipschitz stability estimate for the inverse problem holds for wavespeeds of the form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partitioning increases. To mitigate this growth of the stability constant, we introduce hierarchical compressive approximations of the solution to the inverse problem with piecewise constant functions. We establish an optimal bound of the stability constant, which leads to a condition on the compression rate pertaining to these approximations.

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preprint2014arXiv

Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves

We consider the inverse problem of determining the Lamé parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lamé parameters and the density are assumed to be piecewise constant on a given domain partition.

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preprint2013arXiv

Multi-scale discrete approximations of Fourier integral operators associated with canonical transformations and caustics

We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists in the construction of a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, enabling the application of wave packet driven computation, and in the construction of the associated pseudo-differential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.

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preprint2012arXiv

A convergence analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints

We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a closed, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data. We proceed with developing a multi-level algorithm based on a nested family of closed, convex subsets on which stability holds and the stability constants are ordered. Growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion, after a finite number of steps.

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preprint2012arXiv

Lipschitz stability of an inverse boundary value problem for a Schrödinger type equation

In this paper we study the inverse boundary value problem of determining the potential in the Schrödinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values.

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preprint2012arXiv

Local Analysis of Inverse Problems: Hölder Stability and Iterative Reconstruction

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however, the data space can be an arbitrary Banach space. We study sequences of parameter functions generated by a nonlinear Landweber iteration and conditions under which these strongly converge, locally, to the solutions within an appropriate distance. We express the conditions for convergence in terms of Hölder stability of the inverse maps, which ties naturally to the analysis of inverse problems.

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preprint2012arXiv

Reconstruction of a conformally Euclidean metric from local boundary diffraction travel times

We consider a region $M$ in $\mathbb{R}^n$ with boundary $\partial M$ and a metric $g$ on $M$ conformal to the Euclidean metric. We analyze the inverse problem, originally formulated by Dix, of reconstructing $g$ from boundary measurements associated with the single scattering of seismic waves in this region. In our formulation the measurements determine the shape operator of wavefronts outside of $M$ originating at diffraction points within $M$. We develop an explicit reconstruction procedure which consists of two steps. In the first step we reconstruct the directional curvatures and the metric in what are essentially Riemmanian normal coordinates; in the second step we develop a conversion to Cartesian coordinates. We admit the presence of conjugate points. In dimension $n \geq 3$ both steps involve the solution of a system of ordinary differential equations. In dimension $n=2$ the same is true for the first step, but the second step requires the solution of a Cauchy problem for an elliptic operator which is unstable in general. The first step of the procedure applies for general metrics.

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preprint2012arXiv

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

We analyze the inverse problem, originally formulated by Dix in geophysics, of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of seismic waves. We consider a domain $\tilde M$ with a varying and possibly anisotropic wave speed which we model as a Riemannian metric $g$. For our data, we assume that $\tilde M$ contains a dense set of point scatterers and that in a subset $U\subset \tilde M$, modeling a region containing measurement devices, we can measure the wave fronts of the single scattered waves diffracted from the point scatterers. The inverse problem we study is to recover the metric $g$ in local coordinates anywhere on a set $M \subset \tilde M$ up to an isometry (i.e. we recover the isometry type of $M$). To do this we show that the shape operators related to wave fronts produced by the point scatterers within $\tilde M$ satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming we know $g$ as well as the shape operator of the wave fronts in the region $U$, we may recover $g$ in certain coordinate systems (e.g. Riemannian normal coordinates centered at point scatterers). This generalizes the well-known geophysical method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics.

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preprint2011arXiv

Linearized inverse scattering based on seismic Reverse Time Migration

In this paper we study the linearized inverse problem associated with imaging of reflection seismic data. We introduce an inverse scattering transform derived from reverse-time migration (RTM). In the process, the explicit evaluation of the so-called normal operator is avoided, while other differential and pseudodifferential operator factors are introduced. We prove that, under certain conditions, the transform yields a partial inverse, and support this with numerical simulations. In addition, we explain the recently discussed 'low-frequency artifacts' in RTM, which are naturally removed by the new method.

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