Study solves inverse problems for equations with fractional nonlinearities.
problem Solving inverse problems for semilinear elliptic equations with fractional power nonlinearities.
method Higher order linearization method adapted for fractional order.
result Results of previous studies remain valid for general power nonlinearities.
Gradient descent and SGD solve nonlinear inverse problems efficiently.
problem Solving nonlinear inverse problems with random design.
method Gradient descent and SGD with mini-batching, under classical assumptions.
result Achieves optimal convergence rates in RKHS framework.
In this note, we extend our previous work on the inverse σk problem. Inverse σk problem is a fully nonlinear geometric PDE on compact Kähler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.
We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not…
Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of con…
In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of high-di…
A new method for estimating adversarial strategies in nonlinear systems.
problem Inferring an intelligent adversarial agent's strategy in highly nonlinear systems.
method Formulated inverse cognition as a nonlinear Gaussian state-space model and developed an inverse UKF (IUKF) system.
result The estimation error of IUKF converges and closely follows the recursive Cramér-Rao lower bound.
Method solves Bayesian inverse problems in function space without assuming log-concavity.
problem Bayesian inverse problems in infinite-dimensional nonlinear settings.
method Score-based diffusion models as a prior, Langevin-type MCMC on function spaces.
result Provable convergence bound for posterior sampling, dependent on score approximation.
Diffusion models tackle noisy inverse problems with posterior sampling.
problem Efficiently solving general noisy inverse problems.
method Approximation of posterior sampling for diffusion models.
result Diffusion models can handle various noise statistics and nonlinear problems.
Study inverse boundary value problem for Monge-Ampère equation on convex domains.
problem Determine a positive source function from the Dirichlet-to-Neumann map for Monge-Ampère equation.
method Recover Hessian as Riemannian metric, prove DN map uniqueness, develop asymptotic expansions, solve nonlocal ∂-equation. result DN map uniquely determines positive source function in convex Euclidean plane domains.
This paper tackles regularization parameter learning in inverse problems using data-driven bilevel optimization.
problem Finding optimal regularization parameters in inverse problems.
method Data-driven bilevel optimization approach, analyzing performance in large data samples.
result The approach can reduce computational cost through online numerical schemes based on stochastic gradient descent.
PGD algorithms solve nonlinear inverse problems with generative priors using noisy measurements.
problem Signal estimation from noisy nonlinear measurements with generative priors.
method Projected gradient descent algorithms for two cases: unknown and known nonlinearity.
result PGD algorithms converge linearly to optimal statistical rates using arbitrary initialization.
New methods tackle statistical inverse problems with random data.
problem Statistical inverse problems with random experimental design.
method Spectral regularization, regularization by projection, convex penalties.
result Minimax rates in expectation and probability for convergence.
New approach uses secants to improve sensor placement and feature selection for nonlinear systems.
problem Inadequacy of linear methods for minimal sensor placement and feature selection in nonlinear systems.
method Data-driven approach using secant vectors to develop greedy algorithms for robust, near-minimal reconstruction guarantees.
result Demonstrated on two problems where linear techniques fail, secant-based approach provides robust solutions.
Study tackles inverse problems on low-dimensional manifolds, proving stability and proposing a reconstruction algorithm.
problem Inverse problems in infinite-dimensional spaces with nonlinear and ill-posed nature.
method Assumption of low-dimensional manifold, proving stability, proposing Landweber-type algorithm.
result Global convergence of the proposed algorithm, Lipschitz stability for specific inverse problems.
PnP-CM integrates CMs into PnP frameworks for efficient inverse problem solving.
problem Efficiently solving inverse problems with high-quality reconstructions.
method Reinterpreting CMs as proximal operators and integrating them into PnP frameworks.
result PnP-CM achieves high-quality reconstructions in as few as 4 NFEs.
CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.
problem Generating continuous functions in infinite-dimensional spaces for applications like inverse problems.
method Inspired by DCGAN, CGNNs use wavelet multiresolution analysis with convolutional and nonlinear layers.
result CGNNs can be injective under certain conditions on filters and nonlinearity, leading to Lipschitz stability estimates.
We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determinat…
We study inverse problems consisting on determining medium properties using the responses to probing waves from the machine learning point of view. Based on the understanding of propagation of waves and their nonlinear interactions, we construct a deep convolutional neural network in which the parameters are used to cl…
Inverse problem solved for relativistic Boltzmann equation on spacetime.
problem Determining spacetime from causal measurements.
method Using the nonlinearity of the Boltzmann equation to uniquely determine the spacetime.
result The spacetime is uniquely determined up to isometry in the causal set I+(x−)∩I−(x+). Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.
problem Determining minimal surfaces from boundary data.
method Developed a semiclassical nonlinear calculus for complex geometric optics solutions.
result Minimal surfaces can be recovered from the Dirichlet-to-Neumann map under certain conditions.
The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.
problem Inverse problems on holomorphically separable Kähler manifolds and conformally transversally anisotropic manifolds.
method Application of the Stone-Weierstrass theorem to show uniqueness in inverse problems.
result Generalization and simplification of earlier results in inverse problems.
New framework uses score-based priors to solve ill-conditioned polynomial equations, improving signal recovery from noisy data.
problem Recovering signals from low-order moments in inverse problems, especially ill-conditioned polynomial equations.
method Integrates score-based diffusion priors with moment-based estimators to regularize and solve nonlinear inverse problems.
result Diffusion priors improve recovery from third-order moments and make super-resolution MTD feasible.
The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…
New algorithms solve inverse problems using deep learning, converging faster than traditional methods.
problem Solving inverse problems with deep learning models.
method Simple non-convex algorithm for linear and nonlinear inverse problems, with theoretical and empirical support.
result The proposed algorithms converge faster than conventional techniques for certain inverse problems.
We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study also a variant of the inverse problem of the Cauchy problem and prove that the …
Physics-informed deep learning for PDEs solves forward and inverse problems efficiently.
problem Solving forward and inverse problems in parametric PDEs efficiently and accurately.
method Physics-informed deep latent variable model (PDDLVM) combining deep neural networks, probabilistic modelling, and variational inference.
result Achieves up to three orders of magnitude speed-up compared to traditional FEM while providing coherent uncertainty estimates.
Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map ever…
The paper studies how surfaces evolve in a cone under a specific flow.
problem Investigating the evolution of surfaces in a cone using a special flow.
method Analyzing a fully nonlinear parabolic Neumann problem under inverse curvature flow conditions.
result The evolving surfaces converge to a piece of the round sphere under certain conditions.
Our objective is to estimate the unknown compositional input from its output response through an unknown system after estimating the inverse of the original system with a training set. The proposed methods using artificial neural networks (ANNs) can compete with the optimal bounds for linear systems, where convex optim…
The paper reformulates regression in infinite dimensions as an inverse problem, showing it's equivalent to compact inverse problems.
problem Learning a linear operator between Hilbert spaces from empirical observations.
method Reformulates regression as an inverse problem, proving equivalence to compact inverse problems under specific conditions.
result The inverse problem is equivalent to compact inverse problems in terms of spectral properties and regularisation theory.
The paper analyzes Tikhonov regularization in Hilbert scales for statistical inverse problems.
problem Statistical inverse problems in Hilbert scales with general noise.
method Tikhonov regularization scheme with conditional stability estimates and high probability error bounds.
result Explicit rates of convergence for oversmoothing and regular cases over defined regularity classes.
Proposes an online method for high-dimensional streaming data.
problem Increasing variable dimensions with sample size in online kernel sliced inverse regression.
method Introduces approximate linear dependence condition and dictionary variable sets to address the problem. Transforms into online generalized eigen-decomposition problem and uses stochastic optimization for updates.
result Achieves close performance to batch processing kernel sliced inverse regression.
Study inverse problems with measure samples, improving estimator calibration and recovery.
problem Inverse problems with unknown potentials observed through measure samples.
method Introduced convex empirical objectives and sharpened Fenchel--Young losses for finite-dimensional potential classes.
result High-probability parameter recovery bounds for inverse entropic unbalanced optimal transport and inverse JKO learning.
Neural operators correct PDE residuals to improve BIP solutions.
problem Reducing error in infinite-dimensional Bayesian inverse problems with neural operators.
method Error correction using PDE residuals to improve neural operator approximation.
result Trained neural operators with error correction achieve a quadratic reduction in approximation error.
A new method uses DMs as priors for imaging problems, offering more accurate reconstructions.
problem Accurate probabilistic imaging for complex inverse problems.
method Markov chain Monte Carlo algorithm using DMs as plug-and-play priors for solving Bayesian inverse problems.
result Offers more accurate reconstructions and posterior estimation compared to existing methods.
Proposes a new method for nonlinear Bayesian updates using ensemble kernel regression.
problem Nonlinear and non-Gaussian Bayesian updates for complex systems.
method Combines Kalman filtering for observed components and kernel density estimation for unobserved components, with subsampling and clustering.
result Reduces estimation errors in highly nonlinear scenarios compared to standard linear updates.
Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well a…
Traditionally, nonlinear inversion, direct inversion, or wave estimation methods have been used for reconstructing images from MRE displacement data. In this work, we propose a convolutional neural network architecture that can map MRE displacement data directly into elastograms, circumventing the costly and computatio…
GO-OED maximizes predictive information gain on nonlinear QoIs.
problem Maximizing information gain on nonlinear predictive quantities.
method Nested Monte Carlo estimator, Markov chain Monte Carlo, kernel density estimation, Bayesian optimization.
result GO-OED outperforms conventional OED in nonlinear settings.
Unified framework solves nonlinear PDEs and IPs using Gaussian processes.
problem Solving and identifying parameters in nonlinear PDEs and inverse problems.
method Gaussian process framework approximating solutions as MAP estimators, reducing to finite-dimensional optimization problem.
result Unified method converges in a small number of iterations for various PDEs.
DynNet models dynamic responses of linear and nonlinear systems with fewer variables and higher accuracy.
problem Predicting dynamic responses of linear and nonlinear systems.
method Physics-based recurrent neural network with optimized architecture and training techniques.
result Higher accuracy and fewer trainable variables compared to existing models.
Paired autoencoders solve inverse problems using latent space projections.
problem Solving inverse problems in scientific computing.
method Paired autoencoder framework that projects data and quantity of interest into a latent space.
result Paired autoencoders generate multiple reconstruction metrics and enable latent-space refinement for accurate data fitting.
LazyDINO efficiently solves high-dimensional Bayesian inverse problems with fast and scalable solutions.
problem High-dimensional nonlinear Bayesian inverse problems with expensive parameter-to-observable maps.
method LazyDINO combines derivative-informed neural surrogates and lazy map variational inference for efficient posterior approximation.
result Significant cost reduction in amortized Bayesian inversion, achieving one to two orders of magnitude improvement.
We solve the problem of description for nonsingular pairs of compatible flat metrics in the general N-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. Th…
Improved diffusion models for inverse problems by integrating data consistency constraints.
problem Errors in earlier steps of diffusion models during posterior sampling.
method Guided Decoupled Posterior Sampling (GDPS) with data consistency constraint.
result GDPS achieves state-of-the-art performance, improving accuracy over existing methods.
Blade uses diffusion priors to accurately and calibratedly infer complex systems.
problem Derivative-free Bayesian inversion for high-dimensional, nonlinear problems with costly forward models.
method Blade employs an ensemble of interacting particles and diffusion models as priors, querying forward models only through evaluations.
result Blade produces well-calibrated posterior samples that existing methods cannot, improving with more iterations and particles.
Framework for designing nonlinearities in neural networks with slope constraints.
problem Designing nonlinearities with specific properties for signal processing.
method Variational framework with regularization for slope constraints and optimization of adaptive splines.
result Adaptive nonuniform linear splines achieve global optimum in constrained optimization.