Study Tikhonov regularization for ill-posed surface equations, analyzing perturbations and applications.
problem Solving ill-posed operator equations with solutions on surfaces.
method Error analysis of Tikhonov regularization considering surface perturbations and vector bundle solutions.
result Error analysis and practical applications demonstrated for functions on surfaces.
The Fredholm integral equation of the first kind improves solutions for ill-posed supervised learning problems with limited data.
problem Ill-posed supervised learning problems with insufficient data.
method Using the Fredholm integral equation of the first kind (FIFK) with semi-supervised assumptions and MSDF methods.
result Improved accuracy and stability in solutions for ill-posed problems.
A new mathematical model for the Black-Scholes equation is proposed to forecast option prices. This model includes new interval for the price of the underlying stock as well as new initial and boundary conditions. Conventional notions of maturity time and strike prices are not used. The Black-Scholes equation is solved…
Novel method uses Gaussian process to estimate particle sizes from scattering data.
problem Estimating particle size distributions from noisy optical scattering measurements.
method Constrained Gaussian process regression with normalization constraints.
result Accurately reconstructs particle size distributions from noisy data.
New method forecasts stock option prices accurately.
problem Accurate forecasting of stock option prices.
method Solving the ill-posed Black-Scholes equation using the Quasi-Reversibility Method.
result Good forecasting results demonstrated on market data.
ReTaSA tackles continuous target shift in regression problems.
problem Continuous target shift in regression settings.
method Nonparametric regularized approach to estimate importance weight function.
result The method provides theoretical justification for the estimated importance weight function.
We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We …
In this paper we present nonparametric estimators for coefficients in stochastic differential equation if the data are described by independent, identically distributed random variables. The problem is formulated as a nonlinear ill-posed operator equation with a deterministic forward operator described by the Fokker-Pl…
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.
We introduce a general constructive setting of the density ratio estimation problem as a solution of a (multidimensional) integral equation. In this equation, not only its right hand side is known approximately, but also the integral operator is defined approximately. We show that this ill-posed problem has a rigorous …
Random dictionaries help solve complex inverse problems without strict assumptions.
problem Solving ill-posed linear inverse problems with overcomplete dictionaries.
method Apply random dictionaries to regression problems and study their performance.
result Random dictionaries can solve ill-posed linear inverse problems without stringent compatibility conditions.
New method reconstructs Black-Scholes option prices from current profiles.
problem Reconstructing Black-Scholes prices from current profiles, dealing with ill-posedness.
method Price-dimensional reduction using Legendre polynomials, Tikhonov regularization.
result Reconstructs Black-Scholes prices from noisy initial data, stabilizing the solution.
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estima…
Study rates of convergence for approximate solutions to linear ill-posed problems in Hilbert scales.
problem Linear ill-posed inverse problems with noisy data.
method Approximate reconstructions from random noisy data using regularization schemes in Hilbert scale.
result Explicitly established error bounds for smooth regression functions.
Variational Gaussian Processes solve linear inverse problems efficiently.
problem Solving inverse problems where indirect observations are corrupted by noise.
method Variational Bayesian methods with Gaussian process priors and inducing variables.
result Posterior contraction rates can be attained by correctly tuned variational procedures.
New method tackles ill-posed imaging problems with random projections.
problem Solving inverse problems with limited training data and severe ill-posedness.
method Learn an ensemble of simpler mappings to projections, then combine and deconvolve.
result More robust to noise and unseen corruptions than direct learning.
New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.
problem Comparative analysis of regularization norms in ill-posed problems.
method Small noise analysis framework for Tikhonov and RKHS regularizations.
result Optimal convergence rates achieved with adaptive fractional RKHS, but hyper-parameters decay too fast.
The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow define…
MCGDiff uses SGM to guide SMC for solving ill-posed linear inverse problems.
problem Solving ill-posed linear inverse problems in Bayesian settings.
method Exploiting SGM structure, defining a sequence of intermediate problems, and using SMC methods.
result MCGDiff outperforms competing methods in Bayesian ill-posed inverse problems.
Paper optimizes estimation of quadratic functionals in nonparametric IV models.
problem Optimal estimation of a nonlinear functional in ill-posed inverse regression.
method Adaptive, minimax estimation using leave-one-out, sieve NPIV estimator with data-driven sieve dimension selection.
result Adaptive estimator achieves minimax optimal rate in various ill-posed cases.
Abstract: Translates causal inference into statistical formalism, examines estimability and ill-posedness.
problem What can be estimated from causal inference problems?
method Uses abstract statistical formalism and category theory to analyze identifiability and estimability.
result Identifiability does not guarantee stability, making estimability a stricter condition.
Repulsive ensembles improve uncertainty estimates in PINNs for differential equations.
problem Improving uncertainty estimates in PINNs for differential equations.
method Employing repulsive ensembles (RE-PINN) with a repulsive term in the loss function.
result Repulsive ensembles produce more accurate uncertainty estimates and higher sample diversity.
IAGAN method improves medical image reconstruction by incorporating adaptive GAN priors.
problem Reconstructing high-fidelity medical images from incomplete data.
method Image-adaptive GAN-based reconstruction method (IAGAN).
result IAGAN can recover fine structures relevant for medical diagnosis.
A new method calibrates jump-diffusion models from option prices.
problem Calibrating jump-diffusion models from market data.
method Forward Dupire-type PIDE, Tikhonov regularization.
result Robust method for identifying local volatility and jump size.
New model resolves signal ambiguities in ill-posed systems.
problem Signal retrieval from indirect measurements with known models.
method Variational generative model that captures signal distribution.
result Retrieves consistent signals with high fidelity.
Paper introduces a Gibbs sampler for Bayesian inversion of ill-posed problems.
problem Bayesian inversion of ill-posed problems with linear transformation and additive noise.
method Gibbs algorithm based on prior diffusion model.
result Gibbs algorithm offers a guarantee of convergence in a specific situation.
Noise stabilizes solutions to transport equations, preventing blow-up.
problem Proving global existence and uniqueness of solutions to stochastic transport equations.
method Characteristics-based techniques exploiting the geometric structure of transport equations.
result Noise prevents blow-up in deterministic solutions and ensures global existence and uniqueness of solutions.
Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric inva…
Researchers develop methods to recover agent behavior from sparse data using Gaussian processes.
problem Recovering agent behavior from limited, noisy data in potential mean field games.
method Two Gaussian process-based frameworks: inf-sup formulation and bilevel approach.
result Surrogate MFG models can accurately reproduce observed data, even when prior information is limited.
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.
Score-based models improve diffuse optical tomography accuracy.
problem Improving accuracy in diffuse optical tomography with uncertainty quantification.
method Score-based diffusion models with a mixed score function to prevent overfitting.
result Data-driven prior distribution results in posterior samples with low variance and centred around the ground truth.
Study forward investment performance in semimartingale markets with stochastic factors.
problem Investigate forward investment performance in incomplete semimartingale markets with power risk preferences and stochastic integrated factors.
method Develop necessary and sufficient conditions for FIPP existence, use integral representations, and solve ill-posed HJB equations.
result Explicit constructions for time-monotone FIPPs in semimartingale models, generalizing from Brownian to semimartingale markets.
Physics-informed neural network identifies and characterizes surface cracks in metals.
problem Identifying and characterizing surface-breaking cracks in metals using ultrasound.
method Physics-informed neural network (PINN) trained with ultrasonic surface wave data and adaptive activation functions.
result PINN accurately estimates the speed of sound and identifies crack locations in metals.
We study the ruin problem over a risk process described by a discrete-time Markov model. In contrast to previous studies that focused on the asymptotic behaviour of ruin probabilities for large values of the initial capital, we provide a new technique to compute the quantity of interest for any initial value, and with …
Proves time analyticity for heat and Navier-Stokes equations without decaying conditions.
problem Analyticity of solutions to heat and Navier-Stokes equations without decaying conditions.
method Real variable method and algebraic manipulation of integral kernels.
result First general pointwise time analyticity result for all dimensions.
New method for adaptive estimation and inference in econometric models without knowing smoothness.
problem Adaptive estimation and inference in ill-posed linear inverse problems with unknown smoothness.
method Discrepancy principle-based framework for adaptive hyperparameter selection.
result Achieves optimal rates in weak and strong metrics for linear functionals.
Study on investment and consumption strategy with transaction costs, focusing on a single illiquid asset.
problem Investment and consumption problem with transaction costs.
method Specialized to a case with zero transaction costs except for sales and purchases of a single asset, transformed HJB equation into a boundary value problem.
result Optimal trading strategy involves trading the illiquid asset only when its fraction of the total portfolio value falls outside a fixed interval.
Study identifies unique minimizers for interaction kernels in particle systems.
problem Identifying unique interaction kernels in mean-field equations of interacting particles.
method Data-adaptive L2 spaces, RKHS analysis, regularization. result Characterization of identifiability in both finite and infinite particle systems.
The paper explores deep image priors for solving inverse problems.
problem Solving ill-posed inverse problems in image processing.
method Introduces and analyzes deep image priors as optimization of Tikhonov functionals.
result Analytic results for specific network designs and linear operators.
We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the con…
Paper combines QRM and CNN for better stock option price forecasting.
problem Forecasting stock option prices in a complex market.
method Solves Black-Scholes equation using QRM, trains CNN models on data.
result CNN models improve option price prediction accuracy.
New framework uses deep generative priors for robust phase retrieval.
problem Highly ill-posed and non-linear phase retrieval problem.
method Regularization through deep generative priors with gradient descent.
result Effective for random Gaussian and Fourier friendly measurements.
Deep neural networks improve proximal inference for causal effects.
problem Estimating causal effects in the presence of unmeasured confounders.
method Flexible deep neural network to estimate the bridge function.
result Achieves state-of-the-art performance on benchmarks.
Functional PLS improves prediction and inference for scalar responses from functional predictors.
problem Estimating scalar responses from functional predictors in an ill-posed inverse problem.
method Functional partial least squares (PLS) estimator with adaptive early stopping and new tests.
result PLS attains nearly minimax-optimal convergence rates and detects local alternatives.
The paper tackles inverse uncertainty quantification in neutron noise analysis.
problem Uncertainty in estimating material properties from noisy neutron correlation measurements.
method Surrogate models and inverse uncertainty quantification to account for measurement error and model bias.
result Improved prediction of neutron correlations and quantification of uncertainties.
This work tackles uncertainty quantification in tomography reconstruction.
problem Ill-posed nature of tomographic reconstruction leading to no unique solution.
method Gaussian process modeling to incorporate prior knowledge and experimental noises.
result Efficient uncertainty quantification in tomographic reconstruction.
This paper reviews weighted clustering ensemble methods.
problem Improving clustering results from individual methods.
method Different types of weights and approaches to determining weight values.
result Unified framework for selecting appropriate weighting mechanisms.
WNVI solves inverse problems without forward models using neural networks.
problem Solving high-dimensional Bayesian inverse problems based on PDEs.
method WNVI uses weighted residuals and SVI with neural networks to infer state variables and unknowns.
result WNVI is more accurate and efficient than traditional methods and handles ill-posed problems.