arXiv subjects

Knut Solna

Publications and source records attributed to Knut Solna.

At least 19 recordsLinked to original sources

arXiv2026

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.

stat.ML↗

arXiv2026

Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics

Designing sampling algorithms for multimodal targets that remain stable under refinement of the finite-dimensional approximation of an underlying function-space problem is a central challenge. Annealed Langevin dynamics (ALD) is a natural alternative to classical Langevin in this context, since it is often observed to improve exploration across modes. Yet a gap remains between its empirical success and existing theory: under which conditions can ALD be guaranteed to remain stable across dimensions? In this paper, we bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for Gaussian-mixture targets. Along an explicit annealing path obtained by gradually removing Gaussian smoothing from the target, we identify spectral conditions linking the smoothing covariance to the component covariances under which ALD achieves a prescribed accuracy in Kullback-Leibler divergence within a dimension-uniform time horizon. We then establish stability in a perturbative regime with imperfect initialization and approximate scores. Under a misspecified-mixture score model, we show that preconditioning ALD with an operator whose spectrum decays sufficiently fast prevents error terms from accumulating across coordinates and thereby preserves dimension-uniform control.

math.NA↗

arXiv2025

Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems

Designing algorithms for solving high-dimensional Bayesian inverse problems directly in infinite-dimensional function spaces - where such problems are naturally formulated - is crucial to ensure stability and convergence as the discretization of the underlying problem is refined. In this paper, we contribute to this line of work by analyzing a widely used sampler for linear inverse problems: Langevin dynamics driven by score-based generative models (SGMs) acting as priors, formulated directly in function space. Building on the theoretical framework for SGMs in Hilbert spaces, we give a rigorous definition of this sampler in the infinite-dimensional setting and derive, for the first time, error estimates that explicitly depend on the approximation error of the score. As a consequence, we obtain sufficient conditions for global convergence in Kullback-Leibler divergence on the underlying function space. Preventing numerical instabilities requires preconditioning of the Langevin algorithm and we prove the existence and the form of an optimal preconditioner. The preconditioner depends on both the score error and the forward operator and guarantees a uniform convergence rate across all posterior modes. Our analysis applies to both Gaussian and a general class of non-Gaussian priors. Finally, we present examples that illustrate and validate our theoretical findings.

stat.ML↗

arXiv2024

Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems

This work introduces a sampling method capable of solving Bayesian inverse problems in function space. It does not assume the log-concavity of the likelihood, meaning that it is compatible with nonlinear inverse problems. The method leverages the recently defined infinite-dimensional score-based diffusion models as a learning-based prior, while enabling provable posterior sampling through a Langevin-type MCMC algorithm defined on function spaces. A novel convergence analysis is conducted, inspired by the fixed-point methods established for traditional regularization-by-denoising algorithms and compatible with weighted annealing. The obtained convergence bound explicitly depends on the approximation error of the score; a well-approximated score is essential to obtain a well-approximated posterior. Stylized and PDE-based examples are provided, demonstrating the validity of our convergence analysis. We conclude by presenting a discussion of the method's challenges related to learning the score and computational complexity.

stat.ML↗

arXiv2023

Conditional score-based diffusion models for Bayesian inference in infinite dimensions

Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions - the conditional denoising estimator - can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference.

stat.ML↗

arXiv2022

Paraxial wave propagation in random media with long-range correlations

We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may be integrable or not. We focus attention mostly on the non-integrable case, which corresponds to a random perturbation with long-range correlations, that is relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a high-frequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: The travel time of the wave front is randomized at short distances of propagation and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schroedinger-type equation driven by a standard Brownian field. We use these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications like imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence.

math.AP↗

arXiv2022

Scintillation of Partially Coherent Light in Time Varying Complex Media

We present a theory for wave scintillation in the situation with a time-dependent partially coherent source and a time-dependent randomly heterogeneous medium. Our objective is to understand how the scintillation index of the measured intensity depends on the source and medium parameters. We deduce from an asymptotic analysis of the random wave equation a general form of the scintillation index and we evaluate this in various scaling regimes. The scintillation index is a fundamental quantity that is used to analyze and optimize imaging and communication schemes. Our results are useful to quantify the scintillation index under realistic propagation scenarios and to address such optimization challenges.

physics.optics↗

arXiv2022

Speckle Memory Effect in the Frequency Domain and Stability in Time-Reversal Experiments

When waves propagate through a complex medium like the turbulent atmosphere the wave field becomes incoherent and the wave intensity forms a complex speckle pattern. In this paper we study a speckle memory effect in the frequency domain and some of its consequences. This effect means that certain properties of the speckle pattern produced by wave transmission through a randomly scattering medium is preserved when shifting the frequency of the illumination. The speckle memory effect is characterized via a detailed novel analysis of the fourth-order moment of the random paraxial Green's function at four different frequencies. We arrive at a precise characterization of the frequency memory effect and what governs the strength of the memory. As an application we quantify the statistical stability of time-reversal wave refocusing through a randomly scattering medium in the paraxial or beam regime. Time reversal refers to the situation when a transmitted wave field is recorded on a time-reversal mirror then time reversed and sent back into the complex medium. The reemitted wave field then refocuses at the original source point. We compute the mean of the refocused wave and identify a novel quantitative description of its variance in terms of the radius of the time-reversal mirror, the size of its elements, the source bandwidth and the statistics of the random medium fluctuations.

math.AP↗

arXiv2019

Multiplexing schemes for optical communication through atmospheric turbulence

A central question in free-space optical communications is how to improve the transfer of information between a transmitter and receiver. The capacity of the communication channel can be increased by multiplexing of independent modes using either: (1) the MIMO (Multiple-Input-Multiple- Output) approach, where the communication is done with modes obtained from the singular value decomposition of the transfer matrix from the transmitter array to the receiver array, or (2) the OAM (Orbital Angular Momentum) approach, which uses vortex beams that carry angular momenta. In both cases, the number of usable modes is limited by the finite aperture of the transmitter and receiver, and the effect of the turbulent atmosphere. The goal of this paper is twofold: First, we show that the MIMO and OAM multiplexing schemes are closely related. Specifically, in the case of circular apertures, the usable singular modes of the transfer matrix are essentially the same as the commonly used Laguerre-Gauss vortex beams, provided these have a special radius that depends on the wavelength, the distance from the transmitter to the receiver and the ratio of the radii of their apertures. Second, we study the effect of atmospheric turbulence on the communication modes using the phase screen method put in the mathematical framework of beam propagation in random media.

eess.SP↗

arXiv2018

Optimal hedging under fast-varying stochastic volatility

In a market with a rough or Markovian mean-reverting stochastic volatility there is no perfect hedge. Here it is shown how various delta-type hedging strategies perform and can be evaluated in such markets in the case of European options. A precise characterization of the hedging cost, the replication cost caused by the volatility fluctuations, is presented in an asymptotic regime of rapid mean reversion for the volatility fluctuations. The optimal dynamic asset based hedging strategy in the considered regime is identified as the so-called `practitioners' delta hedging scheme. It is moreover shown that the performances of the delta-type hedging schemes are essentially independent of the regularity of the volatility paths in the considered regime and that the hedging costs are related to a vega risk martingale whose magnitude is proportional to a new market risk parameter. It is also shown via numerical simulations that the proposed hedging schemes which derive from option price approximations in the regime of rapid mean reversion, are robust: the `practitioners' delta hedging scheme that is identified as being optimal by our asymptotic analysis when the mean reversion time is small seems to be optimal with arbitrary mean reversion times.

q-fin.PR↗

arXiv2018

Chaos and Order in the Bitcoin Market

The bitcoin price has surged in recent years and it has also exhibited phases of rapid decay. In this paper we address the question to what extent this novel cryptocurrency market can be viewed as a classic or semi-efficient market. Novel and robust tools for estimation of multi-fractal properties are used to show that the bitcoin price exhibits a very interesting multi-scale correlation structure. This structure can be described by a power-law behavior of the variances of the returns as functions of time increments and it can be characterized by two parameters, the volatility and the Hurst exponent. These power-law parameters, however, vary in time. A new notion of generalized Hurst exponent is introduced which allows us to check if the multi-fractal character of the underlying signal is well captured. It is moreover shown how the monitoring of the power-law parameters can be used to identify regime shifts for the bitcoin price. A novel technique for identifying the regimes switches based on a goodness of fit of the local power-law parameters is presented. It automatically detects dates associated with some known events in the bitcoin market place. A very surprising result is moreover that, despite the wild ride of the bitcoin price in recent years and its multi-fractal and non-stationary character, this price has both local power-law behaviors and a very orderly correlation structure when it is observed on its entire period of existence.

q-fin.ST↗

arXiv2018

Emergence of Turbulent Epochs in Oil Prices

Oil price data have a complicated multi-scale structure that may vary with time. We use time-frequency analysis to identify the main features of these variations and, in particular, the regime shifts. The analysis is based on a wavelet-based decomposition and analysis of the associated scale spectrum. The joint estimation of the local Hurst exponent and volatility is the key to detect and identify regime shifting and switching of the oil price. The framework involves in particular modeling in terms of a process of `multi-fractional' type so that both the roughness and the volatility of the price process may vary with time. Special epochs then emerge as a result of these degrees of freedom, moreover, as a result of the special type of spectral estimator used. These special epochs are discussed and related to historical events. Some of them are not detected by standard analysis based on maximum likelihood estimation. The paper presents a novel algorithm for robust detection of such special epochs and multi-fractional behavior in financial or other types of data. In the financial context insight about such behavior of the asset price is important to evaluate financial contracts involving the asset.

q-fin.ST↗

arXiv2018

Wave propagation and imaging in moving random media

We present a study of sound wave propagation in a time dependent random medium and an application to imaging. The medium is modeled by small temporal and spatial random fluctuations in the wave speed and density, and it moves due to an ambient flow. We develop a transport theory for the energy density of the waves, in a forward scattering regime, within a cone (beam) of propagation with small opening angle. We apply the transport theory to the inverse problem of estimating a stationary wave source from measurements at a remote array of receivers. The estimation requires knowledge of the mean velocity of the ambient flow and the second-order statistics of the random medium. If these are not known, we show how they may be estimated from additional measurements gathered at the array, using a few known sources. We also show how the transport theory can be used to estimate the mean velocity of the medium. If the array has large aperture and the scattering in the random medium is strong, this estimate does not depend on the knowledge of the statistics of the random medium.

math.AP↗

arXiv2018

Imaging through a scattering medium by speckle intensity correlations

In this paper we analyze an imaging technique based on intensity speckle correlations over incident field position proposed in [J. A. Newmann and K. J. Webb, Phys. Rev. Lett. 113, 263903 (2014)]. Its purpose is to reconstruct a field incident on a strongly scattering random medium. The thickness of the complex medium is much larger than the scattering mean free path so that the wave emerging from the random section forms an incoherent speckle pattern. Our analysis clarifies the conditions under which the method can give a good reconstruction and characterizes its performance. The analysis is carried out in the white-noise paraxial regime, which is relevant for the applications in optics that motivated the original paper.

math.AP↗

arXiv2018

Non-invasive imaging through random media

When waves propagate through a strongly scattering medium the energy is transferred to the incoherent wave part by scattering. The wave intensity then forms a random speckle pattern seemingly without much useful information. However, a number of recent physical experiments show how one can extract useful information from this speckle pattern. Here we present the mathematical analysis that explains the quite stunning performance of such a scheme for speckle imaging. Our analysis identifies a scaling regime where the scheme works well. This regime is the white-noise paraxial regime, which leads to the Ito-Schrodinger model for the wave amplitude. The results presented in this paper conform with the sophisticated physical intuition that has motivated these schemes, but give a more detailed characterization of the performance. The analysis gives a description of (i) the information that can be extracted and with what resolution (ii) the statistical stability or signal-to-noise ratio with which the information can be extracted.

physics.optics↗

arXiv2017

Option Pricing under Fast-varying and Rough Stochastic Volatility

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.

q-fin.PR↗

arXiv2017

Global acoustic daylight imaging in a stratified Earth-like model

We present an analysis of acoustic daylight imaging in an Earth-like model assuming a random distribution of noise sources spatially supported in an annulus located away from the surface. We assume a situation with scalar wave propagation and that the measurements are of the wave field at the surface. Then, we obtain a relation between the autocorrelation function of the measurements and the trace of the scattered field generated by an impulsive source localized just below the surface. From this relation it is, for example, clear that the eigenfrequencies can be recovered from the autocorrelation. Moreover, the complete scattering operator can be extracted under the additional assumption that the annulus is close to the surface and has a thickness smaller than the typical wavelength.

math.AP↗

arXiv2016

Focusing Waves Through a Randomly Scattering Medium in the White-Noise Paraxial Regime

When waves propagate through a complex or heterogeneous medium the wave field is corrupted by the heterogeneities. Such corruption limits the performance of imaging or communication schemes. One may then ask the question: is there an optimal way of encoding a signal so as to counteract the corruption by the medium? In the ideal situation the answer is given by time reversal: for a given target or focusing point, in a first step let the target emit a signal and then record the signal transmitted to the source antenna, time reverse this and use it as the source trace at the source antenna in a second step. This source will give a sharply focused wave at the target location if the source aperture is large enough. Here we address this scheme in the more practical situation with a limited aperture, time-harmonic signal, and finite-sized elements in the source array. Central questions are then the focusing resolution and signal-to-noise ratio at the target, their dependence on the physical parameters, and the capacity to focus selectively in the neighborhood of the target point and therefore to transmit images. Sharp results are presented for these questions.

math.AP↗