arXiv subjects

Filip Tronarp

Publications and source records attributed to Filip Tronarp.

At least 19 recordsLinked to original sources

arXiv2026

Causal Spatio-Temporal Sound Field Reconstruction

In sound field control applications, it is commonly assumed that one has access to an accurate representation of the sound field in the region of interest. This is a problematic assumption since the reconstruction of a sound field from available microphone measurements is especially challenging in real-time applications where only causal measurements are available. Notably, causal time-windowed observations introduce correlation between frequency components, making sound field reconstruction methods that process each frequency band independently sub-optimal. In this work, we formulate a causal finite-window spatio-temporal linear minimum mean-square error estimator for sound field reconstruction. The sound field is modeled as the solution to the wave equation driven by a stationary stochastic spatio-temporal source distribution, which induces a physically interpretable covariance function. It is shown that this covariance function is closely related to the classical diffuse-field coherence model. Since the computational complexity grows rapidly with the number of spatio-temporal observations, we formulate a budget-constrained spatio-temporal sample selection approach to minimize the posterior reconstruction variance. The proposed estimator and sampling strategy are evaluated using both simulated and measured sound fields, demonstrating improved short-window reconstruction compared to frequency domain finite-window baselines.

eess.AS↗

arXiv2026

Barron-Wiener-Laguerre models

We propose a probabilistic extension of Wiener-Laguerre models for causal operator learning. Classical Wiener-Laguerre models parameterize stable linear dynamics using orthonormal Laguerre bases and apply a static nonlinear map to the resulting features. While structurally efficient and interpretable, they provide only deterministic point estimates. We reinterpret the nonlinear component through the lens of Barron function approximation, viewing two-layer networks, random Fourier features, and extreme learning machines as discretizations of integral representations over parameter measures. This perspective naturally admits Bayesian inference on the nonlinear map and yields posterior predictive uncertainty. By combining Laguerre-parameterized causal dynamics with probabilistic Barron-type nonlinear approximators, we obtain a structured yet expressive class of causal operators equipped with uncertainty quantification. The resulting framework bridges classical system identification and modern measure-based function approximation, providing a principled approach to time-series modeling and nonlinear systems identification.

stat.ME↗

arXiv2025

Recursive state estimation via approximate modal paths

In this paper, a method for recursively computing approximate modal paths is developed. A recursive formulation of the modal path can be obtained either by backward or forward dynamic programming. By combining both methods, a ``two-filter'' formula is demonstrated. Both method involves a recursion over a so-called value function, which is intractable in general. This problem is overcome by quadratic approximation of the value function in the forward dynamic programming paradigm, resulting in both a filtering and smoothing method. The merit of the approach is verified in a simulation experiments, where it is shown to be on par or better than other modern algorithms.

stat.ME↗

arXiv2025

A Recursive Theory of Variational State Estimation: The Dynamic Programming Approach

In this article, variational state estimation is examined from the dynamic programming perspective. This leads to two different value functional recursions depending on whether backward or forward dynamic programming is employed. The result is a theory of variational state estimation that corresponds to the classical theory of Bayesian state estimation. More specifically, in the backward method, the value functional corresponds to a likelihood that is upper bounded by the state likelihood from the Bayesian backward recursion. In the forward method, the value functional corresponds to an unnormalized density that is upper bounded by the unnormalized filtering density. Both methods can be combined to arrive at a variational two-filter formula. Additionally, it is noted that optimal variational filtering is generally of quadratic time-complexity in the sequence length. This motivates the notion of sub-optimal variational filtering, which also lower bounds the evidence but is of linear time-complexity. Another problem is the fact that the value functional recursions are generally intractable. This is briefly discussed and a simple approximation is suggested that retrieves the filter proposed by Courts et. al (2021).The methodology is examined in (i) a jump Gauss-Markov system under a certain factored Markov process approximation, and (ii) in a Gauss-Markov model with log-polynomial likelihoods under a Gauss--Markov constraint on the variational approximation. It is demonstrated that the value functional recursions are tractable in both cases. The resulting estimators are examined in simulation studies and are found to be of adequate quality in comparison to sensible baselines.

stat.ME↗

arXiv2025

Numerically robust Gaussian state estimation with singular observation noise

This article proposes numerically robust algorithms for Gaussian state estimation with singular observation noise. Our approach combines a series of basis changes with Bayes' rule, transforming the singular estimation problem into a nonsingular one with reduced state dimension. In addition to ensuring low runtime and numerical stability, our proposal facilitates marginal-likelihood computations and Gauss-Markov representations of the posterior process. We analyse the proposed method's computational savings and numerical robustness and validate our findings in a series of simulations.

stat.ME↗

arXiv2025

Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical applications, however, the underlying dynamical system often contains uncertain parameters, requiring the propagation of this model uncertainty to the ODE solution. In this paper, we demonstrate that ODE filters, despite their probabilistic nature, do not automatically solve this uncertainty propagation problem. To address this limitation, we present a novel approach that combines ODE filters with numerical quadrature to properly marginalize over uncertain parameters, while accounting for both parameter uncertainty and numerical solver uncertainty. Experiments across multiple dynamical systems demonstrate that the resulting uncertainty estimates closely match reference solutions. Notably, we show how the numerical uncertainty from the ODE solver can help prevent overconfidence in the propagated uncertainty estimates, especially when using larger step sizes. Our results illustrate that probabilistic numerical methods can effectively quantify both numerical and parametric uncertainty in dynamical systems.

stat.ML↗

arXiv2025

The two filter formula reconsidered: Smoothing in partially observed Gauss--Markov models without information parametrization

In this article, the two filter formula is re-examined in the setting of partially observed Gauss--Markov models. It is traditionally formulated as a filter running backward in time, where the Gaussian density is parametrized in ``information form''. However, the quantity in the backward recursion is strictly speaking not a distribution, but a likelihood. Taking this observation seriously, a recursion over log-quadratic likelihoods is formulated instead, which obviates the need for ``information'' parametrization. In particular, it greatly simplifies the square-root formulation of the algorithm. Furthermore, formulae are given for producing the forward Markov representation of the a posteriori distribution over paths from the proposed likelihood representation.

stat.ME↗

arXiv2024

Numerically robust square root implementations of statistical linear regression filters and smoothers

In this article, square-root formulations of the statistical linear regression filter and smoother are developed. Crucially, the method uses QR decompositions rather than Cholesky downdates. This makes the method inherently more numerically robust than the downdate based methods, which may fail in the face of rounding errors. This increased robustness is demonstrated in an ill-conditioned problem, where it is compared against a reference implementation in both double and single precision arithmetic. The new implementation is found to be more robust, when implemented in lower precision arithmetic as compared to the alternative.

stat.ME↗

arXiv2024

Probabilistic ODE Solvers for Integration Error-Aware Numerical Optimal Control

Appropriate time discretization is crucial for real-time applications of numerical optimal control, such as nonlinear model predictive control. However, if the discretization error strongly depends on the applied control input, meeting accuracy and sampling time requirements simultaneously can be challenging using classical discretization methods. In particular, neither fixed-grid nor adaptive-grid discretizations may be suitable, when they suffer from large integration error or exceed the prescribed sampling time, respectively. In this work, we take a first step at closing this gap by utilizing probabilistic numerical integrators to approximate the solution of the initial value problem, as well as the computational uncertainty associated with it, inside the optimal control problem (OCP). By taking the viewpoint of probabilistic numerics and propagating the numerical uncertainty in the cost, the OCP is reformulated such that the optimal input reduces the computational uncertainty insofar as it is beneficial for the control objective. The proposed approach is illustrated using a numerical example, and potential benefits and limitations are discussed.

math.OC↗

arXiv2023

Computing the matrix exponential and the Cholesky factor of a related finite horizon Gramian

In this article, an efficient numerical method for computing both the matrix exponential and a finite horizon controllability Gramian in Cholesky-factored form is proposed. The method is applicable to general dense matrices of moderate size and produces a Cholesky factor of the Gramian without computing the full product. It is a generalization of the scaling-and-squaring approach for approximating the matrix exponential and exploits a similar doubling formula for the Gramian to compute both at once. The required computational effort is thereby kept modest. Most importantly, a rigorous backward error analysis is provided, which guarantees that the approximation is accurate to the round-off error level in double precision. This accuracy is illustrated in practice on a large number of standard test examples. The method has been implemented in the Julia package FiniteHorizonGramians.jl, which is available online under the MIT license. Code for reproducing the experimental results is included in this package, as well as code for determining the optimal method parameters. The analysis can thus easily be adapted to a different finite-precision arithmetic.

math.NA↗

arXiv2023

Parallel-in-Time Probabilistic Numerical ODE Solvers

Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic flexibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this flexibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the span cost from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.

math.NA↗

arXiv2023

The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions

Inference and simulation in the context of high-dimensional dynamical systems remain computationally challenging problems. Some form of dimensionality reduction is required to make the problem tractable in general. In this paper, we propose a novel approximate Gaussian filtering and smoothing method which propagates low-rank approximations of the covariance matrices. This is accomplished by projecting the Lyapunov equations associated with the prediction step to a manifold of low-rank matrices, which are then solved by a recently developed, numerically stable, dynamical low-rank integrator. Meanwhile, the update steps are made tractable by noting that the covariance update only transforms the column space of the covariance matrix, which is low-rank by construction. The algorithm differentiates itself from existing ensemble-based approaches in that the low-rank approximations of the covariance matrices are deterministic, rather than stochastic. Crucially, this enables the method to reproduce the exact Kalman filter as the low-rank dimension approaches the true dimensionality of the problem. Our method reduces computational complexity from cubic (for the Kalman filter) to \emph{quadratic} in the state-space size in the worst-case, and can achieve \emph{linear} complexity if the state-space model satisfies certain criteria. Through a set of experiments in classical data-assimilation and spatio-temporal regression, we show that the proposed method consistently outperforms the ensemble-based methods in terms of error in the mean and covariance with respect to the exact Kalman filter. This comes at no additional cost in terms of asymptotic computational complexity.

stat.ML↗

arXiv2023

Probabilistic Exponential Integrators

Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator -- with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics.

math.NA↗

arXiv2022

Orthonormal Expansions for Translation-Invariant Kernels

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of $\mathscr{L}_2(\mathbb{R})$. This allows us to derive explicit expansions on the real line for (i) Mat\'ern kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

math.CA↗

arXiv2022

Fenrir: Physics-Enhanced Regression for Initial Value Problems

We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches.

cs.LG↗

arXiv2021

Pick-and-Mix Information Operators for Probabilistic ODE Solvers

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions.

stat.ML↗

arXiv2021

A probabilistic Taylor expansion with Gaussian processes

We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. We discuss and prove some results on maximum likelihood estimation of parameters of Taylor kernels. The proposed framework is a special case of Gaussian process regression based on data that is orthogonal in the reproducing kernel Hilbert space of the covariance kernel.

cs.LG↗

arXiv2020

Calibrated Adaptive Probabilistic ODE Solvers

Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver's step size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.

math.NA↗