Workshop Schedule

Speaker: Rafael Villanueva Micó (IMM- Universitat Politècnica de València- IMM)

(joint with Juan Carlos Cortés, Francesca González-Marcet )

Abstract:

Many real-world phenomena, from the spread of infectious diseases to engineering processes, cannot be described precisely because both the data and the models involve uncertainty. One way to represent this uncertainty is through random differential equations, in which some system parameters are treated as random variables. However, a key challenge lies in identifying the parameter values that allow the model to adequately capture the observed data. This calibration problem has received little attention so far, and the available techniques often rely on very restrictive assumptions about the underlying distributions. In this work, we present a new method to calibrate random differential equation models, offering greater flexibility and applicability across different contexts. We illustrate its usefulness with a concrete example: a model of influenza transmission dynamics in the Valencian Community .

Speaker: Carlos Mora (Universidad de Concepción)

(Joint with J. Carlos Jimenez, Monica Selva)

Abstract:

The talk addresses the numerical solution of stochastic differential equations driven by independent Brownian motions. We present a general method for constructing variable step-size weak schemes for stochastic differential equations, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. Using the new method we design a variable step-size Euler scheme, and a variable step-size second order weak scheme. We present numerical experiments illustrating the satisfactory performance of the new adaptive schemes.

Speaker: Nuno Crokidakis (Universidade Federal Fluminense)

Abstract:

This talk explores the dynamics of papal conclaves through the lens of agent-based modeling. We develop two computational models to investigate how social influence, strategic voting, and ideological alignment affect the time and outcome of the election of a new pope. In the first model, cardinals interact through imitation and responsiveness to the leading candidate, while also engaging in useful voting by abandoning hopeless candidates. The second model incorporates ideological blocs, conservatives and progressives, based on realistic distributions from recent conclaves. We simulate both scenarios and show that ideological polarization can delay consensus, especially when strategic flexibility is limited. Our results reproduce the short duration of the 2025 conclave and align well with historical data from earlier conclaves, suggesting that informal consensus-building, possibly even before the voting begins, plays a crucial role. This study offers insights into the dynamics of collective decision-making under strict consensus rules, with broader implications for political and organizational behavior.

Speaker: Mario Muñoz (FGV-Escola de Matemática Aplicada)

(joint with Hugo de la Cruz)

Abstract:

We propose a numerical approach for estimating the stationary distribution of ergodic stochastic systems. The method relies on simulating a long-horizon sample path and approximating the first moments of each component of the system. Specifically, by combining sample-path averages with numerical integration techniques such as the trapezoidal rule and employing extrapolation procedures, we obtain accurate approximations of the ergodic limits of the first N moments. These estimated moments are then used to reconstruct the marginal distributions of the stationary law via the maximum entropy principle. The methodology is flexible and requires only a single sample path, making it suitable for a wide class of systems where explicit stationary measures are unavailable. As an application, we investigate stochastic predator-prey models with different functional responses and analyze how the choice of functional response affects the marginals of the stationary distribution.

Speaker: Daniel Barci (Universidade do Estado do Rio de Janeiro)

Abstract:

We investigate the collective dynamics of two-dimensional interacting Brownian particles in the hydrodynamic regime. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we construct a generating functional for correlation functions. In the continuum limit, we identify an exact symmetry under area-preserving diffeomorphisms, which defines a liquid state and enforces the conservation of local vorticity. Evaluating the generating functional within the saddle-point approximation supplemented by Gaussian fluctuations, we demonstrate the emergence of an effective U(1) gauge symmetry, enabling the description of density fluctuations within a gauge-theoretical framework. We explicitly solve the resulting equations of motion for a repulsive soft-core pair potential and uncover a competition between stochastic noise and particle interactions. This interplay gives rise to multiple dynamical regimes and associated non-equilibrium phase transitions, which manifest as distinct pattern formation processes at the critical points.

Speaker:Ana Navarro (Universitat Politècnica de València- IMM)

(Joint with C. Andreu-Vilarroig, J.-C. Cortés, Š. Papáček, C.-L. Pérez)

Abstract:

The three-state photosynthetic model is widely applied to describe microalgal growth in photobioreactors, where cells are constantly cycling between illuminated and shaded regions. Experimental observations reveal that hydrodynamic mixing generates irregular, non-uniform intervals between successive light-dark transitions. To account for this feature, we reformulate the deterministic framework into a random differential equation, treating the switching period as a positive random variable. From this formulation, we derive closed-form expressions for the long-term mean and variance, showing that the averaged behavior of the stochastic model departs from the quasi-steady trajectory predicted by the periodic deterministic case. Monte Carlo simulations are then employed to investigate how both the distribution of switching times and the fraction of time spent in darkness affect productivity. This type of stochastic modeling can provide deeper insights into how mixing-induced intermittent lighting affects microalgae growth, offering valuable guidance for designing more energy-efficient photobioreactors in industrial settings.

Speaker: Carol Santos Almonte (INSA-Rouen, France)

Abstract:

(joint with O. Sanchez Gimenez, E. Souza de Cursi, E. Pagnacco)

In the framework of Bayesian methods, many works attempt to develop objective methods for the generation of priors, such as Jeffrey's priors or Reference priors. In this line of research, a possible approach consists in using available data to generate a prior - such an approach can be effective including for limited data if Uncertainty Quantification (UQ) Hilbert Expansions (HE) are introduced. In this work, we present the use of UQ-HE in Bayesian optimization: HE allows the generation of probability distributions from limited numerical data, which are considered as a prior, and combined to an acquisition function to guide the selection of new candidate points for the optimization procedure. Since UQ-HE allows the use of expansions involving almost arbitrary random variables, we examine the effects of the choice of the distribution used in HE, and we compare their performance, including with classical Bayesian optimization. The results show that the approach is effective to calculate, and competitive, with significant improvements in some classes of problems. An application to a structural shape optimization problem is made to illustrate the performance in real-world problems

Speaker: Clara Burgos (Universitat Politècnica de València- IMM)

(joint with A. Navarro-Quiles, A. Sánchez-Sánchez. L. Villafuerte)

Abstract:

In this work we study the following linear fractional differential equation described by impulsive inputs and random parameters:

$$\begin{equation}\label{eq:Model} \left.\begin{array}{rcl} \left(^CD^{\alpha} X\right)(t)+AX(t)&=&\displaystyle \sum_{j=1}^M C_j \delta(t-t_j),\quad 0<\alpha< 1,\quad t>0, \\ X(t_0)&=&X_0. \end{array}\right\} \end{equation}$$

The term $\left(^CD^{\alpha} X\right)(t)$ represents the Caputo derivative of order $\alpha \in (0,1)$ and $\delta(t)$ is the Dirac delta function. These functions describe instantaneous jumps at prescribed time points. The initial condition, $X_0$, coefficient, $A$, and impulse magnitudes, $C_j$, are modeled as random variables defined on a common probability space. The main objective is to construct a mean-square (m.s.) convergent solution, compute its first and second moments and its first probability density function.

To this end, we will use the Laplace transform for second-order stochastic processes, under suitable conditions that ensure its existence. Our analysis provides a mathematically rigorous framework for understanding the behavior of fractional systems with impulsive and random features. The results contribute to the growing theory of random fractional differential equations and extend classical techniques to encompass impulsive effects and randomness simultaneously.

Speaker: Pablo de Maio (FGV-Escola de Matemática Aplicada)

(joint with Hugo de la Cruz and J. C. Jimenez)

Abstract:

This work presents an adaptive method for stochastic differential equations (SDEs) with additive noise. We introduce techniques for embedding exponential-based schemes, leading to an embedded A-stable explicit numerical method. In addition, we propose optimization strategies for the Padé algorithm to efficiently compute the matrix exponentials required by these methods. A comprehensive framework for adaptive time-stepping is also developed, providing a strategy that effectively handles SDEs with varying noise levels throughout the integration interval. The resulting methods yield explicit, numerically stable adaptive integrators that achieve the same level of accuracy as conventional explicit unstable schemes, while being particularly efficient for stiff SDEs, offering substantially reduced computational cost compared to their unstable counterparts.

Speaker: Zochil González (Universidade do Estado do Rio de Janeiro)

Abstract:

Stochastic differential equations with multiplicative noise are essential for modeling nonlinear systems with multiple stable states, where noise can induce switching and resonance phenomena. In this work, we study how Kramers' escape rate is modified when the dynamics is driven by a general multiplicative white noise. Using the generalized Stratonovich prescription, which includes Itô and Stratonovich calculus, we derive corrections to the escape rate arising from the diffusion function for arbitrary values of the stochastic prescription. Our analysis, carried out for a bistable system, combines a path-integral approach to overdamped Langevin dynamics with numerical simulations in the weak-noise regime. The excellent agreement between theory and simulations highlights the impact of stochastic prescriptions on escape processes and provides further insights into stochastic resonance.

Speaker: Americo Barbosa da Cunha Junior (Laboratório Nacional de Computação Cientifica e UERJ)

Abstract:

The Cross-Entropy (CE) method, a robust stochastic optimization technique, has garnered attention for its efficacy in addressing complex optimization problems across various disciplines. Originating from the field of rare event simulation, the CE method has evolved into a versatile tool for combinatorial optimization, continuous optimization, and machine learning tasks. Its core strategy involves generating sample solutions and iteratively refining probability distributions to hone in on the optimal regions of the solution space. This seminar introduces the CE method and its practical implementation through the CEopt code, a MATLAB/Python-based framework designed to simplify the application of CE techniques. The CEopt code encapsulates the method's adaptability and effectiveness, featuring support for both constrained and unconstrained optimization problems. Its modular architecture, equipped with input validation, adaptive sampling mechanisms, and dynamic parameter adjustment, enables users to tackle a wide array of optimization challenges without deep dives into algorithmic intricacies. Participants from machine learning and computational mechanics backgrounds will find particular interest in how the CEopt code can integrate into their workflows to optimize performance metrics and system designs. This seminar will cover the theoretical foundations, practical considerations, and potential applications of the CE method, demonstrating its utility with real-world examples and discussing future directions in optimization technology.

Speaker: Francisco Navarro (Universitat d'Alacant)

(joint with Y. Villacampa, P. Matos Castañeda)

Abstract:

The construction of civil engineering projects using reinforced concrete gives structures built with this material the ability to withstand heavy loads. The stiffness of the material is an essential parameter in the physical characteristics of the structure. However, there are a number of factors, such as the passage of time, environmental factors, and loads supported, which can affect the stiffness value and, therefore, the performance of the associated structures. There are different ways to determine the stiffness in reinforced concrete constructions, the most interesting being non-destructive methods, such as the study of displacements or deformations and natural frequencies, without compromising the integrity of the structure. However, the existence of uncertainties in the measurements of the dynamic properties of the material may imply a lack of certainty in the results for the stiffness and behaviour of the structure under real conditions. This work develops a model of the temporal evolution of a structure's stiffness by studying the evolution of the uncertainty associated with physical measurements taken on the structure at specific times. To achieve this, the Random Variable Transform (RVT) technique and structural modelling using Kirchhoff's equation and the Navier method are used.

Speaker: Jairon Batista (FGV-Escola de Matemática Aplicada)

Abstract:

Sampling from complex distributions with intractable normalizing constants remains a central challenge in Bayesian statistics and computational methods. Recent advances in generative modeling - particularly diffusion-based approaches grounded in stochastic differential equations (SDEs) - have shown remarkable success in handling high-dimensional, highly structured data. Motivated by these developments, we explore how the diffusion framework of Song et al. (2021), originally proposed for image generation, can be reinterpreted for sampling in a statistical context.

Our key contribution is a reformulation of this framework in terms of coupled Forward-Backward Stochastic Differential Equations (FBSDEs). This perspective removes the need for explicit score estimation, a critical bottleneck in diffusion models, while retaining their flexibility. We propose a deep-learning-based numerical solver tailored to this setting.

Preliminary results in low-dimensional examples demonstrate the potential of this approach, suggesting that FBSDE-driven diffusion sampling could provide a new pathway for Bayesian computation in scenarios where traditional MCMC methods struggle.