Weekly Seminar

Abstract

This work proposes a new mathematical formulation for flow measurement based on the inverse source problem for wave equations with partial boundary measurement. Inspired by the design of acoustic Doppler current profilers (ADCPs), we formulate an inverse source problem that can recover the flow field from the observation data on boundary receivers. To our knowledge, this is the first mathematical model of flow measurement using partial differential equations. This model is proved well-posed, and the corresponding algorithm is derived to compute the velocity field efficiently. Extensive numerical simulations are performed to demonstrate the accuracy and robustness of our model. The comparison results demonstrate that our model is ten times more accurate than ADCP. Our formulation is capable of simulating a variety of practical measurement scenarios.

Abstract

Stochastic Galerkin formulations to random hyperbolic equations are based on the idea that the functional dependence on the stochastic input is described a priori by a polynomial chaos expansion and a Galerkin projection is used to obtain deterministic evolution equations for the coefficients in the series. Then, all envolved mathematical operations, e.g. products and norms, must be adopted and applied to the variables in the governing equations. In general, results for hyperbolic systems are not available, since desired properties like hyperbolicity and the existence of entropies are not transferred to the intrusive formulation. To this end, auxiliary variables have been introduced to establish wellposedness results. For instance, entropy-entropy flux pairs can be obtained by an expansion in entropy variables, i.e. the gradient of the deterministic entropy. Roe variables, which include the square root of the density, preserve hyperbolicity for Euler equations. The drawback of introducing auxiliary variables is an additional computational overhead that arises from an optimization problem, which is required to calculate the auxiliary variables. These results exploit quadratic relationships that are expressed efficiently by the Galerkin product. Extensions of the classical polynomial chaos expansions to general nonlinearities, e.g. with Legendre or Hermite polynomials, however, are not straightforward. More general nonlinearities, which occur for instance in isentropic Euler, level-set equations or when including source terms in shallow water equations, can be expressed by wavelet families that are generated by Haar-type matrices. Those are widely used in signal processing, e.g. for the Hadamard, Walsh, Chebyshev matrices and for the discrete cosine transform. In this talk we give an overview on methods to circumvent the loss of hyperbolicity and discuss briefly numerical challenges.

Abstract

Originally developed to model reaction-diffusion systems, Gradient Random Walk methods are particle-based techniques that track the evolution of spatial derivatives of a solution. Inspired by vortex methods for the Navier-Stokes equations, these approaches emerged in the 1990s and attracted significant interest due to their advantageous properties: (i) they eliminate the need for a computational grid, (ii) they naturally adapt to solution features by concentrating particles where gradients are steep, and (iii) they offer a notable reduction in variance.

In this seminar, we revisit and extend these ideas by demonstrating how the Gradient Random Walk framework can be adapted to a wider class of partial differential equations. To achieve this goal, we first extend the classical Monte Carlo method to the relaxation approximation of systems of conservation laws, and subsequently introduce a novel particle dynamics approach based on the spatial derivatives of the solution. This methodology, when combined with an asymptotic-preserving splitting discretization, enables the development of a new class of gradient-based Monte Carlo methods tailored for hyperbolic systems of conservation laws.

Abstract: The mathematical modeling of various real life phenomena involving the interplay of formation and consumption processes leads to non-linear Production--Destruction Systems (PDS) of the form y'(t) = (P(y(t)) - D(y(t))) e, where P,D:R^N \to \mathbb{R}^{N\times N}, \qquad \bm{e}=[1,\dots,1]^\mathsf{T}\in \mathbb{R}^N. In many practical applications, the underlying physics naturally yields PDS that are both positive and fully conservative for which $\Omega=\{ [x_1,\dots,x_N]^\mathsf{T} : 0 \leq x_i \leq \bm{e}^\mathsf{T}\bm{y}(t_0),  \ i=1,\dots,N \} $ is a positively-invariant set and the conservation law $\bm{e}^\mathsf{T}(\bm{y}(t)-\bm{y}(t_0))=0, \ \forall t\geq t_0$ holds true. As a result, there is an increasing demand for numerical integrators that retain both the positivity of the solution and the system's linear invariant with no limitations on the discretization steplength. In this context, \textit{Modified Patankar} (MP) methods have proven particularly beneficial (see \cite{Buchard,Huang2019,Izgin2024,Offner} and references therein). The purpose of this contribution is twofold. First, we present the results of \cite{Izzo2025}, where a novel class of $k$-step, unconditionally positive and conservative \textit{Modified Patankar Linear Multistep} (MPLM) methods is introduced. A thorough theoretical analysis is conducted, with emphasis placed upon the conditions on the \textit{Patankar Weight Denominators} (PWDs) required to achieve arbitrarily high orders of convergence. Additionally, the \(\sigma\)-embedding technique is proposed as a practical tool for the effective computation of PWDs. A comparison with other well-established MP discretizations reveals the competitiveness of the proposed methods for the simulation of PDS. \\ The second part of the present work explores the findings of \cite{MPSL}, where the framework of a \textit{Controlled PDS} (CPDS) \bm{y}'(t) = \bigl(P(\bm{y}(t))\odot \mathcal{P}(\bm{\alpha}(t)) - D(\bm{y}(t))\odot \mathcal{D}(\bm{\alpha}(t))\bigr)\bm{e},\quad \bm{\alpha}: \mathbb{R}^+ \rightarrow A\subset\mathbb{R}^M, \; \qquad \; \mathcal{P},\mathcal{D}: A \rightarrow \mathbb{R}^{N\times N} is outlined with the aim of influencing the dynamics of a PDS without altering its inherent properties. A general finite horizon \textit{Optimal Control Problem} (OCP) is formulated and addressed via the dynamic programming approach. Specifically, the OCP is restated in terms of a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation, whose unique viscosity solution corresponds to the value function \cite{Crandall}. We then propose a parallel-in-space \textit{Modified Patankar Semi-Lagrangian} (MPSL) approximation scheme for the HJB equation \cite{Falcone} and design an uncondionally positive and conservative reconstruction procedure for the optimal control in feedback and open-loop forms. The application to two case studies, specifically enzyme catalyzed biochemical reactions and infectious diseases, highlights the advantages of the proposed methodology over classical semi-Lagrangian discretizations.