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Weak Error of Dean-Kawasaki Equation with Smooth Mean-Field Interactions
No full textWe consider the weak-error rate of the SPDE approximation by regularized Dean-Kawasaki equation with Itô noise for particle systems with mean-field interactions both on the drift and the noise. The global existence and uniqueness of the corresponding SPDEs are established using the variational approach to SPDEs, and the weak-error rate is estimated using the technique of Kolmogorov equations on the space of probability measures. In particular, the rate derived in this paper coincides with that is the previous work arXiv:2212.11714, which considered free Brownian particles using Laplace duality
Solution concepts for a model of visco-elasto-plasticity with slight compressibility
No full textWe study a model for the deformation of a visco-elasto-plastic material that is nearly incompressible. It originates from geophysics, is given in the Eulerian description and combines a Kelvin-Voigt rheology in the spherical part with a Jeffreys-type rheology in the deviatoric part. Despite a constant density, the model allows for non-isochoric deformation and the propagation of pressure waves. An additive decomposition of the strain rate into elastic and inelastic parts leads to an evolution equation for the small elastic strain, which is coupled with an adapted momentum equation. As plasticity is modeled through a non-smooth dissipation potential, we introduce a weak formulation in terms of a variational inequality. Since the well-posedness in such a weak setting is out of reach, we study two possible modifications: the regularization in terms of stress diffusion, and the relaxation of the solvability concept by transition to energy-variational solutions. In both cases, solutions are constructed by the same time-discrete scheme, consisting of solving a saddle-point problem in each time step
Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator
No full textWe study a two-variable dynamical system modeling pH oscillations in the urea-urease reaction within giant lipid vesicles -- a problem that intrinsically contains multiple, well-separated timescales. Building on an existing, deterministic formulation via ordinary differential equations, we resolve different orders of magnitude within a small parameter and analyze the system's limit cycle behavior using geometric singular perturbation theory (GSPT). By introducing two different coordinate scalings -- each valid in a distinct region of the phase space -- we resolve the local dynamics near critical fold points, using the extension of GSPT through such singular points due to Krupa and Szmolyan. This framework enables a geometric decomposition of the periodic orbits into slow and fast segments and yields closed-form estimates for the period of oscillation. In particular, we link the existence of such oscillations to an underlying biochemical asymmetry, namely, the differential transport across the vesicle membrane
Approximation of time-periodic flow past a translating body by flows in bounded domains
No full textWe consider a time-periodic incompressible three-dimensional Navier-Stokes flow past a translating rigid body. In the first part of the paper, we establish the existence and uniqueness of strong solutions in the exterior domain
Ω ⊂ R3 that satisfy pointwise estimates for both the velocity and pressure. The fundamental solution of the time-periodic Oseen equations plays a central role in obtaining these estimates. The second part focuses on approximating
this exterior flow within truncated domains Ω∩BR, incorporating appropriate artificial boundary conditions on ∂BR. For these bounded domain problems, we prove the existence and uniqueness of weak solutions. Finally, we estimate the error in the velocity component as a function of the truncation radius R, showing that, as R → ∞, the velocities of the truncated problems converge, in an appropriate norm, to the velocity of the exterior flow
TetRex: a novel algorithm for index-accelerated search of highly conserved motifs
No full textThe scale of modern datasets has necessitated innovations to solve even the most traditional and fundamental of computational problems. Set membership and set cardinality are both examples of simple queries that, for large enough datasets, quickly become challenging using traditional approaches. Interestingly, we find a need for these innovations within the field of biology. Despite the vast differences among living organisms, there exist functions so critical to life that they are conserved in the genomes and proteomes across seemingly unrelated species. Regular expressions (regexes) can serve as a convenient way to represent these conserved sequences compactly. However, despite the strong theoretical foundation and maturity of tools available, the state-of-the-art regex search falls short of what is necessary to quickly scan an enormous database of biological sequences. In this work, we describe a novel algorithm for regex search that reduces the search space by leveraging a recently developed compact data structure for set membership, the hierarchical interleaved bloom filter. We show that the runtime of our method combined with a traditional search outperforms state-of-the-art tools
Asymptotic stability of the equilibrium for the free boundary problem of a compressible atmospheric primitive model with physical vacuum
No full textThis paper concerns the large time asymptotic behavior of solutions to the free boundary problem of the compressible primitive equations in atmospheric dynamics with physical vacuum. Up to second order of the perturbations of an equilibrium, we have introduced a model of the compressible primitive equations with a specific viscosity and shown that the physical vacuum free boundary problem for this model system has a global-in-time solution converging to an equilibrium exponentially, provided that the initial data is a small perturbation of the equilibrium. More precisely, we introduce a new coordinate system by choosing the enthalpy (the square of sound speed) as the vertical coordinate, and thanks to the hydrostatic balance, the degenerate density at the free boundary admits a representation with separation of variables in the new coordinates. Such a property allows us to establish horizontal derivative estimates without involving the singular vertical derivative of the density profile, which plays a key role in our analysis
A Hybrid ABM-PDE Framework for Real-World Infectious Disease Simulations
No full textThis paper presents a hybrid modeling approach that couples an Agent-Based Model (ABM) with a partial differential equation (PDE) model in an epidemic setting to simulate the spatial spread of infectious diseases using a compartmental structure with seven health states. The goal is to reduce the computational complexity of a full-ABM by introducing a coupled ABM-PDE model that offers significantly faster simulations while maintaining comparable accuracy. Our results demonstrate that the hybrid model not only reduces the overall simulation runtime (defined as the number of runs required for stable results multiplied by the duration of a single run) but also achieves smaller errors across both 25% and 100% population samples. The coupling mechanism ensures consistency at the model interface: agents crossing from the ABM into the PDE domain are removed and represented as density contributions at the corresponding grid node, while surplus density in the PDE domain is used to generate agents with plausible trajectories derived from mobile phone data. We evaluate the hybrid model using real-world mobility and infection data for the Berlin-Brandenburg region in Germany, showing that it captures the core epidemiological dynamics while enabling efficient large-scale simulations
Weak reservoirs are superexponentially irrelevant for misanthrope processes
No full textWe provide a short proof for the exponential equivalence between misanthrope processes in contact with weak reservoirs and those with impermeable boundaries. As a consequence, we can derive both the hydrodynamic limit and the large deviations of the totally asymmetric simple exclusion process (TASEP) in contact with weak reservoirs. This extends a recent result which proved the hydrodynamic behaviour of a vanishing viscocity approximation of the TASEP in contact with weak reservoirs. Further applications to a class of asymmetric exclusion processes with long jumps are
discussed
Coherent set identification via direct low rank maximum likelihood estimation
No full textWe analyze connections between two low rank modeling approaches from the last decade for treating dynamical data. The first one is the coherence problem (or coherent set approach), where groups of states are sought that evolve under the action of a stochastic transition matrix in a way maximally distinguishable from other groups. The second one is a low rank factorization approach for stochastic matrices, called Direct Bayesian Model Reduction (DBMR), which estimates the low rank factors directly from observed data. We show that DBMR results in a low rank model that is a projection of the full model, and exploit this insight to infer bounds on a quantitative measure of coherence within the reduced model. Both approaches can be formulated as optimization problems, and we also prove a bound between their respective objectives. On a broader scope, this work relates the two classical loss functions of nonnegative matrix factorization, namely the Frobenius norm and the generalized Kullback--Leibler divergence, and suggests new links between likelihood-based and projection-based estimation of probabilistic models
Reference Map Approach to Eulerian Thermomechanics Using GENERIC
No full textAn Eulerian GENERIC model for thermo-viscoelastic materials with diffusive components is derived based on a transformation framework that maps a Lagrangian formulation to corresponding Eulerian coordinates. The key quantity describing the deformation in Eulerian coordinates is the inverse of the deformation, i.e., the reference map. The Eulerian model is formally constructed, and by reducing the GENERIC system to a damped Hamiltonian system, the isothermal limit is derived. A structure-preserving weak formulation is developed. As an example, the coupling of finite strain viscoelasticity and diffusion in a multiphase system governed by Lagrangian indicator functions is demonstrated