arXiv subjects

André Schlichting

Publications and source records attributed to André Schlichting.

35 records · Page 2Linked to original sources

arXiv2021

Ergodicity of the infinite swapping algorithm at low temperature

Sampling Gibbs measures at low temperatures is an important task but computationally challenging. Numerical evidence suggests that the infinite-swapping algorithm (isa) is a promising method. The isa can be seen as an improvement of the replica methods. We rigorously analyze the ergodic properties of the isa in the low temperature regime, deducing an Eyring-Kramers formula for the spectral gap (or Poincaré constant) and an estimate for the log-Sobolev constant. Our main results indicate that the effective energy barrier can be reduced drastically using the isa compared to the classical overdamped Langevin dynamics. As a corollary, we derive a deviation inequality showing that sampling is also improved by an exponential factor. Finally, we study simulated annealing for the isa and prove that the isa again outperforms the overdamped Langevin dynamics.

math.PR↗

arXiv2021

Optimal stability estimates and a new uniqueness result for advection-diffusion equations

This paper contains two main contributions. First, it provides optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial variable. This estimate is formulated with the help of Kantorovich--Rubinstein distances with logarithmic cost functions. Second, the stability estimates are extended to the advection-diffusion equations with velocity fields whose gradients are singular integrals of $L^1$ functions entailing a new well-posedness result.

math.AP↗

arXiv2021

The Scharfetter--Gummel scheme for aggregation-diffusion equations

In this paper, we propose a finite-volume scheme for aggregation-diffusion equations based on a Scharfetter--Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzed concerning well-posedness and convergence towards solutions to the continuous problem. Also, it is proven that the numerical scheme has several structure-preserving features. More specifically, it is shown that the discrete solutions satisfy a free-energy dissipation relation analogous to the continuous model. Consequently, the numerical solutions converge in the large time limit to stationary solutions, for which we provide a thermodynamic characterization. Numerical experiments complement the study.

math.NA↗

arXiv2021

Self-similar behavior of the exchange-driven growth model with product kernel

We study the self-similar behavior of the exchange-driven growth model, which describes a process in which pairs of clusters, consisting of an integer number of monomers, interact through the exchange of a single monomer. The rate of exchange is given by an interaction kernel $K(k,l)$ which depends on the sizes $k$ and $l$ of the two interacting clusters and is assumed to be of product form $(k\,l)^λ$ for $λ\in [0,2)$. We rigorously establish the coarsening rates and convergence to the self-similar profile found by Ben-Naim and Krapivsky [7]. For the explicit kernel, the evolution is linked to a discrete weighted heat equation on the positive integers by a nonlinear time-change. For this equation, we establish a new weighted Nash inequality that yields scaling-invariant decay and continuity estimates. Together with a replacement identity that links the discrete operator to its continuous analog, we derive a discrete-to-continuum scaling limit for the weighted heat equation. Reverting the time-change under the use of additional moment estimates, the analysis of the linear equation yields coarsening rates and self-similar convergence of the exchange-driven growth model.

math.AP↗

arXiv2021

Barriers of the McKean--Vlasov energy via a mountain pass theorem in the space of probability measures

We show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean--Vlasov free energy, which, for suitable attractive interaction potentials, has at least two distinct global minimisers at the critical parameter value $β=β_c$. On the torus, one of these states is the spatially homogeneous constant state, and the other is a clustered state. We show that a third critical point exists at this value. As a result, we obtain that the probability of transition of the empirical process from the constant state scales like $\exp(-N Δ)$, with $Δ$ the energy gap at $β=β_c$. The proof is based on a version of the mountain pass theorem for lower semicontinuous and $λ$-geodesically convex functionals on the space of probability measures $\mathcal{P}_2(M)$ equipped with the $2$-Wasserstein metric, where $M$ is a complete, connected, and smooth Riemannian manifold.

math.AP↗

arXiv2020

Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou-Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of "vertices" is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$^2$IE). We develop the existence theory for the solutions of the NL$^2$IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$^2$IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

math.AP↗

arXiv2019

The exchange-driven growth model: basic properties and longtime behavior

The exchange-driven growth model describes a process in which pairs of clusters interact through the exchange of single monomers. The rate of exchange is given by an interaction kernel $K$ which depends on the size of the two interacting clusters. Well-posedness of the model is established for kernels growing at most linearly and arbitrary initial data. The longtime behavior is established under a detailed balance condition on the kernel. The total mass density $\varrho$, determined by the initial data, acts as an order parameter, in which the system shows a phase transition. There is a critical value $\varrho_c\in (0,\infty]$ characterized by the rate kernel. For $\varrho \leq \varrho_c$, there exists a unique equilibrium state $ω^\varrho$ and the solution converges strongly to $ω^\varrho$. If $\varrho > \varrho_c$ the solution converges only weakly to $ω^{\varrho_c}$. In particular, the excess $\varrho - \varrho_c$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the Becker-Döring equation. The main ingredient for the longtime behavior is the free energy acting as Lyapunov function for the evolution. It is also the driving functional for a gradient flow structure of the system under the detailed balance condition.

math-ph↗

arXiv2019

Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces

We consider non-linear evolution equations arising from mean-field limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics based on convexity of the free energy along interpolations in a discrete transportation distance related to the gradient flow structure of the dynamics. This notion extends the one for linear Markov chain dynamics studied by Erbar and Maas. We show that positive curvature bounds entail several functional inequalities controlling the convergence to equilibrium of the dynamics. We establish explicit curvature bounds for several examples of mean-field limits of various classical models from statistical mechanics.

math.PR↗

arXiv2019

Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities

We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz'ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new definition in the context of the random field Curie-Weiss model, where metastability and the additional regularity assumptions are verifiable.

math.PR↗

arXiv2018

Poincaré and log-Sobolev inequalities for mixtures

This work studies mixtures of probability measures on $\mathbb{R}^n$ and gives bounds on the Poincaré and the log-Sobolev constant of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the $χ^2$-distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter whereas the log-Sobolev may blow up as the mixture ratio goes to $0$ or $1$. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

math.PR↗

arXiv2018

A non-local problem for the Fokker-Planck equation related to the Becker-Döring model

This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield Hölder regularity of the non-local order parameter, which is improved by an iteration argument. The asymptotic behavior of solutions depends on some order parameter $ρ$ depending on the initial data. The system shows different behavior depending on a value $ρ_s>0$, determined from the potentials and diffusion coefficient. For $ρ\leq ρ_s$, there exists an equilibrium solution $c^{\text{eq}}_{(ρ)}$. If $ρ\leρ_s$ the solution converges strongly to $c^{\text{eq}}_{(ρ)}$, while if $ρ> ρ_s$ the solution converges weakly to $c^{\text{eq}}_{(ρ_s)}$. The excess $ρ- ρ_s$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-Döring equation. The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case $ρ<ρ_s$ the entropy method, based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates, is used to obtain explicit convergence rates to the equilibrium solution. The close connection of the presented model and the Becker-Döring model is outlined by a family of discrete Fokker-Planck type equations interpolating between both of them. This family of models possesses a gradient flow structure, emphasizing their commonality.

math.AP↗

arXiv2018

Macroscopic limit of the Becker-Döring equation via gradient flows

This work considers gradient structures for the Becker-Döring equation and its macroscopic limits. The result of Niethammer [17] is extended to prove the convergence not only for solutions of the Becker-Döring equation towards the Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the associated gradient structures. We establish the gradient structure of the nonlocal coarsening equation rigorously and show continuous dependence on the initial data within this framework. Further, on the considered time scale the small cluster distribution of the Becker--Döring equation follows a quasistationary distribution dictated by the monomer concentration.

math.AP↗

arXiv2017

Analysis of the implicit upwind finite volume scheme with rough coefficients

We study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation in the low regularity DiPerna-Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at least 1/2. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.

math.AP↗

arXiv2017

Convergence rates for upwind schemes with rough coefficients

This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for numerically solving continuity equations. We are interested in the case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable. We estimate the error between approximate solutions constructed by the upwind scheme and distributional solutions of the continuous problem in a Kantorovich-Rubinstein distance, which was recently used for stability estimates for the continuity equation by Seis [23]. Restricted to Cartesian meshes, our estimate shows that the rate of weak convergence is at least of order $1/2$ in the mesh size. The proof relies on a probabilistic interpretation of the upwind scheme Delarue and Lagoutière [9]. We complement the weak convergence result with an example that illustrates that for rough initial data no rates can be expected in strong norms. The same example suggests that the weak order $1/2$ rate is optimal.

math.AP↗

arXiv2016

Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation

We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this paper we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy ${\mathcal{F}}$ on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an explicit Euler scheme obtained by minimizing $h {\mathcal{F}}(\varrho)+W_2^2(\varrho^0,\varrho)$ among all $\varrho$ satisfying the constraint for some $\varrho^0$ and time-step $h>0$. Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities.

math.AP↗

arXiv2016

Gradient flow structure for McKean-Vlasov equations on discrete spaces

In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity.

math.PR↗

arXiv2014

Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian $H:\mathbb {R}^n\to \mathbb {R}$ in the regime of low temperature $\varepsilon$. We proof the Eyring-Kramers formula for the optimal constant in the Poincaré (PI) and logarithmic Sobolev inequality (LSI) for the associated generator $L=\varepsilon Δ-\nabla H\cdot\nabla$ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 302-351] and of the mean-difference estimate introduced by Chafaï and Malrieu [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 72-96]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to metastable regions corresponding to the local minima scales well in $\varepsilon$. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.

math.PR↗