اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExact targeting of Gibbs distributions using velocity-jump processes
80
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This work introduces and studies a new family of velocity jump Markov processes directly amenable to exact simulation with the following two properties: i) trajectories converge in law when a time-step parameter vanishes towards a given Langevin or Hamil-tonian dynamics; ii) the stationary distribution of the process is always exactly given by the product of a Gaussian (for velocities) by any target log-density whose gradient is pointwise computabe together with some additional explicit appropriate upper bound. The process does not exhibit any velocity reflections (jump sizes can be controlled) and is suitable for the factorization method. We provide a rigorous mathematical proof of: i) the small time-step convergence towards Hamiltonian/Langevin dynamics, as well as ii) the exponentially fast convergence towards the target distribution when suitable noise on velocity is present. Numerical implementation is detailed and illustrated.
قيم البحث
اقرأ أيضاً
We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $pi$ on R d , based on (over-damped) Langevin diffusions. This method both inspired by [PP18] and [GMS + 20] relies on a multilevel occupation measur
e, i.e. on an appropriate combination of R occupation measures of (constant-step) discretized schemes of the Langevin diffusion with respective steps $gamma$r = $gamma$02 --r , r = 0,. .. , R. For a given diffusion, we first state a result under general assumptions which guarantees an $epsilon$-approximation (in a L 2-sense) with a cost proportional to $epsilon$ --2 (i.e. proportional to a Monte-Carlo method without bias) or $epsilon$ --2 | log $epsilon$| 3 under less contractive assumptions. This general result is then applied to over-damped Langevin diffusions in a strongly convex setting, with a study of the dependence in the dimension d and in the spectrum of the Hessian matrix D 2 U of the potential U : R d $rightarrow$ R involved in the Gibbs distribution. This leads to strategies with cost in O(d$epsilon$ --2 log 3 (d$epsilon$ --2)) and in O(d$epsilon$ --2) under an additional condition on the third derivatives of U. In particular, in our last main result, we show that, up to universal constants, an appropriate choice of the diffusion coefficient and of the parameters of the procedure leads to a cost controlled by ($lambda$ U $lor$1) 2 $lambda$ 3 U d$epsilon$ --2 (where$lambda$U and $lambda$ U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D 2 U). In our numerical illustrations, we show that our theoretical bounds are confirmed in practice and finally propose an opening to some theoretical or numerical strategies in order to increase the robustness of the procedure when the largest and smallest eigenvalues of D 2 U are respectively too large or too small.
We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals
and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted $ell^1$ minimization (sparse signals).
What are the face-probabilities of a cuboidal die, i.e. a die with different side-lengths? This paper introduces a model for these probabilities based on a Gibbs distribution. Experimental data produced in this work and drawn from the literature supp
ort the Gibbs model. The experiments also reveal that the physical conditions, such as the quality of the surface onto which the dice are dropped, can affect the face-probabilities. In the Gibbs model, those variations are condensed in a single parameter, adjustable to the physical conditions.
Almost all materials are anisotropic. In this paper, interface relations of anisotropic elliptic partial differential equations involving discontinuities across interfaces are derived in two and three dimensions. Compared with isotropic cases, the in
variance of partial differential equations and the jump conditions under orthogonal coordinates transformation is not valid anymore. A systematic approach to derive the interface relations is established in this paper for anisotropic elliptic interface problems, which can be important for deriving high order accurate numerical methods.
We revisit the traditional upwind schemes for linear conservation laws in the viewpoint of jump processes, allowing studying upwind schemes using probabilistic tools. In particular, for Fokker-Planck equations on $mathbb{R}$, in the case of weak conf
inement, we show that the solution of upwind scheme converges to a stationary solution. In the case of strong confinement, using a discrete Poincare inequality, we prove that the $O(h)$ numeric error under $ell^1$ norm is uniform in time, and establish the uniform exponential convergence to the steady states. Compared with the traditional results of exponential convergence of upwind schemes, our result is in the whole space without boundary. We also establish similar results on torus for which the stationary solution of the scheme does not have detailed balance. This work shows an interesting connection between standard numerical methods and time continuous Markov chains, and could motivate better understanding of numerical analysis for conservation laws.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
