Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

We establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form $$(-\Delta )^{\beta /2}(\log u)\le C/t$$ ( - Δ ) β / 2 ( log u ) ≤ C / t , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) . We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

A note on the high-dimensional sparse Fourier transform in the continuous setting

In this paper, we theoretically propose a new hashing scheme to establish the sparse Fourier transform in high-dimension space. The estimation of the algorithm complexity shows that this sparse Fourier transform can overcome the curse of dimensionality. To the best of our knowledge, this is the first polynomial-time algorithm to recover the high-dimensional continuous frequencies.

Overhang control in topology optimization: a comparison of continuous front propagation-based and discrete layer-by-layer overhang control

Although additive manufacturing (AM) allows for a large design freedom, there are some manufacturing limitations that have to be taken into consideration. One of the most restricting design rules is the minimum allowable overhang angle. To make topology optimization suitable for AM, several algorithms have been published to enforce a minimum overhang angle. In this work, the layer-by-layer overhang filter proposed by Langelaar (Struct Multidiscip Optim 55(3):871–883, 2017), and the continuous, front propagation-based, overhang filter proposed by van de Ven et al. (Struct Multidiscipl Optim 57(5):2075–2091, 2018) are compared in detail. First, it is shown that the discrete layer-by-layer filter can be formulated in a continuous setting using front propagation. Then, a comparison is made in which the advantages and disadvantages of both methods are highlighted. Finally, the continuous overhang filter is improved by incorporating complementary aspects of the layer-by-layer filter: continuation of the overhang filter and a parameter that had to be user-defined are no longer required. An implementation of the improved continuous overhang filter is provided.

Continuous frames for unbounded operators

Few years ago Găvruţa gave the notions of K-frame and atomic system for a linear bounded operator K in a Hilbert space $$\mathcal {H}$$ H in order to decompose $$\mathcal {R}(K)$$ R ( K ) , the range of K, with a frame-like expansion. These notions are here generalized to the case of a densely defined and possibly unbounded operator A on a Hilbert space in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.

Non-testability of instrument validity under continuous treatments

This note presents a proof of the conjecture in Pearl (1995) about testing the validity of an instrumental variable in hidden variable models. It implies that instrument validity cannot be tested in the case where the endogenous treatment is continuously distributed. This stands in contrast to the classical testability results for instrument validity when the treatment is discrete. However, imposing weak structural assumptions on the model, such as continuity between the observable variables, can re-establish theoretical testability in the continuous setting.

The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures

Effective Hamiltonians arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry–Mather theory. In Aubry–Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

Multiplicative Decomposition of Heterogeneity in Mixtures of Continuous Distributions

A system’s heterogeneity (diversity) is the effective size of its event space, and can be quantified using the Rényi family of indices (also known as Hill numbers in ecology or Hannah–Kay indices in economics), which are indexed by an elasticity parameter q≥0. Under these indices, the heterogeneity of a composite system (the γ-heterogeneity) is decomposable into heterogeneity arising from variation within and between component subsystems (the α- and β-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that γ≥α. There exists a multiplicative decomposition for Rényi heterogeneity of composite systems with discrete event spaces, but less attention has been paid to decomposition in the continuous setting. We therefore describe multiplicative decomposition of the Rényi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. Under non-parametric pooling, the γ-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that γ≥α for q>0. Conversely, under parametric pooling, γ-heterogeneity can be computed efficiently in closed-form, but the γ≥α condition holds reliably only at q=1. Our findings will further contribute to heterogeneity measurement in continuous systems.

A regularized logistic regression based model for supervised learning

In this work, we introduce a new regularized logistic model for the supervised classification problem. Current logistic models have become the preferred tools for supervised classification in many situations. They mostly use either L1 or L2 regularization of the weight vector of parameters. Here we take a different approach by applying regularization not to the weight vector but to the gradient vector of the function representing the separating hyper-surface. We present the mathematical analysis of the model in its continuous setting and provide experimental evidence to show that the new model is competitive with state of the art models.

Volume Growth, Curvature, and Buser-Type Inequalities in Graphs

We study the volume growth of metric balls as a function of the radius in discrete spaces and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the 1st author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, $\lambda _2$ of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the 2nd eigenvalue (i.e. the 1st nonzero eigenvalue). We also describe a method for proving Buser’s Inequality in graphs, particularly under a lower bound assumption on curvature.