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We study the problem of training diffusion models to sample from a distribution with a given unnormalized density or energy function. We benchmark several diffusion-structured inference methods, including simulation-based variational approaches and off-policy methods (continuous generative flow networks). Our results shed light on the relative advantages of existing algorithms while bringing into question some claims from past work. We also propose a novel exploration strategy for off-policy methods, based on local search in the target space with the use of a replay buffer, and show that it improves the quality of samples on a variety of target distributions. Our code for the sampling methods and benchmarks studied is made public at //github.com/GFNOrg/gfn-diffusion as a base for future work on diffusion models for amortized inference.

The functional logit regression model was proposed by Escabias et al. (2004) with the objective of modeling a scalar binary response variable from a functional predictor. The model estimation proposed in that case was performed in a subspace of L2(T) of squared integrable functions of finite dimension, generated by a finite set of basis functions. For that estimation it was assumed that the curves of the functional predictor and the functional parameter of the model belong to the same finite subspace. The estimation so obtained was affected by high multicollinearity problems and the solution given to these problems was based on different functional principal component analysis. The logitFD package introduced here provides a toolbox for the fit of these models by implementing the different proposed solutions and by generalizing the model proposed in 2004 to the case of several functional and non-functional predictors. The performance of the functions is illustrated by using data sets of functional data included in the fda.usc package from R-CRAN.

We present a graph-based discretization method for solving hyperbolic systems of conservation laws using discontinuous finite elements. The method is based on the convex limiting technique technique introduced by Guermond et al. (SIAM J. Sci. Comput. 40, A3211--A3239, 2018). As such, these methods are mathematically guaranteed to be invariant-set preserving and to satisfy discrete pointwise entropy inequalities. In this paper we extend the theory for the specific case of discontinuous finite elements, incorporating the effect of boundary conditions into the formulation. From a practical point of view, the implementation of these methods is algebraic, meaning, that they operate directly on the stencil of the spatial discretization. This first paper in a sequence of two papers introduces and verifies essential building blocks for the convex limiting procedure using discontinuous Galerkin discretizations. In particular, we discuss a minimally stabilized high-order discontinuous Galerkin method that exhibits optimal convergence rates comparable to linear stabilization techniques for cell-based methods. In addition, we discuss a proper choice of local bounds for the convex limiting procedure. A follow-up contribution will focus on the high-performance implementation, benchmarking and verification of the method. We verify convergence rates on a sequence of one- and two-dimensional tests with differing regularity. In particular, we obtain optimal convergence rates for single rarefaction waves. We also propose a simple test in order to verify the implementation of boundary conditions and their convergence rates.

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

Partitioned methods for coupled problems rely on data transfers between subdomains to synchronize the subdomain equations and enable their independent solution. By treating each subproblem as a separate entity, these methods enable code reuse, increase concurrency and provide a convenient framework for plug-and-play multiphysics simulations. However, accuracy and stability of partitioned methods depends critically on the type of information exchanged between the subproblems. The exchange mechanisms can vary from minimally intrusive remap across interfaces to more accurate but also more intrusive and expensive estimates of the necessary information based on monolithic formulations of the coupled system. These transfer mechanisms are separated by accuracy, performance and intrusiveness gaps that tend to limit the scope of the resulting partitioned methods to specific simulation scenarios. Data-driven system identification techniques provide an opportunity to close these gaps by enabling the construction of accurate, computationally efficient and minimally intrusive data transfer surrogates. This approach shifts the principal computational burden to an offline phase, leaving the application of the surrogate as the sole additional cost during the online simulation phase. In this paper we formulate and demonstrate such a \emph{dynamic flux surrogate-based} partitioned method for a model advection-diffusion transmission problem by using Dynamic Mode Decomposition (DMD) to learn the dynamics of the interface flux from data. The accuracy of the resulting DMD flux surrogate is comparable to that of a dual Schur complement reconstruction, yet its application cost is significantly lower. Numerical results confirm the attractive properties of the new partitioned approach.

Based on the expectile loss function and the adaptive LASSO penalty, the paper proposes and studies the estimation methods for the accelerated failure time (AFT) model. In this approach, we need to estimate the survival function of the censoring variable by the Kaplan-Meier estimator. The AFT model parameters are first estimated by the expectile method and afterwards, when the number of explanatory variables can be large, by the adaptive LASSO expectile method which directly carries out the automatic selection of variables. We also obtain the convergence rate and asymptotic normality for the two estimators, while showing the sparsity property for the censored adaptive LASSO expectile estimator. A numerical study using Monte Carlo simulations confirms the theoretical results and demonstrates the competitive performance of the two proposed estimators. The usefulness of these estimators is illustrated by applying them to three survival data sets.

The joint bidiagonalization (JBD) process iteratively reduces a matrix pair $\{A,L\}$ to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of $\{A,L\}$. The process has a nested inner-outer iteration structure, where the inner iteration usually can not be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiagonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depend on the computing accuracy of inner iterations and condition number of $(A^T,L^T)^T$ while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.

In pure integer linear programming it is often desirable to work with polyhedra that are full-dimensional, and it is well known that it is possible to reduce any polyhedron to a full-dimensional one in polynomial time. More precisely, using the Hermite normal form, it is possible to map a non full-dimensional polyhedron to a full-dimensional isomorphic one in a lower-dimensional space, while preserving integer vectors. In this paper, we extend the above result simultaneously in two directions. First, we consider mixed integer vectors instead of integer vectors, by leveraging on the concept of "integer reflexive generalized inverse." Second, we replace polyhedra with convex quadratic sets, which are sets obtained from polyhedra by enforcing one additional convex quadratic inequality. We study structural properties of convex quadratic sets, and utilize them to obtain polynomial time algorithms to recognize full-dimensional convex quadratic sets, and to find an affine function that maps a non full-dimensional convex quadratic set to a full-dimensional isomorphic one in a lower-dimensional space, while preserving mixed integer vectors. We showcase the applicability and the potential impact of these results by showing that they can be used to prove that mixed integer convex quadratic programming is fixed parameter tractable with parameter the number of integer variables. Our algorithm unifies and extends the known polynomial time solvability of pure integer convex quadratic programming in fixed dimension and of convex quadratic programming.

We propose a model for the coupling of flow and transport equations with porous membrane-type conditions on part of the boundary. The governing equations consist of the incompressible Navier--Stokes equations coupled with an advection-diffusion equation, and we employ a Lagrange multiplier to enforce the coupling between penetration velocity and transport on the membrane, while mixed boundary conditions are considered in the remainder of the boundary. We show existence and uniqueness of the continuous problem using a fixed-point argument. Next, an H(div)-conforming finite element formulation is proposed, and we address its a priori error analysis. The method uses an upwind approach that provides stability in the convection-dominated regime. We showcase a set of numerical examples validating the theory and illustrating the use of the new methods in the simulation of reverse osmosis processes.

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babu\v{s}ka--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.