Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations are a natural tool to help gain understanding into elliptic equations, it is natural to ask if such principles also hold at the discrete level. In this work, we prove a version of the unique continuation principle for piecewise-linear and -bilinear finite-element discretizations of the Laplacian eigenvalue problem on polygonal domains in $\mathbb{R}^2$. Namely, we show that any solution to the discretized equation $-\Delta u = \lambda u$ with vanishing Dirichlet and Neumann traces must be identically zero under certain geometric and topological assumptions on the resulting triangulation. We also provide a counterexample, showing that a nonzero \emph{inner solution} exists when the topological assumptions are not satisfied. Finally, we give an application to an eigenvalue interlacing problem, where the space of inner solutions makes an explicit appearance.
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Generative diffusion models apply the concept of Langevin dynamics in physics to machine leaning, attracting a lot of interests from engineering, statistics and physics, but a complete picture about inherent mechanisms is still lacking. In this paper, we provide a transparent physics analysis of diffusion models, formulating the fluctuation theorem, entropy production, equilibrium measure, and Franz-Parisi potential to understand the dynamic process and intrinsic phase transitions. Our analysis is rooted in a path integral representation of both forward and backward dynamics, and in treating the reverse diffusion generative process as a statistical inference, where the time-dependent state variables serve as quenched disorder akin to that in spin glass theory. Our study thus links stochastic thermodynamics, statistical inference and geometry based analysis together to yield a coherent picture about how the generative diffusion models work.
The SE and DE formulas are known as efficient quadrature formulas for integrals with endpoint singularities. Especially for integrals with algebraic singularity, explicit error bounds in a computable form have been given, which are useful for computation with guaranteed accuracy. Such explicit error bounds have also given for integrals with logarithmic singularity. However, the error bounds have two points to be discussed. The first point is on overestimation of divergence speed of logarithmic singularity. The second point is on the case where there exist both logarithmic and algebraic singularity. To remedy these points, this study provides new error bounds for integrals with logarithmic and algebraic singularity. Although existing and new error bounds described above handle integrals over the finite interval, the SE and DE formulas may be applied to integrals over the semi-infinite interval. On the basis of the new results, this study provides new error bounds for integrals over the semi-infinite interval with logarithmic and algebraic singularity at the origin.
We consider a mixed variational formulation recently proposed for the coupling of the Brinkman--Forchheimer and Darcy equations and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D version of the associated conforming mixed finite element scheme. For the reliability analysis, due to the nonlinear nature of the problem, we make use of the inf-sup condition and the strong monotonicity of the operators involved, along with a stable Helmholtz decomposition in Hilbert spaces and local approximation properties of the Raviart--Thomas and Cl\'ement interpolants. On the other hand, inverse inequalities, the localization technique through bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms are reported. In particular, the case of flow through a heterogeneous porous medium is considered.
Block majorization-minimization (BMM) is a simple iterative algorithm for constrained nonconvex optimization that sequentially minimizes majorizing surrogates of the objective function in each block while the others are held fixed. BMM entails a large class of optimization algorithms such as block coordinate descent and its proximal-point variant, expectation-minimization, and block projected gradient descent. We first establish that for general constrained nonsmooth nonconvex optimization, BMM with $\rho$-strongly convex and $L_g$-smooth surrogates can produce an $\epsilon$-approximate first-order optimal point within $\widetilde{O}((1+L_g+\rho^{-1})\epsilon^{-2})$ iterations and asymptotically converges to the set of first-order optimal points. Next, we show that BMM combined with trust-region methods with diminishing radius has an improved complexity of $\widetilde{O}((1+L_g) \epsilon^{-2})$, independent of the inverse strong convexity parameter $\rho^{-1}$, allowing improved theoretical and practical performance with `flat' surrogates. Our results hold robustly even when the convex sub-problems are solved as long as the optimality gaps are summable. Central to our analysis is a novel continuous first-order optimality measure, by which we bound the worst-case sub-optimality in each iteration by the first-order improvement the algorithm makes. We apply our general framework to obtain new results on various algorithms such as the celebrated multiplicative update algorithm for nonnegative matrix factorization by Lee and Seung, regularized nonnegative tensor decomposition, and the classical block projected gradient descent algorithm. Lastly, we numerically demonstrate that the additional use of diminishing radius can improve the convergence rate of BMM in many instances.
We study a circular-circular multiplicative regression model, characterized by an angular error distribution assumed to be wrapped Cauchy. We propose a specification procedure for this model, focusing on adapting a recently proposed goodness-of-fit test for circular distributions. We derive its limiting properties and study the power performance of the test through extensive simulations, including the adaptation of some other well-known goodness-of-fit tests for this type of data. To emphasize the practical relevance of our methodology, we apply it to several small real-world datasets and wind direction measurements in the Black Forest region of southwestern Germany, demonstrating the power and versatility of the presented approach.
We present a simple universal algorithm for high-dimensional integration which has the optimal error rate (independent of the dimension) in all weighted Korobov classes both in the randomized and the deterministic setting. Our theoretical findings are complemented by numerical tests.
We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based formulas are based on the inverse Laplace transform or the Fourier transform. We show these expressions are essentially equivalent in terms of complex integrals and choose the former as a starting point to reduce computational cost. By choosing a simple integral path, we derive an integral expression mentioned above. Then, we can easily apply the double-exponential formula and the Gauss-Legendre formula, which have rigorous error bounds. As numerical experiments show, the proposed formula outperforms the existing formulas when the imaginary parts of the eigenvalues of matrices have large absolute values.
The problem of distributed matrix multiplication with straggler tolerance over finite fields is considered, focusing on field sizes for which previous solutions were not applicable (for instance, the field of two elements). We employ Reed-Muller-type codes for explicitly constructing the desired algorithms and study their parameters by translating the problem into a combinatorial problem involving sums of discrete convex sets. We generalize polynomial codes and matdot codes, discussing the impossibility of the latter being applicable for very small field sizes, while providing optimal solutions for some regimes of parameters in both cases.
This paper investigates logical consequence defined in terms of probability distributions, for a classical propositional language using a standard notion of probability. We examine three distinct probabilistic consequence notions, which we call material consequence, preservation consequence, and symmetric consequence. While material consequence is fully classical for any threshold, preservation consequence and symmetric consequence are subclassical, with only symmetric consequence gradually approaching classical logic at the limit threshold equal to 1. Our results extend earlier results obtained by J. Paris in a SET-FMLA setting to the SET-SET setting, and consider open thresholds beside closed ones. In the SET-SET setting, in particular, they reveal that probability 1 preservation does not yield classical logic, but supervaluationism, and conversely positive probability preservation yields subvaluationism.
We develop randomized matrix-free algorithms for estimating partial traces, a generalization of the trace arising in quantum physics and chemistry. Our algorithm improves on the typicality-based approach used in [T. Chen and Y-C. Cheng, \emph{Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems}, J. Chem. Phys. 157, 064106 (2022)] by deflating important subspaces (e.g. corresponding to the low-energy eigenstates) explicitly. This results in a significant variance reduction, leading to several order-of-magnitude speedups over the previous state of the art. We then apply our algorithm to study the thermodynamics of several Heisenberg spin systems, particularly the entanglement spectrum and ergotropy.
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