Negative Sobolev Norms Research Articles

Negative order sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness

We present a variational approach aimed at enhancing the training of physics-informed neural networks (PINNs) and more general surrogate models for learning partial differential equations (PDE). In particular, we extend our formerly introduced notion of Sobolev cubatures to negative orders, enabling the approximation of negative order Sobolev norms. We mathematically prove the effect of negative order Sobolev cubatures in improving the condition number of discrete PDE learning problems, providing balancing scalars that mitigate numerical stiffness issues caused by loss imbalances. Additionally, we consider polynomial surrogate models (PSMs), which maintain the flexibility of PINN formulations while preserving the convexity structure of the PDE operators. The combination of negative order Sobolev cubatures and PSMs delivers well-conditioned discrete optimization problems, solvable via an exponentially fast convergent gradient descent for λ-convex losses. Our theoretical contributions are supported by numerical experiments, addressing linear and non-linear, forward and inverse PDE problems. These experiments show that the Sobolev cubature-based PSMs emerge as the superior state-of-the-art PINN technique.

Logarithmic type stability for the simultaneous identification of Robin coefficient and heat flux in an elliptic equation

In this paper, we aim to reconstruct the unknown Robin coefficient and unknown heat flux simultaneously in an elliptic system from Cauchy data on arbitrarily small portion of accessible boundary. A novel logarithmic type stability in terms of the negative Sobolev norms is established.

Global existence and the time decay estimates of solutions to the compressible quantum Navier–Stokes–Maxwell system in R3$\mathbb {R}^3$

AbstractWe consider the Cauchy problem of the compressible quantum Navier–Stokes–Maxwell equations with the linear damping in the isentropic case under the small perturbation of the constant equilibrium state in three dimensions. Based on the refined energy method, we establish the classical solution globally in time in Sobolev space. By the combination of the energy estimates with the interpolation between the positive Sobolev norms and the negative Sobolev norms with , we also obtain the algebraic decay rates of the classical solution. What is more, the –L2 types of the time decay rates of the solution are obtained without small assumption on the initial data in L1 norm.

Decay of the 3D Lüst model

In this paper, we consider the time-decay rate of the strong solution to the Cauchy problem for the three-dimensional Lüst model. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. The dot{H}^{-s} (0leq s<frac{3}{2}) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.

Minimal residual methods in negative or fractional Sobolev norms

For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.

Well‐posedness and decay for 3D non‐isothermal Navier–Stokes–Allen–Cahn system in ℝ3

In this paper, we investigate the local existence, global existence and time decay estimates for the nonisentropic Navier–Stokes–Allen–Cahn system in . This system describes the flow of a two‐phase immiscible heat‐conducting viscous incompressible mixture. In order to overcome the difficulties caused by the derivatives of double‐well potential and the nonlinear terms, we rewrite the Cahn–Hilliard part of the system as a new equivalent equation with “good” linear principle parts. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well‐posedness of strong solutions. Moreover, by using pure energy method and standard continuity argument together with negative Sobolev norm estimates, one proves the global well‐posedness and time decay estimates provided that the initial data are sufficiently small.

Global existence and time-decay rates of classical solutions to the generalized incompressible Oldroyd-B model in [formula omitted

In this paper, we investigate the initial value problem for the three dimensional generalized incompressible Oldroyd-B model. Based on the energy estimates and the tricky analytical skills, we prove that the problem always exists a unique global classical solution without any smallness restriction on the higher order spatial derivatives of initial data. Moreover, the optimal time-decay rates of solutions itself and the higher-order spatial derivatives of solutions are obtained without linear decay analysis. The result implies that the negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rate.

Global existence and time-decay rates of solutions to the generalized Boussinesq equation with weak damping

In this paper, we study the initial value problem for the generalized Boussineq equation with weak damping. The existence and time-decay rates of global solutions and its derivatives are established for all space dimensions d ≥ 1, provided that the norm of the initial data is suitably small. The negative Sobolev norms of the initial data in low frequency are shown to be preserved along time evolution and enhance the decay rates of global solutions. The proof is based on the energy method and flexible interpolation trick without investigating the corresponding linear equation.

A class of three dimensional Cahn-Hilliard equation with nonlinear diffusion

In this paper, we study the well-posedness and asymptotic behavior for a class of Cahn-Hilliard equation with nonlinear diffusion in R3. In order to overcome the difficulties caused by the derivatives of multi-well potential and the nonlinear terms, we “borrow” a linear principle part from the derivatives of multi-well potential, rewrite the equation as an equivalent equation with linear principle part. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Moreover, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H4-norm of initial data is sufficiently small. In the end, based on parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also establish the space-time decay estimates of strong solutions.

Quadratic behaviors of the 1D linear Schrödinger equation with bilinear control

We consider a 1D linear Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear control. We study its controllability around the ground state when the linearized system is not controllable and wonder whether the quadratic term can help to recover the directions lost at the first order. More precisely, in this paper, we formulate assumptions under which the quadratic term induces a drift which prevents the small-time local controllability (STLC) of the system in appropriate spaces.For finite-dimensional systems, quadratic terms induce coercive drifts in the dynamic, quantified by integer negative Sobolev norms, along explicit Lie brackets which prevent STLC.In the context of the bilinear Schrödinger equation, the first drift, quantified by the H−1-norm of the control, was already observed in [8] and used to deny STLC with controls small in L∞. In this article, we improve this result by denying STLC with controls small in W−1,∞.Furthermore, for any positive integer n, we formulate assumptions under which one may observe a quadratic drift quantified by the H−n-norm of the control and we use it to deny STLC in suitable spaces.

Stability estimates for singular SDEs and applications

We prove several stability estimates, comparing solutions driven by different (bi,σi), both for Itô and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference b1−b2. We then discuss several applications of these results to McKean–Vlasov SDEs, criteria for strong compactness of solutions and Wong–Zakai type theorems.

Quantifying mixing in arbitrary fluid domains: a Padé approximation approach

We consider the model problem of mixing of passive tracers by an incompressible viscous fluid. Addressing questions of optimal control in realistic geometric settings or alternatively the design of fluid-confining geometries that successfully effect mixing requires a meaningful norm in which to quantify mixing that is also suitable for easy and efficient computation (as is needed, e.g., for use in gradient-based optimization methods). We use the physically inspired reasonable surrogate of a negative index Sobolev norm over the complex fluid mixing domain $$\Omega$$ , a task which could be seen as computationally expensive since it requires the computation of an eigenbasis for $$L^{2}(\Omega )$$ by definition. Instead, we compute a representant of the scalar concentration field in an appropriate Sobolev space in order to obtain an equivalent definition of the Sobolev surrogate norm. The representant, in turn, can be computed to high-order accuracy by a Padé approximation to certain fractional pseudo-differential operators, which naturally leads to a sequence of elliptic problems with an inhomogeneity related to snapshots of the time-varying concentration field. Fast and accurate potential theoretic methods are used to efficiently solve these problems, with rapid per-snapshot mix-norm computation made possible by recent advances in numerical methods for volume potentials. We couple the methodology to existing solvers for Stokes and advection equations to obtain a unified framework for simulating and quantifying mixing in arbitrary fluid domains. We provide numerical results demonstrating the convergence of the new approach as the approximation order is increased.

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Convergence rate for the incompressible limit of nonlinear diffusion–advection equations

The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cellpopulation models to free boundary problems of Hele–Shaw type. Although a vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV-uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.

Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$($0\leq s&lt;\frac {3}{2}$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

Multilevel decompositions and norms for negative order Sobolev spaces

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in H − s H^{-s} for s ∈ ( 0 , 1 ) s\in (0,1) . Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.

Decay of strong solution for the compressible Navier–Stokes equations with large initial data

In this paper, we investigate the convergence of the global large solution to its associated constant equilibrium state with an explicit decay rate for the compressible Navier–Stokes equations in three-dimensional whole space. Suppose the initial data belongs to some negative Sobolev space instead of Lebesgue space, we not only prove the negative Sobolev norms of the solution being preserved along time evolution, but also obtain the convergence of the global large solution to its associated constant equilibrium state with algebra decay rate. Besides, we shall show that the decay rate of the first order spatial derivative of large solution of the full compressible Navier–Stokes equations converging to zero in L2−norm is (1+t)−54, which coincides with the heat equation. This extends the previous decay rate (1+t)−34 obtained in He et al. (2020).

On mix-norms and the rate of decay of correlations

Two quantitative notions of mixing are the decay of correlations and the decay of a mix-norm—a negative Sobolev norm—and the intensity of mixing can be measured by the rates of decay of these quantities. From duality, correlations are uniformly dominated by a mix-norm; but can they decay asymptotically faster than the mix-norm? We answer this question by constructing an observable with correlation that comes arbitrarily close to achieving the decay rate of the mix-norm. Therefore the mix-norm is the sharpest rate of decay of correlations in both the uniform sense and the asymptotic sense. Moreover, there exists an observable with correlation that decays at the same rate as the mix-norm if and only if the rate of decay of the mix-norm is achieved by its projection onto low-frequency Fourier modes. In this case, the function being mixed is called q -recurrent; otherwise it is q-transient. We use this classification to study several examples and raise questions for future investigations.

Sequences of Well-Distributed Vertices on Graphs and Spectral Bounds on Optimal Transport

Given a graph $$G=(V,E)$$ , suppose we are interested in selecting a sequence of vertices $$(x_j)_{j=1}^n$$ such that $$\left\{ x_1, \dots , x_k\right\} $$ is ‘well-distributed’ uniformly in k. We describe a greedy algorithm motivated by potential theory and corresponding developments in the continuous setting. The algorithm performs nicely on graphs and may be of use for sampling problems. We can interpret the algorithm as trying to greedily minimize a negative Sobolev norm; we explain why this is related to Wasserstein distance by establishing a purely spectral bound on the Wasserstein distance on graphs that mirrors R. Peyre’s estimate in the continuous setting. We illustrate this with many examples and discuss several open problems.

Decay estimates for three-dimensional Navier–Stokes equations with damping

In this paper, the optimal decay rates of solutions for three-dimensional incompressible Navier–Stokes equations with damping term uβ−1u are established. Bounds on the H.−s negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.