Parameter-dependent Operators Research Articles

Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition

POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solutions us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs.

Stochastic collocation method for computing eigenspaces of parameter-dependent operators

We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces can in fact be extended to complex-analytic functions of the parameters and quantify this analytic dependence in a way that leads to convergence of sparse polynomial approximations. A stochastic collocation method on an anisoptropic sparse grid in the parameter domain is proposed for computing a basis for the eigenspace of interest. The convergence of this method is verified in a series of numerical examples based on the eigenvalue problem of a stochastic diffusion operator.

Eta-Invariants for Parameter-Dependent Operators Associated with an Action of a Discrete Group

Eta-Invariants for Parameter-Dependent Operators Associated with an Action of a Discrete Group

Parametric pseudodifferential operators with point-singularity in the covariable

Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb–Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential operators originally proved by Grubb–Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents.

Fibre homogenisation

In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by ‘fibre-wise’ homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods.

Bounding stability constants for affinely parameter-dependent operators

In this article we introduce new possibilities of bounding the stability constants that play a vital role in the reduced basis method. By bounding stability constants over a neighborhood we make it possible to guarantee stability at more than a finite number of points and to do that in the offline stage. We additionally show that Lyapunov stability of dynamical systems can be handled in the same framework.

Reduced Basis Methods: From Low-Rank Matrices to Low-Rank Tensors

We propose a novel combination of the reduced basis method with low-rank tensor techniques for the efficient solution of parameter-dependent linear systems in the case of several parameters. This combination, called rbTensor, consists of three ingredients. First, the underlying parameter-dependent operator is approximated by an explicit affine representation in a low-rank tensor format. Second, a standard greedy strategy is used to construct a problem-dependent reduced basis. Third, the associated reduced parametric system is solved for all parameter values on a tensor grid simultaneously via a low-rank approach. This allows us to explicitly represent and store an approximate solution for all parameter values at a time. Once this approximation is available, the computation of output functionals and the evaluation of statistics of the solution become a cheap online task, without requiring the solution of a linear system.

Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates

In the process of discrete and nonlocal aggregation, the major problem arises when each fragmentation rate becomes infinite at infinity. In this paper, a discrete Cauchy problem describing multiple fragmentation processes is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. We focus on the case where fragmentation rates are size and position dependent and where new particles are spatially randomly distributed according to a certain probabilistic law. Unlike [8], where the discrete model with bounded fragmentation rates is treated, we use, in this paper, Katoʼs theorem in L1[2] and the dominated convergence theorem [4] to show the existence of the infinitesimal generator of a positive semigroup of contractions and give sufficient conditions for honesty in the case of unbounded fragmentation rates. Essentially, we demonstrate that, even in the discrete and nonlocal case, the process is conservative if at infinity daughter particles tend to go back into the system with a high probability.

On the Honesty in Nonlocal and Discrete Fragmentation Dynamics in Size and Random Position

A discrete initial-value problem describing multiple fragmentation processes, where the fragmentation rate is size and position dependent and where new particles are spatially randomly distributed according to some probabilistic law, is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. The existence of semigroups is established for each parameter and “glued” together so as to obtain a semigroup to the full space. Under certain conditions on each fragmentation rate, we used Kato’s Theorem in to show the existence of the generator and we provide sufficient conditions for honesty.

Ellipticity in pseudodifferential algebras of Toeplitz type

Let L⋆ be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T⋆ be a subalgebra of operators of the form P1AP0, where P0,P1∈L⋆ are projections, i.e., Pj2=Pj. The elements of L⋆ act as linear continuous operators in a scale of abstract Sobolev spaces, those of T⋆ in the corresponding subspaces determined by the projections. We study how the ellipticity in L⋆ descends to T⋆, focusing on parametrix construction, equivalence with the Fredholm property, characterisation in terms of invertibility of principal symbols, and spectral invariance. Applications concern SG-pseudodifferential operators, operators on manifolds with conical singularities, and Boutet de Monvelʼs algebra for boundary value problems. In particular, we derive invertibility of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvelʼs algebra. We indicate how the concept generalizes to parameter-dependent operators.

Parameter-dependent edge operators

We study parameter-dependent operators on a manifold with edge and construct new classes of elliptic elements in the corner calculus on an infinite cone with a singular base.

Reduction of orders in boundary value problems without transmission property

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

Parameter-dependent operators and resolvent expansions on conic manifolds

The goal of this paper is to present, from the $b$-calculus perspective, a program for analyzing generalized resolvent families and heat kernels of pseudodifferential operators on conic manifolds. To carry this out, a new class of parameter-dependent cone pseudodifferential operators is developed and studied.

MATRIX-FREE NUMERICAL CONTINUATION AND BIFURCATION1*

A numerical continuation method traces solution branches of a nonlinear system which is typically obtained by discretizing a parameter dependent operator equation. The method is called matrix-free if the Jacobian of H is not calculated explicitly, but its action on a vector is given via a difference approximation of a directional derivative. In connection with modern (transpose-free) iterative linear solvers, this is a suitable approach for large systems. We give an introduction into the technique of matrix-free numerical continuation and analyze those recent approaches to numerical bifurcation which permit a matrix-free approach. A sequence of MATLAB codes, which can be viewed as a blueprint for the numerical implementation of this approach, is currently under construction and will be made available on the internet. Parts of this manuscript are based on the 200 page article [2]. *This work was supported in part by the NSF under contect number DMS-9870274.

The approximation of generalized turning points\newline by Krylov subspace methods

Certain types of singular solutions of nonlinear parameter-dependent operator equations were characterized by Griewank and Reddien [5, 6] as regular solutions of suitable augmented systems. For their numerical approximation an approach based on the use of Krylov subspaces is here presented. The application to boundary value problems is illustrated by numerical examples.

Parameter‐dependent operator equations of the Riccati type with application to transport theory

AbstractWe obtain an existence result for global solutions to initial‐value problems for Riccati equations of the form R′(t) + TR(t) + R(t)T = Tρ A(t)T1−ρ + Tρ B(t)T1−ρ R(t) + R(t)TρC(t) T1−ρ + R(t)TρD(t)T1−ρ R(t), R(0)=R0, where 0 ⩽ ρ ⩽ 1 and where the functions R and A through D take on values in the cone of non‐negative bounded linear operators on L1 (0, W; μ). T is an unbounded multiplication operator. This problem is of particular interest in case ρ = 1 since it arisess in the theories of particle transport and radiative transfer in a slab. However, in this case there are some serious difficulties associated with this equation, which lead us to define a solution for the case ρ = 1 as the limit of solutions for the cases 0 < ρ < 1.

Influence of thyrotropin releasing hormone on recovery function of somatosensory evoked potentials in the rabbit : N. Sakai, T. Yamamori, H. Yamada, Y. Tanabe and T. Funakoshi (Gifu, Japan)

Let L⋆ be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T⋆ be a subalgebra of operators of the form P1AP0, where P0,P1∈L⋆ are projections, i.e., Pj2=Pj. The elements of L⋆ act as linear continuous operators in a scale of abstract Sobolev spaces, those of T⋆ in the corresponding subspaces determined by the projections. We study how the ellipticity in L⋆ descends to T⋆, focusing on parametrix construction, equivalence with the Fredholm property, characterisation in terms of invertibility of principal symbols, and spectral invariance. Applications concern SG-pseudodifferential operators, operators on manifolds with conical singularities, and Boutet de Monvelʼs algebra for boundary value problems. In particular, we derive invertibility of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvelʼs algebra. We indicate how the concept generalizes to parameter-dependent operators.

The Convergence of Operator Approximations at Turning Points

We consider the convergence of solution curves of approximations to parameter-dependent operator equations of the form G(λ, x) = 0. Provided Gx(λ, x) remains non-singular this problem is catered for by a simple extension to standard theory. In this paper, however, attention is concentrated on solution curves through certain singular points (λ0, x0), and the main result is that convergence depends on consistency and stability results for the linear eigenvalue problem Gx(λ0, x0)0 = 0.