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Abstract This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function, meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter-dependent fibre problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form the loss function is given as a dual test norm of the residual; a central objective is to develop equivalent computable expressions. The first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, for example, by neural networks. Second, working with first-order SVFs we distinguish two scenarios: (i) the test space can be chosen as an $$L_{2}$$-space (such as for elliptic or parabolic problems) so that residuals can be evaluated directly as elements of $$L_{2}$$; (ii) when trial and test spaces for the fibre problems depend on the parameters (as for transport equations) we use ultra-weak formulations. In combination with discontinuous Petrov–Galerkin concepts the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter-dependent convection fields. 

Abstract In this article, using that the fractional Laplacian can be factored into a product of the divergence operator, a Riesz potential operator and the gradient operator, we introduce an anomalous fractional diffusion operator, involving a matrixK(x), suitable when anomalous diffusion is being studied in a non homogeneous medium. For the case ofK(x) a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function. 

Abstract Approximating functions of a large number of variables poses particular challenges often subsumed under the term “Curse of Dimensionality” (CoD). Unless the approximated function exhibits a very high level of smoothness the CoD can be avoided only by exploiting some typically hiddenstructural sparsity. In this paper we propose a general framework for new model classes of functions in high dimensions. They are based on suitable notions ofcompositional dimension-sparsityquantifying, on a continuous level, approximability by compositions with certain structural properties. In particular, this describes scenarios where deep neural networks can avoid the CoD. The relevance of these concepts is demonstrated forsolution manifoldsof parametric transport equations. For such PDEs parameter-to-solution maps do not enjoy the type of high order regularity that helps to avoid the CoD by more conventional methods in other model scenarios. Compositional sparsity is shown to serve as the key mechanism for proving that sparsity of problem data is inherited in a quantifiable way by the solution manifold. In particular, one obtains convergence rates for deep neural network realizations showing that the CoD is indeed avoided. 

Abstract Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings. 

Abstract We introduce a new approach to the numerical simulation of Scanning Transmission Electron Microscopy images. The Lattice Multislice Algorithm takes advantage of the fact that the electron waves passing through the specimen have limited bandwidth and therefore can be approximated very well by a low-dimensional linear space spanned by translations of a well-localized function. Just like in the PRISM algorithm recently published by C. Ophus, we utilize the linearity of the Schrödinger equation but perform the approximations with functions that are well localized in real space instead of Fourier space. This way, we achieve a similar computational speedup as PRISM, but at a much lower memory consumption and reduced numerical error due to avoiding virtual copies of the probe waves interfering with the result. Our approach also facilitates faster recomputations if local changes are made to the specimen such as changing a single atomic column. 

Abstract The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus.