Linear Variational Problems Research Articles

A multiscale approach to a linear programming problem for calculating a refractive optical element forming a given far-field illumination distribution

We consider an approach to the calculation of a refractive optical element generating a prescribed irradiance distribution in the far field for a plane incident beam based on a certain variational problem. We consider an explicit formulation of this problem in the form of the Monge-Kantorovich mass transfer problem. We also demonstrate a connection between the mass transfer problem and the dual linear variational problem. A numerical solution of the linear variational problem is also considered. The direct solution of this type of problem presents a huge computational complexity. To overcome this difficulty we use the so-called multiscale approach based on constructing a chain of approximations that are solutions on refining grids.

Residual-based error corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction–diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction–diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.

On [H^{1}]^{3 times 3}, [H(text {curl})]^3 and H(text {sym Curl}) finite elements for matrix-valued Curl problems

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [{{,mathrm{textit{H}^1},}}]^{3times 3}, [ H (textrm{curl}{})]^3 and H (textrm{sym}, textrm{Curl}{}) for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [ H (textrm{curl}{})]^3 in place of the classical [{{,mathrm{textit{H}^1},}}]^{3 times 3}. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H (textrm{sym}, textrm{Curl}{}). The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory.

An anisotropic adaptive method for the numerical approximation of orthogonal maps

Orthogonal maps are two-dimensional mappings that are solutions of the so-called origami problem obtained when folding a paper. These mappings are piecewise linear, and the discontinuities of their gradient form a singular set composed of straight lines representing the folding edges. The proposed algorithm relies on the minimization of a variational principle discussed in Caboussat et al. (2019). A splitting algorithm for the corresponding flow problem derived from the first-order optimality conditions alternates between local nonlinear problems and linear elliptic variational problems at each time step. Anisotropic adaptive techniques allow to obtain refined triangulations near the folding edges while keeping the number of vertices as low as possible. Numerical experiments validate the accuracy and efficiency of the adaptive method in various situations. Appropriate convergence properties are exhibited, and solutions with sharp edges are recovered.

Application of Bernstein Polynomial Multiwavelets for Solving Non Linear Variational Problems with Moving and Fixed Boundaries

Background: In this article, an efficient direct method has been proposed in order to solve physically significant variational problems. The proposed technique finds its basis in Bernstein polynomials multiwavelets (BPMWs). The mechanism of the proposed method is to transform the variational problem into an algebraic equation system through the use of BPMWs. Objective: Since the necessary condition of extremization consists of a differential equation that cannot be easily integrated into complex cases, an approximated numerical solution becomes a necessity. Our primary objective was to establish a wavelet-based method for solving variational problems of physical interest. Besides being computationally more effective, the proposed approach yields relatively more accurate results than other comparable methods. The approach employs fewer basis elements, which in turn increase the simplicity, decrease the calculation time, and furnishes better results. Methods: An operational matrix of integration, which is based on the BPMWs, is presented. We substituted the approximated values of x , unknown function ξ (x) and their derivative functions ( ), ( ),..... ( ) ξ ′ x ξ ′′ x ξ n x with BPMWs operational matrix of integration and BPMWs. On substituting the respective values in the given variational problem, it gets converted into a system of algebraic equations. The obtained system is further solved using the Lagrange multiplier. Results: The results obtained yield a greater degree of convergence as compared to other existing numerical methods. Numerical illustrations based on physical variational problems and the comparisons of outcomes with exact solutions demonstrate that the proposed method yields better efficiency, applicability, and accuracy. Conclusion: The proposed method gives better results than other comparable methods, even with the use of a fewer number of basis elements. The large order of matrices, such as 32, 64, and 512, obtained by using other available methods is far too high to achieve accuracy in results in comparison to the ones we obtained by using matrices of relatively lower orders, such as 7, 8 and 13, in the proposed method. This method can also be used for extremization functional occurring in electrical circuits and mechanical physical problems.

Linear Integrable Systems on Quad-Graphs

Abstract In the first part of the paper, we classify linear integrable (multidimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb{C}$, enjoying natural symmetries and the property that the restriction of their solutions to the black vertices satisfies a Laplace type equation. The classification reduces to solving a functional equation. Under certain restriction, we give a complete solution of the functional equation, which is expressed in terms of elliptic functions. We find two real analytic reductions, corresponding to the cases when the underlying complex torus is of a rectangular type or of a rhombic type. The solution corresponding to the rectangular type was previously found by Boutillier, de Tilière, and Raschel. Using the multidimensional consistency, we construct the discrete exponential function, which serves as a basis of solutions of the quad-equation. In the second part of the paper, we focus on the integrability of discrete linear variational problems. We consider discrete pluri-harmonic functions, corresponding to a discrete two-form with a quadratic dependence on the fields at black vertices only. In an important particular case, we show that the problem reduces to a two-field generalization of the classical star-triangle map. We prove the integrability of this novel 3D system by showing its multidimensional consistency. The Laplacians from the first part come as a special solution of the two-field star-triangle map.

The double adaptivity paradigm: (How to circumvent the discrete inf–sup conditions of Babuška and Brezzi)

We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in the context of the convection-dominated diffusion problem. The presented ideas, however, apply to virtually any well-posed system of first-order partial differential equations, including singular perturbation problems.

FFT-based homogenisation accelerated by low-rank tensor approximations

Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier–Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. Here these kinds of methods are accelerated by low-rank tensor approximation techniques for a solution field using canonical polyadic, Tucker, and tensor train formats. This reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem. It significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation. The advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients. This approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material.

Nonlocal Models with Heterogeneous Localization and Their Application to Seamless Local-Nonlocal Coupling

Motivated by recent development on nonlocal mechanical models like peridynamics, we consider nonlocal integral models with a spatially varying horizon that allows the finite range of nonlocal interactions to be position-dependent. In particular, we focus on linear variational problems of such nonlocal models with heterogeneous localization on co-dimension one interfaces. The well-posedness is established for several classes of variational problems involving homogeneous and inhomogeneous Neumann- and Dirichlet-type constraints. We also study their seamless coupling with local models. In addition, we present numerical studies of the nonlocal models and local-nonlocal coupled models based on the asymptotically compatible Galerkin finite element discretizations.

Finite Element Exterior Calculus for Parabolic Evolution Problems on Riemannian Hypersurfaces

Over the last ten years, the Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Holst and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Holst, and Zhu extended the FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces (spaces of Banach-space valued parametrized curves). Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through the use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in the flat (non-criminal) setting. Some numerical examples are also presented. Date: September 21, 2015.

Generalized FPK equations corresponding to systems of nonlinear random differential equations excited by colored noise. Revisitation and new directions

The solution of systems of nonlinear random differential equations (RDEs) excited by Gaussian colored noise is an important question in physics and engineering. An established way of work, which is revisited in the present paper, is to formulate approximate equations governing the probability density function of the RDE system response, called the generalized Fokker-Planck-Kolmogorov (genFPK) equations. These genFPK equations are derived from the stochastic Liouville equation corresponding to the RDE system, which is an exact, yet not closed equation, due to a non-local term, the presence of which is a manifestation the non-Markovian character of the response. The novelty of the present work lies in the identification of the said non-local term as the transition matrix of the linear variational problem associated with the nonlinear RDE system. This identification enables us to easily rederive the small correlation time genFPK equation, as well as to derive a new, exact in the linear case, genFPK equation that constitutes the multidimensional counterpart of Fox’s genFPK equation, by approximating the transition matrix using Peano-Baker and Magnus expansion respectively. Last, from Fox’s genFPK equation, Hänggi’s ansatz for RDE systems is stated, which is the first step in generalizing, for systems of RDEs, Volterra adjustable decoupling approximation, a technique we developed recently for the derivation of novel genFPK equations corresponding to scalar RDEs.

Schwarz Iterative Methods: Infinite Space Splittings

We prove the convergence of greedy and randomized versions of Schwarz iterative methods for solving linear elliptic variational problems based on infinite space splittings of a Hilbert space. For the greedy case, we show a squared error decay rate of \(O((m+1)^{-1})\) for elements of an approximation space \(\mathscr {A}_1\) related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of \(O((m+1)^{-1})\) on a class \(\mathscr {A}_{\infty }^{\pi }\subset \mathscr {A}_1\) depending on the probability distribution.

Realtime Reconstruction of an Animating Human Body from a Single Depth Camera.

We present a method for realtime reconstruction of an animating human body,which produces a sequence of deforming meshes representing a given performance captured by a single commodity depth camera. We achieve realtime single-view mesh completion by enhancing the parameterized SCAPE model.Our method, which we call Realtime SCAPE, performs full-body reconstruction without the use of markers.In Realtime SCAPE, estimations of body shape parameters and pose parameters, needed for reconstruction, are decoupled. Intrinsic body shape is first precomputed for a given subject, by determining shape parameters with the aid of a body shape database. Subsequently, per-frame pose parameter estimation is performed by means of linear blending skinning (LBS); the problem is decomposed into separately finding skinning weights and transformations. The skinning weights are also determined offline from the body shape database,reducing online reconstruction to simply finding the transformations in LBS. Doing so is formulated as a linear variational problem;carefully designed constraints are used to impose temporal coherence and alleviate artifacts. Experiments demonstrate that our method can produce full-body mesh sequences with high fidelity.

Uncertain Lambert Problem

This analysis addresses the problem of constructing uncertainties in initial and final satellite orbital state vectors that are dynamically consistent in Lambert’s boundary-value problem, given uncertainties in the initial and final position vectors. The structure of the general nonlinear problem is discussed in terms of transformations of the probability density functions of the given positions. For the linear (first-order) variational version of the problem, the covariance matrices of the state variations are presented in explicit terms of the covariance matrix of the given position variations and partitions of the state transition matrix of the nominal orbit. Equivalence is demonstrated between the linear variational results, which involve no iteration, and a weighted batch least-squares differential-correction solution of the orbit determination problem. To illustrate the utility of the working formulas in the linear variational problem, two approaches are taken: 1) an analytic approach to solve directly for the covariance terms, based on linearized dynamics, and 2) a numerical Monte Carlo-based approach that generates a sample covariance. The numerical results for state covariance from the analytic approach agree with results obtained by batch least-squares differential correction, and both agree with results from the Monte Carlo approach to within differences explainable by the limitations of sampling.

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015

Unification of distance inequalities for linear variational problems

In this work a unifying approach is presented that leads to bounds for the distance in natural norms between solutions belonging to different spaces, of well-posed linear variational problems with the same input data. This is done in a general hilbertian framework, and in this sense, well-known inequalities such as Cea’s or Babuska’s for coercive and non-coercive problems are extended and/or refined, as mere by-products of this unified setting. More particularly such an approach gives rise to both an improvement and a generalization to the weakly coercive case, of second Strang’s inequality for abstract coercive problems. Additionally several aspects specific to linear variational problems are the subject of a thorough analysis beforehand, which also allows for clarifications and further refinements about the concept of weak coercivity.

The bond-based peridynamic system with Dirichlet-type volume constraint

In this paper, the bond-based peridynamic system is analysed as a non-local boundary-value problem with volume constraint. The study extends earlier works in the literature on non-local diffusion and non-local peridynamic models, to include non-positive definite kernels. We prove the well-posedness of both linear and nonlinear variational problems with volume constraints. The analysis is based on some non-local Poincaré-type inequalities and the compactness of the associated non-local operators. It also offers careful characterizations of the associated solution spaces, such as compact embedding, separability and completeness. In the limit of vanishing non-locality, the convergence of the peridynamic system to the classical Navier equations of elasticity with Poisson ratio ¼ is demonstrated.

Variational relation problems: existence of solutions and fixed points of contraction mappings

We propose a new approach to study variational relation problems. Namely, we apply Mizoguchi and Takahashi’s fixed point theorem of contraction mappings and an error bound of a system of linear inequalities to establish existence conditions for a variational relation problem in which the variational relation linearly depends on the decision variable. Then we develop an algorithm to compute a solution of a linear variational relation problem. MSC:49J52, 47H10.

A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems

A variational inequality (VI) and a mixed formulation for an elliptic obstacle problem are considered. Both formulations are discretized by an hp-FE interior penalty discontinuous Galerkin (IPDG) method. In the case of the mixed method, the discrete Lagrange multiplier is a linear combination of biorthogonal basis functions. In particular, also the discrete problems are equivalent. For these formulations a residual based a posteriori error estimate and a hierarchical a posteriori error estimate are derived. For the mixed method the residual based estimate is constructed explicitly, for which the approximation error is split into a discretization error of a linear variational equality problem and additional consistency and obstacle condition terms. For the VI-method a hierarchical estimate based on an overkill solution is derived explicitly. This includes the h–h/2 and p–(p+1)-estimates. The numerical experiments show exponential convergence up to the desired tolerance.