We study the generalization error of functions that interpolate prescribed data points and are selected by minimizing a weighted norm. Under natural and general conditions, we prove that both the interpolants and their generalization errors converge as the number of parameters grow, and the limiting interpolant belongs to a reproducing kernel Hilbert space. This rigorously establishes an implicit bias of minimum weighted norm interpolation and explains why norm minimization may either benefit or suffer from over-parameterization. As special cases of this theory, we study interpolation by trigonometric polynomials and spherical harmonics. Our approach is from a deterministic and approximation theory viewpoint, as opposed to a statistical or random matrix one.
In this paper we investigate polynomial interpolation using orthogonal polynomials. We use weight functions associated with orthogonal polynomials to define a weighted form of Lagrange interpolation. We introduce an upper bound of error estimation for such kinds of approximations. Later, we introduce the sufficient condition of Stenger's conjecture for orthogonal polynomials and numerical verification for such conjecture.
This Technical Note outlines an adjoint complement to a critical building block of pressure-based Finite-Volume (FV) flow solvers that employ a collocated variable arrangement to simulate virtually incompressible fluids. The focal point is to strengthen the adjoint pressure-velocity coupling by using an adjoint Momentum-Weighted Interpolation (MWI) strategy. To this end, analogies of established primal MWI techniques are derived and investigated. The study reveals the merits of an adjoint MWI but also highlights the importance of its careful implementation.
We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grids methods constructed with the help of quasi-interpolation operators. The class of such operators includes classical interpolation and sampling operators, Kantorovich-type operators, scaling expansions associated with wavelet constructions, and others. We obtain the rate of convergence of the corresponding sparse grids methods in weighted Wiener norms as well as analogues of the Littlewood-Paley-type characterizations in terms of families of quasi-interpolation operators.
Interpolation in Jacobi-weighted spaces and its application to a posteriori error estimations of the p-version of the finite element method
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, IMAS - Conicet（布宜诺斯艾利斯大学精确与自然科学学院数学系，IMAS-CONICET）；Universidad de Buenos Aires（布宜诺斯艾利斯大学）；IMAS - Conicet（IMAS-CONICET）
For the Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the interpolation associated to the Fujino--Morley FEM space and the convex-hull property of the Bernstein representation of func…
We consider the extremal problem of interpolation of convex scattered data in $\mathbb{R}^3$ by smooth edge convex curve networks with minimal $L_p$-norm of the second derivative for $1<p\leq\infty$. The problem for $p=2$ was set and solved by Andersson et al. (1995). Vlachkova (2019) extended the results in (Andersson et al., 1995) and solved the problem for $1<p<\infty$. The minimum edge convex $L_p$-norm network for $1<p<\infty$ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case $1<p<\infty$ is unique for stric…
In this work we consider the problem of semi-active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the $\mathcal{H}_\infty$-norm of the transfer function from the exogenous inputs to the performance outputs. We make use of a new greedy method for computing the $\mathcal{H}_\infty$-norm of a transfer function based on rational interpolation. In this paper, this approach is adapted to parameter-dependent transfer functions. The interpolation leads to parametric reduced-order models that can be optim…
The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{β=0}^NC_β(z)φ(x_β)$ of given values of a function $φ$ from the space $K_2(P_2)$. The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients $C_β(z)$ of the i…
Equivalence of anchored and ANOVA spaces via interpolation
Institut für Analysis, Johannes Kepler Universität Linz（林茨约翰·开普勒大学分析研究所）；Institut für Mathematik, Friedrich Schiller Universität Jena（耶拿弗里德里希·席勒大学数学研究所）
In this work, a space-time scheme for goal-oriented a posteriori error estimation is proposed. The error estimator is evaluated using a partition-of-unity dual-weighted residual method. As application, a low mach number combustion equation is considered. In some numerical tests, different interpolation variants are investigated, while observing convergence orders and effectivity indices between true errors (obtained on a sufficiently refined mesh) and the error estimator.
