Degree Of Approximating Polynomial Research Articles

A New Minimax Theorem for Randomized Algorithms

The celebrated minimax principle of Yao says that for any Boolean-valued function f with finite domain, there is a distribution μ over the domain of f such that computing f to error ε against inputs from μ is just as hard as computing f to error ε on worst-case inputs. Notably, however, the distribution μ depends on the target error level ε: the hard distribution which is tight for bounded error might be trivial to solve to small bias, and the hard distribution which is tight for a small bias level might be far from tight for bounded error levels. In this work, we introduce a new type of minimax theorem which can provide a hard distribution μ that works for all bias levels at once. We show that this works for randomized query complexity, randomized communication complexity, some randomized circuit models, quantum query and communication complexities, approximate polynomial degree, and approximate logrank. We also prove an improved version of Impagliazzo’s hardcore lemma. Our proofs rely on two innovations over the classical approach of using Von Neumann’s minimax theorem or linear programming duality. First, we use Sion’s minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as “forecasting algorithms” evaluated by a certain proper scoring rule. The expected score of the forecasting version of a randomized algorithm appears to be a more fine-grained way of analyzing the bias of the algorithm. We show that such expected scores have many elegant mathematical properties—for example, they can be amplified linearly instead of quadratically. We anticipate forecasting algorithms will find use in future work in which a fine-grained analysis of small-bias algorithms is required.

On the Development of an Implicit Discontinuous Galerkin Solver for Turbulent Real Gas Flows

The aim of this work is to describe an efficient implementation of cubic and multiparameter real gas models in an existing discontinuous Galerkin solver to extend its capabilities to the simulation of turbulent real gas flows. The adopted thermodynamic models are van der Waals, Peng–Robinson, and Span–Wagner, which differ from each other in terms of accuracy and computational cost. Convective numerical fluxes across elements interfaces are calculated with a thermodynamic consistent linearized Riemann solver, whereas for boundary conditions, a linearized expression of the generalized Riemann invariants is employed. Transport properties are treated as temperature- and density-dependent quantities through multiparameter correlations. An implicit time integration is adopted; Jacobian matrix and thermodynamic derivatives are obtained with the automatic differentiation tool Tapenade. The solver accuracy is assessed by computing both steady and unsteady real gas test cases available in the literature, and the effect of the mesh size and polynomial degree of approximation on the solution accuracy is investigated. A good agreement with experimental and numerical reference data is observed and specific non-classical phenomena are well reproduced by the solver.

Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model

We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements. We consider the mathematical analysis and the numerical approximation of the problem. More precisely, we carry out firstly the well-posedness analysis of the model. Then, we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation; stability and numerical error estimates are proved.Particular attention is dedicated to the study of the saddle-point structure of the problem, that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi-incompressibility constraint. Our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity, solid displacement and pressure, to obtain a stable and robust discretization based on Taylor–Hood type finite element spaces. Interestingly, we show how this choice depends on the porosity of the mixture, i.e. the volume fraction of the fluid phase.

Approximation of the matrix exponential for matrices with a skinny field of values

The backward error analysis is a great tool which allows selecting in an effective way the scaling parameter s and the polynomial degree of approximation m when the action of the matrix exponential $$\exp (A)v$$ has to be approximated by $$\left( p_m(s^{-1}A)\right) ^sv=\exp (A+\varDelta A)v$$ . We propose here a rigorous bound for the relative backward error $$\left\Vert \varDelta A\right\Vert _{2}/\left\Vert A\right\Vert _{2}$$ , which is of particular interest for matrices whose field of values is skinny, such as the discretization of the advection–diffusion or the Schrodinger operators. The numerical results confirm the superiority of the new approach with respect to methods based on the classical power series expansion of the backward error for the matrices of our interest, both in terms of computational cost and achieved accuracy.

Multiscale mortar expanded mixed discretization of nonlinear elliptic problems

We consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method. This is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non-overlapping subdomains. An unknown (Lagrange multiplier) is introduced on the interfaces which serve as pressure Dirichlet boundary condition for local subdomain problems. A finite element space is defined on interface to approximate the pressure boundary such that the normal fluxes match weakly on interface.We demonstrated solvability of discretization, and established a priori L2-error estimates for both vector and scalar approximations. We proved the optimal order convergence rates by an appropriate choice of mortar space and polynomial degree of approximation. The uniqueness of discrete problem is shown for sufficiently small values of mesh size. An error estimate for the mortar pressure is derived via linear interface formulation with pressure dependent coefficient. We also present the analysis of the linear second order elliptic problem and prove the similar results. The computational experiments are presented to validate theory.

Quantum Query Algorithms Are Completely Bounded Forms

We prove a characterization of $t$-query quantum algorithms in terms of the unit ball of a space of degree-$2t$ polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials. Revision note: A mistake was found in the proof of the second result on degree-4 polynomials far from 2-query quantum algorithms. An explanation of the issue, a corrected proof and stronger examples are presented in work of Escudero Guti\'errez and the second author.

Refined Isogeometric Analysis for a preconditioned conjugate gradient solver

Starting from a highly continuous Isogeometric Analysis (IGA) discretization, refined Isogeometric Analysis (rIGA) introduces C0 hyperplanes that act as separators for the direct LU factorization solver. As a result, the total computational cost required to solve the corresponding system of equations using a direct LU factorization solver dramatically reduces (up to a factor of 55) (Garcia et al., 2017). At the same time, rIGA enriches the IGA spaces, thus improving the best approximation error. In this work, we extend the complexity analysis of rIGA to the case of iterative solvers. We build an iterative solver as follows: we first construct the Schur complements using a direct solver over small subdomains (macro-elements). We then assemble those Schur complements into a global skeleton system. Subsequently, we solve this system iteratively using Conjugate Gradients (CG) with an incomplete LU (ILU) preconditioner. For a 2D Poisson model problem with a structured mesh and a uniform polynomial degree of approximation, rIGA achieves moderate savings with respect to IGA in terms of the number of Floating Point Operations (FLOPs) and computational time (in seconds) required to solve the resulting system of linear equations. For instance, for a mesh with four million elements and polynomial degree p=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGA system than to the IGA one. These savings occur because the skeleton rIGA system contains fewer non-zero entries than the IGA one. The opposite situation occurs for 3D problems, and as a result, 3D rIGA discretizations provide no gains with respect to their IGA counterparts when considering iterative solvers.

Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations

A multiscale mortar mixed finite element method is established to approximate non-linear second order elliptic equations. The method is based on non-overlapping domain decomposition and mortar finite element methods. The existence and uniqueness of the approximation are demonstrated, and a priori L2-error estimates for the velocity and pressure are derived. An error bound for mortar pressure is proved. Convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete-pressure dependent coefficient. Optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation. The quadratic convergence of the Newton–Raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem. Numerical experiments are performed to support theoretic results.

A comparison of hybridized and standard DG methods for target-based hp-adaptive simulation of compressible flow

Objective: We present a comparison between hybridized and non-hybridized discontinuous Galerkin methods in the context of target-based hp-adaptation for compressible flow problems. The aim is to provide a critical assessment of the computational efficiency of hybridized DG methods.Method: Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. Using a discrete-adjoint approach, sensitivities with respect to output functionals are computed to drive the adaptation. From the error distribution given by the adjoint-based error estimator, h- or p-refinement is chosen based on the smoothness of the solution which can be quantified by properly-chosen smoothness indicators.Results: Numerical results are shown for subsonic, transonic, and supersonic flow around the NACA0012 airfoil. $hp$-adaptation proves to be superior to pure h-adaptation if discontinuous or singular flow features are involved. In all cases, a higher polynomial degree turns out to be beneficial. We show that for polynomial degree of approximation p=2 and higher, and for a broad range of test cases, HDG performs better than DG in terms of runtime and memory requirements.

The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers

We study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis. The problem of choice is the canonical Laplace equation in three dimensions. From this study we conclude that for a fixed number of unknowns and polynomial degree of approximation, a higher degree of continuity k drastically increases the CPU time and RAM needed to solve the problem when using a direct solver. This paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause.

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

AbstractThis article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (−1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ∓½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = −β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N8) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to Nd+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD FOR SLIGHTLY COMPRESSIBLE FLOWS IN POROUS MEDIA

We consider multiscale mortar mixed finite element discretizations for slightly compressible Darcy flows in porous media. This paper is an extension of the formulation introduced by Arbogast et al. for the incompressible problem [2]. In this method, flux continuity is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. Optimal fine scale convergence is obtained by an appropriate choice of mortar grid and polynomial degree of approximation. Parallel numerical simulations on some multiscale benchmark problems are given to show the efficiency and effectiveness of the method.

A Levinson--Galerkin Algorithm for Regularized Trigonometric Approximation

Trigonometric polynomials are widely used for the approximation of a smooth function from a set of nonuniformly spaced samples. If the samples are perturbed by noise, a good choice for the polynomial degree of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Standard methods for trigonometric leastsquares approximation assume that the degree for the approximating polynomial is known a priori, which is usually not the case in practice. We derive a multilevel algorithm that recursively adapts to the least squares solution of suitable degree. We analyze under which conditions this multilevel approach yields the optimal solution. The proposed algorithm computes the solution in at most $\mathcal{O}(rM + M^2)$ operations (M being the polynomial degree of the approximation and r being the number of samples) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the left ventricle of the heart.