arXiv subjects

David W. Zingg

Publications and source records attributed to David W. Zingg.

At least 19 recordsLinked to original sources

arXiv2026

Construction and Optimization of Summation-by-Parts Operators for General Function Spaces Using an Improved Generalized Gaussian Quadrature Algorithm

We construct optimized summation-by-parts (SBP) operators for general function spaces with provably minimal degrees of freedom on open, closed, and half-open nodal distributions. These operators rely on generalized Gaussian quadrature rules, for which we present an improved algorithm that is flexible, efficient, and provably convergent. In cases where free parameters are available, we further introduce two operator optimization strategies. We test our operators on a handful of numerical examples that contain large or unbounded gradients, in which some a priori knowledge of the solution has been assumed to select an appropriate basis. The novel operators are found to outperform standard polynomial operators by several orders of magnitude in solution accuracy relative to degrees of freedom. Furthermore, our novel operators significantly outperform function-space SBP operators with equispaced nodal distributions, which require significantly more nodes for the same operator basis. Finally, we demonstrate that the operator optimization procedures are critical to achieving accurate and efficient discretizations, as the standard SBP construction procedure can lead to nullspace-inconsistent and poorly-conditioned operators.

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arXiv2026

Efficient Gradient-Enhanced Bayesian Optimizer with Comparisons to Conjugate-Gradient and Quasi-Newton Optimizers for Unconstrained Local Optimization

The probabilistic surrogates used by Bayesian optimizers make them popular methods when function evaluations are noisy or expensive to evaluate. While Bayesian optimizers are traditionally used for global optimization, their benefits are also valuable for local optimization. In this paper, a framework for gradient-enhanced unconstrained local Bayesian optimization is presented. It involves selecting a subset of the evaluation points to construct the surrogate and using a probabilistic trust region for the minimization of the acquisition function. The Bayesian optimizer is compared to conjugate-gradient and quasi-Newton optimizers from MATLAB and SciPy for unimodal problems with 2 to 40 dimensions. The Bayesian optimizer converges the optimality as deeply as the optimizers used for comparison and often does so using significantly fewer function evaluations. For the minimization of the 40-dimensional Rosenbrock function for example, the Bayesian optimizer requires half as many function evaluations as the MATLAB and SciPy optimizers to reduce the optimality by 10 orders of magnitude. For test cases with noisy gradients, the probabilistic surrogate of the Bayesian optimizer enables it to converge the optimality several additional orders of magnitude relative to the conjugate-gradient and quasi-Newton optimizers. The final test case involves the chaotic Lorenz 63 model and inaccurate gradients. For this problem, the Bayesian optimizer achieves a lower final objective evaluation than the SciPy quasi-Newton optimizer for all initial starting solutions. The results demonstrate that a Bayesian optimizer can be competitive with quasi-Newton and conjugate-gradient optimizers when accurate gradients are available, and significantly outperforms them when the gradients are innacurate.

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arXiv2026

A Framework for Nonlinearly-Constrained Gradient-Enhanced Local Bayesian Optimization with Comparisons to Quasi-Newton Optimizers

Bayesian optimization is a popular and versatile approach that is well suited to solve challenging optimization problems. Their popularity comes from their effective minimization of expensive function evaluations, their capability to leverage gradients, and their efficient use of noisy data. Bayesian optimizers have commonly been applied to global unconstrained problems, with limited development for many other classes of problems. In this paper, two alternative methods are developed that enable rapid and deep convergence of nonlinearly-constrained local optimization problems using a Bayesian optimizer. The first method uses an exact augmented Lagrangian and the second augments the minimization of the acquisition function to contain additional constraints. Both of these methods can be applied to nonlinear equality constraints, unlike most previous methods developed for constrained Bayesian optimizers. The new methods are applied with a gradient-enhanced Bayesian optimizer and enable deeper convergence for three nonlinearly-constrained unimodal optimization problems than previously developed methods for constrained Bayesian optimization. In addition, both new methods enable the Bayesian optimizer to reach a desired tolerance with fewer function evaluations than popular quasi-Newton optimizers from SciPy and MATLAB for unimodal problems with 2 to 30 variables. The Bayesian optimizer had similar results using both methods. It is recommended that users first try using the second method, which adds constraints to the acquisition function minimization, since its parameters are more intuitive to tune for new problems.

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arXiv2026

On the Practical Impact of Local Linear Instabilities in Entropy-Stable Schemes

Local linear instability refers to the linearized discrete operator exhibiting perturbation growth exceeding that of the corresponding continuous linearized problem. In the context of nonlinear entropy-stable discretizations, we argue that local linear instabilities should be interpreted as a source of numerical error whose practical impact is often negligible compared with other discretization errors. For split-form discretizations of the variable-coefficient linear advection equation, such as those resulting from linearizations of entropy-stable discretizations of the Burgers equation, perturbations can indeed exhibit unphysical modal growth. However, we demonstrate that this growth satisfies physically interpretable bounds and is typically small. Furthermore, through modified-equation analysis and numerical experiments, we show that the growth is dominated by highly oscillatory and boundary-localized unphysical modes, and can therefore be readily controlled by small amounts of numerical dissipation. More generally, this modal perturbation growth does not extend directly to nonlinear two-point-flux discretizations of the type used in entropy-stable discretizations of the Euler equations. Floquet analysis demonstrates that unstable spectra of frozen-baseflow Jacobians need not lead to unstable perturbation growth. Using the geometric flux for the variable-coefficient linear advection equation, we derive a sharp perturbation growth bound predicting negligible growth, then show analogous behaviour for the logarithmic flux numerically. Finally, we argue that robustness issues observed for entropy-stable schemes in density-wave problems are better attributed to poor near-vacuum behaviour of the logarithmic mean than to local linear instabilities. Overall, our results suggest that local linear instabilities do not pose a practical obstacle to the use of high-order entropy-stable schemes.

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arXiv2026

Convergence of entropy-stable continuous summation-by-parts discretizations of symmetric hyperbolic conservation laws

The Lax equivalence theorem guarantees convergence of stable and consistent discretizations for linear hyperbolic partial differential equations (PDEs). For nonlinear problems, however, stability and consistency alone do not generally guarantee convergence, even for smooth solutions, and existing convergence results typically rely either on projection-based error decompositions or on linearization arguments that do not directly extend to entropy-stable split-form discretizations. In particular, general convergence results for entropy-stable discretizations of hyperbolic PDEs are currently lacking, despite their widespread use. In this work, we prove convergence under smoothness assumptions on the exact solution and fluxes for entropy-stable split-form discretizations of scalar and symmetric hyperbolic systems with homogeneous flux functions within the continuous summation-by-parts (C-SBP) framework. The scalar inviscid Burgers equation is presented as a canonical example. The analysis is based on a stability-consistency argument that yields a nonlinear error evolution inequality whose solution provides an explicit upper bound on the numerical error. We show that, for sufficiently small mesh spacing, and for degree-$p$ C-SBP discretizations in $d$ spatial dimensions with $p>1+d/2$, this bound remains finite on any finite time interval and tends to zero as the mesh is refined, implying convergence despite the presence of local linear instabilities. The results help clarify the relationship between consistency, entropy stability, nonlinear error growth, and convergence for discretizations of nonlinear hyperbolic problems.

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arXiv2026

Stable Volume Dissipation for High-Order Finite-Difference and Spectral-Element Methods with the Summation-by-Parts Property

The construction of stable, conservative, and accurate volume dissipation is extended to discretizations that possess a generalized summation-by-parts (SBP) property within a tensor-product framework. The dissipation operators can be applied to any finite-difference or spectral-element scheme that uses the SBP framework, including high-order entropy-stable schemes. Additionally, we clarify the incorporation of a variable coefficient within the operator structure and analyze the impact of a boundary correction matrix on operator structure and accuracy. Following the theoretical development and construction of novel dissipation operators, we relate the presented volume dissipation to the use of upwind SBP operators. When applied to spectral-element methods, the presented approach yields unique dissipation operators that can also be derived through alternative approaches involving orthogonal polynomials. Numerical examples featuring the linear convection, Burgers, and Euler equations verify the properties of the constructed dissipation operators and assess their performance compared to existing upwind SBP schemes, including linear stability behaviour. When applied to entropy-stable schemes, the presented approach results in accurate and robust methods that can solve a broader range of problems where comparable existing methods fail.

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arXiv2024

Very high-order symmetric positive-interior quadrature rules on triangles and tetrahedra

We present novel fully-symmetric quadrature rules with positive weights and strictly interior nodes of degrees up to 84 on triangles and 40 on tetrahedra. Initial guesses for solving the nonlinear systems of equations needed to derive quadrature rules are generated by forming tensor-product structures on quadrilateral/hexahedral subdomains of the simplices using the Legendre-Gauss nodes on the first half of the line reference element. In combination with a methodology for node elimination, these initial guesses lead to the development of highly efficient quadrature rules, even for very high polynomial degrees. Using existing estimates of the minimum number of quadrature points for a given degree, we show that the derived quadrature rules on triangles and tetrahedra are more than 95% and 80% efficient, respectively, for almost all degrees. The accuracy of the quadrature rules is demonstrated through numerical examples.

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arXiv2024

Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra

We present a new class of efficient and robust discontinuous spectral-element methods of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. Such discretizations employ a recently introduced family of sparse tensor-product summation-by-parts (SBP) operators in collapsed coordinates within an entropy-stable modal formulation. The proposed algorithms exploit the structure of such SBP operators alongside that of the Proriol-Koornwinder-Dubiner polynomial basis, and a weight-adjusted approximation is used to efficiently invert the local mass matrix for curvilinear elements. Using such techniques, the number of required entropy-conservative two-point flux evaluations between pairs of quadrature nodes is significantly reduced relative to existing entropy-stable formulations using (non-tensor-product) multidimensional SBP operators, particularly for high polynomial degrees, with an improvement in time complexity from $\mathcal{O}(p^{2d})$ to $\mathcal{O}(p^{d+1})$, where $p$ is the polynomial degree of the approximation and $d$ is the number of spatial dimensions. In numerical experiments involving smooth solutions to the compressible Euler equations, the proposed tensor-product schemes demonstrate similar levels of accuracy for a given mesh and polynomial degree to those using multidimensional SBP operators based on symmetric quadrature rules. Furthermore, both operator families are shown to give rise to entropy-stable methods which exhibit excellent robustness for under-resolved problems. Such results suggest that the algorithmic advantages resulting from the use of tensor-product operators are obtained without compromising accuracy or robustness, enabling the efficient extension of the benefits of entropy stability to higher polynomial degrees than previously considered for triangular and tetrahedral elements.

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arXiv2024

Tensor-Product Split-Simplex Summation-By-Parts Operators

We present an approach to construct efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with a tensor-product structure. The operators are constructed by splitting the simplices into quadrilateral or hexahedral subdomains, mapping tensor-product SBP operators onto the subdomains, and assembling back using a continuous-Galerkin-type procedure. These tensor-product split-simplex operators do not have repeated degrees of freedom at the interior interfaces between the split subdomains. Furthermore, they satisfy the SBP property by construction, leading to stable discretizations. The accuracy and sparsity of the operators substantially enhance the efficiency of SBP discretizations on simplicial meshes. The sparsity is particularly important for entropy-stable discretizations based on two-point flux functions, as it reduces the number of two-point flux computations. We demonstrate through numerical experiments that the operators exhibit efficiency surpassing that of the existing dense multidimensional SBP operators by more than an order of magnitude in many cases. This superiority is evident in both accuracy per degree of freedom and computational time required to achieve a specified error threshold.

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arXiv2024

Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra

We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.

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arXiv2024

Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations

High-order Hadamard-form entropy stable multidimensional summation-by-parts discretizations of the Euler and compressible Navier-Stokes equations are considerably more expensive than the standard divergence-form discretization. In search of a more efficient entropy stable scheme, we extend the entropy-split method for implementation on unstructured grids and investigate its properties. The main ingredients of the scheme are Harten's entropy functions, diagonal-$ \mathsf{E} $ summation-by-parts operators with diagonal norm matrix, and entropy conservative simultaneous approximation terms (SATs). We show that the scheme is high-order accurate and entropy conservative on periodic curvilinear unstructured grids for the Euler equations. An entropy stable matrix-type interface dissipation operator is constructed, which can be added to the SATs to obtain an entropy stable semi-discretization. Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta method. Entropy stable viscous SATs, applicable to both the Hadamard-form and entropy-split schemes, are developed for the compressible Navier-Stokes equations. In the absence of heat fluxes, the entropy-split scheme is entropy stable for the compressible Navier-Stokes equations. Local conservation in the vicinity of discontinuities is enforced using an entropy stable hybrid scheme. Several numerical problems involving both smooth and discontinuous solutions are investigated to support the theoretical results. Computational cost comparison studies suggest that the entropy-split scheme offers substantial efficiency benefits relative to Hadamard-form multidimensional SBP-SAT discretizations.

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arXiv2023

Quadrature Rules on Triangles and Tetrahedra for Multidimensional Summation-By-Parts Operators

Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-$ \mathsf{E} $ operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However, there is a limited number of such operators, and those currently in existence often have a relatively high node count for a given polynomial order due to a scarcity of suitable quadrature rules. We present several new symmetric positive-weight quadrature rules on triangles and tetrahedra that are suitable for construction of diagonal-$ \mathsf{E} $ SBP operators. For triangles, quadrature rules of degree one through twenty with facet nodes that correspond to the Legendre-Gauss-Lobatto (LGL) and Legendre-Gauss (LG) quadrature rules are derived. For tetrahedra, quadrature rules of degree one through ten are presented along with the corresponding facet quadrature rules. All of the quadrature rules are provided in a supplementary data repository. The quadrature rules are used to construct novel SBP diagonal-$ \mathsf{E} $ operators, whose accuracy and maximum timestep restrictions are studied numerically.

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arXiv2023

A Solution to the Ill-Conditioning of Gradient-Enhanced Covariance Matrices for Gaussian Processes

Gaussian processes provide probabilistic surrogates for various applications including classification, uncertainty quantification, and optimization. Using a gradient-enhanced covariance matrix can be beneficial since it provides a more accurate surrogate relative to its gradient-free counterpart. An acute problem for Gaussian processes, particularly those that use gradients, is the ill-conditioning of their covariance matrices. Several methods have been developed to address this problem for gradient-enhanced Gaussian processes but they have various drawbacks such as limiting the data that can be used, imposing a minimum distance between evaluation points in the parameter space, or constraining the hyperparameters. In this paper a new method is presented that applies a diagonal preconditioner to the covariance matrix along with a modest nugget to ensure that the condition number of the covariance matrix is bounded, while avoiding the drawbacks listed above. Optimization results for a gradient-enhanced Bayesian optimizer with the Gaussian kernel are compared with the use of the new method, a baseline method that constrains the hyperparameters, and a rescaling method that increases the distance between evaluation points. The Bayesian optimizer with the new method converges the optimality, ie the $\ell_2$ norm of the gradient, an additional 5 to 9 orders of magnitude relative to when the baseline method is used and it does so in fewer iterations than with the rescaling method. The new method is available in the open source python library GpGradPy, which can be found at https://github.com/marchildon/gpgradpy/tree/paper_precon. All of the figures in this paper can be reproduced with this library.

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arXiv2021

Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators

Several types of simultaneous approximation term (SAT) for diffusion problems discretized with diagonal-norm multidimensional summation-by-parts (SBP) operators are analyzed based on a common framework. Conditions under which the SBP-SAT discretizations are consistent, conservative, adjoint consistent, and energy stable are presented. For SATs leading to primal and adjoint consistent discretizations, the error in output functionals is shown to be of order $h^{2p}$ when a degree $p$ multidimensional SBP operator is used to discretize the spatial derivatives. SAT penalty coefficients corresponding to various discontinuous Galerkin fluxes developed for elliptic partial differential equations are identified. We demonstrate that the original method of Bassi and Rebay, the modified method of Bassi and Rebay, and the symmetric interior penalty method are equivalent when implemented with SBP diagonal-E operators that have diagonal norm matrix, e.g., the Legendre-Gauss-Lobatto SBP operator in one space dimension. Similarly, the local discontinuous Galerkin and the compact discontinuous Galerkin schemes are equivalent for this family of operators. The analysis remains valid on curvilinear grids if a degree $\le p+1$ bijective polynomial mapping from the reference to physical elements is used. Numerical experiments with the two-dimensional Poisson problem support the theoretical results.

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arXiv2021

A unifying algebraic framework for discontinuous Galerkin and flux reconstruction methods based on the summation-by-parts property

We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. Within such an algebraic framework, the multidimensional summation-by-parts (SBP) property is used to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties of a broad class of DG and FR schemes. Specifically, the analysis enables the extension of the equivalence between the strong and weak forms of the discontinuous Galerkin collocation spectral-element method demonstrated by Kopriva and Gassner (J Sci Comput 44:136-155, 2010) to more general nodal and modal DG formulations, as well as to the Vincent-Castonguay-Jameson-Huynh (VCJH) family of FR methods. Moreover, new algebraic proofs of conservation and energy stability for DG and VCJH schemes with respect to suitable quadrature rules and discrete norms are presented, in which the SBP property serves as a unifying mechanism for establishing such results. Numerical experiments are provided for the two-dimensional linear advection and Euler equations, highlighting the design choices afforded for methods within the proposed framework and corroborating the theoretical analysis.

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arXiv2021

Stability and Functional Superconvergence of Narrow-Stencil Second-Derivative Generalized Summation-By-Parts Discretizations

We analyze the stability and functional superconvergence of discretizations of diffusion problems with the narrow-stencil second-derivative generalized summation-by-parts (SBP) operators coupled with simultaneous approximation terms (SATs). Provided that the primal and adjoint solutions are sufficiently smooth and the SBP-SAT discretization is primal and adjoint consistent, we show that linear functionals associated with the steady diffusion problem superconverge at a rate of $ 2p $ when a degree $ p+1 $ narrow-stencil or a degree $ p $ wide-stencil generalized SBP operator is used for the spatial discretization. Sufficient conditions for stability of adjoint consistent discretizations with the narrow-stencil generalized SBP operators are presented. The stability analysis assumes nullspace consistency of the second-derivative operator and the invertibility of the matrix approximating the first derivative at the element boundaries. The theoretical results are verified by numerical experiments with the one-dimensional Poisson problem.

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arXiv2017

Conservative and Stable Degree Preserving SBP Operators for Non-Conforming Meshes

Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP-SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new \emph{degree preserving} discretizations require an ansatz that the norm matrix of the SBP operator is of a degree $\geq 2p$, in contrast to, for example, existing finite difference SBP operators, where the norm matrix is $2p-1$ accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.

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arXiv2016

Simultaneous Approximation Terms for Multi-Dimensional Summation-by-Parts Operators

This paper is concerned with the accurate, conservative, and stable imposition of boundary conditions and inter-element coupling for multi-dimensional summation-by-parts (SBP) finite-difference operators. More precisely, the focus is on diagonal-norm SBP operators that are not based on tensor products and are applicable to unstructured grids composed of arbitrary elements. We show how penalty terms --- simultaneous approximation terms (SATs) --- can be adapted to discretizations based on multi-dimensional SBP operators to enforce boundary and interface conditions. A general SAT framework is presented that leads to conservative and stable discretizations of the variable-coefficient advection equation. This framework includes the case where there are no nodes on the boundary of the SBP element at which to apply penalties directly. This is an important generalization, because elements analogous to Legendre-Gauss collocation, \ie without boundary nodes, typically have higher accuracy for the same number of degrees of freedom. Symmetric and upwind examples of the general SAT framework are created using a decomposition of the symmetric part of an SBP operator; these particular SATs enable the pointwise imposition of boundary and inter-element conditions. We illustrate the proposed SATs using triangular-element SBP operators with and without nodes that lie on the boundary. The accuracy, conservation, and stability properties of the resulting SBP-SAT discretizations are verified using linear advection problems with spatially varying divergence-free velocity fields.

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