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Solving differential‐algebraic equations in power system dynamic analysis with quantum computing
Abstract Power system dynamics are generally modeled by high dimensional non‐linear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system non‐linearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.
Nonperturbative gravity corrections to bulk reconstruction
Abstract We introduce a new algebraic framework for understanding nonperturbative gravitational aspects of bulk reconstruction with a finite or infinite-dimensional boundary Hilbert space. We use relative entropy equivalence between bulk and boundary with an inclusion of nonperturbative gravitational errors, which give rise to approximate recovery. We utilize the privacy/correctability correspondence to prove that the reconstruction wedge, the intersection of all entanglement wedges in pure and mixed states, manifestly satisfies bulk reconstruction. We explicitly demonstrate that local operators in the reconstruction wedge of a given boundary region can be recovered in a state-independent way for arbitrarily large code subspaces, up to nonperturbative errors in G N . We further discuss state-dependent recovery beyond the reconstruction wedge and the use of the twirled Petz map as a universal recovery channel. We discuss our setup in the context of quantum islands and the information paradox.
A High Performance Sparse Tensor Algebra Compiler in MLIR
Sparse tensor algebra is widely used in many applications, including scientific computing, machine learning, and data analytics. The performance of sparse tensor algebra kernels strongly depends on the intrinsic characteristics of the input tensors, hence many storage formats are designed for tensors to achieve optimal performance for particular applications/architectures, which makes it challenging to implement and optimize every tensor operation of interest on a given architecture. We propose a tensor algebra domain-specific language (DSL) and compiler framework to automatically generate kernels for mixed sparse-dense tensor algebra operations. The proposed DSL provides high-level programming abstractions that resemble the familiar Einstein notation to represent tensor algebra operations. The compiler introduces a new Sparse Tensor Algebra dialect built on top of LLVM's extensible MLIR compiler infrastructure for efficient code generation while covering a wide range of tensor storage formats. Our compiler also leverages input-dependent code optimization to enhance data locality for better performance. Our results show that the performance of automatically generated kernels outperforms the state-of-the-art sparse tensor algebra compiler, with up to 20.92x, 6.39x, and 13.9x performance improvement over state-of-the-art tensor algebra compilers, for parallel SpMV, SpMM, and TTM, respectively.
Learning Algebraic Multigrid Prolongation with Residual Graph Neural Networks.
Abstract not provided.
On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations
An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included.
An Algebraic Monolithic Method for Volume-coupled Multiphysics Problems.
Abstract not provided.
An Algebraic Monolithic Method for Volume-coupled Multiphysics Problems.
Abstract not provided.
Smoothed aggregation for difficult stretched mesh and coefficient variation problems
Abstract Four adaptations of the smoothed aggregation algebraic multigrid (SA‐AMG) method are proposed with an eye toward improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill‐in within the coarse discretization matrices and can also give rise to suboptimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi‐like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.
A Monolithic Algebraic Multigrid Approach for Coupled Multiphysics Problems using the MueLu Framework.
Abstract not provided.
Algebraic Multigrid for mixed finite element discretizations of the Stokes Equations.
Abstract not provided.
Vertex algebra of extended operators in 4d N=2 superconformal field theories. Part I
Abstract We construct a class of extended operators in the cohomology of a pair of twisted Schur supercharges of 4d$$ \mathcal{N} $$ N =2 SCFTs. The extended operators are constructed from the local operators in this cohomology — the Schur operators — by a version of topological descent. They are line, surface, and domain wall world volume integrals of certain super descendants of Schur operators. Their world volumes extend in directions transverse to a spatial plane in Minkowski space-time. As operators in the cohomology of these twisted Schur supercharges, their correlators are (locally) meromorphic functions only of the positions where they intersect this plane. This implies the extended operators enlarge the vertex operator algebra of the Schur operators. We illustrate this enlarged vertex algebra by computing some extended-operator product expansions within a subalgebra of it for the free hypermultiplet SCFT.
Variational approach to relative entropies with an application to QFT
Abstract We define a new divergence of von Neumann algebras using a variational expression similar in nature to Kosaki’s formula for Umegaki’s relative entropy. Our divergence satisfies several of the usual desirable properties, upper bounds the sandwiched Renyi entropy and reduces to the fidelity in a limit. As an illustration, we use the formula in quantum field theory to compute our divergence between the vacuum in a bipartite system and an “orbifolded”—in the sense of a conditional expectation—system in terms of the Jones index. We take the opportunity to point out an entropic certainty relation associated with an inclusion of von Neumann factors related to the relative entropy. This certainty relation has an equivalent formulation in terms of error correcting codes.
Scalar fields in 3D asymptotically flat higher-spin gravity
Abstract In this work we construct a novel associative algebra and use it to define a theory of higher-spin gravity in (2 + 1)-dimensional asymptotically flat spacetimes. Our construction is based on a quotient of the universal enveloping algebra of i s l ( 2 , R ) with respect to the ideal generated by its Casimir elements, the mass squared M 2 and the three-dimensional analogue of the square of the Pauli–Lubanski vector S and propose to call the resulting associative algebra i h s ( M 2 , S ) . We provide a definition of its generators and even though we are not yet able to provide the complete set of multiplication rules of this algebra our analysis allows us to study many interesting and relevant sub-structures of i h s ( M 2 , S ) . We then show how to consistently couple a scalar field to an i h s ( M 2 , S ) higher-spin gauge theory.
Surface science insight note: A linear algebraic approach to elucidate native films on Fe 3 O 4 surface
Standard materials are often used to obtain spectra that can be compared to those from unknown samples. Spectra measured from these known substances are also used as a means of computing sensitivity factors to allow quantification by X-ray photoelectron spectroscopy (XPS) of less well-defined materials. Spectra from known materials also provide line shapes suitable for inclusion in spectral models which, when fitted to spectra, permit the chemical state for a sample to be assessed. Both types of information depend on isolating photoemission signals from the inelastically scattered signal. In this Insight note, technical issues associated with the use of XPS of as received Fe 3 O 4 powder sample surface are discussed. The Insight note is designed to show how linear algebraic techniques applied to data collected from a sample marketed as pure Fe 3 O 4 powder are used to verify that XPS has been performed on chemistry representative of the sample. The methods described in this Insight note can further be utilized in elucidating complex XPS data obtained from thin films formed or evolved during cyclic/non-steady use of complex (electro)catalyst surfaces, especially in the presence of contaminants.
Introduction to abstract analysis
This book, which grew out of lectures given at the NASA Lewis Research Center, introduces the scientist and engineer with the usual background in applied mathematics to the concepts of abstract analysis. The emphasis is not on preparing the reader to do research in the field but on giving him some of the background necessary for reading the literature of pure mathematics. Although the material here is by no means original, the presentation differs in some respects from texts on material of this nature. The proofs are more detailed herein and quite easy to follow. We have attempted to indicate how the material relates to and serves as a foundation for more advanced sub- jects. We have also attempted at several places to show how the material covered here relates to the more familiar “real mathematics.” Enough examples are included to illustrate the concepts. No attempt is made to indicate the original sources of the material or even to point out the originators of all the concepts. Contrary to the usual practice, the relation between convergence and continuity on the one hand and algebraic operations on the other is dis- cussed in the abstract setting of linear spaces. This is done principally to familiarize the reader with these very important concepts in a reasonably simple way.
A symbolic framework to obtain mid-fidelity models of flexible multibody systems with application to horizontal-axis wind turbines
Abstract. The article presents a symbolic framework (also called computer algebra program) that is used to obtain, in symbolic mathematical form, the linear and nonlinear equations of motion of a mid-fidelity multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The shape function approach does not represent the state of the art for flexible multibody dynamics but is an effective trade-off to obtain mid-fidelity models with few degrees of freedom, taking advantage of the separation of space and time. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitates the creation and manipulation of models with various levels of complexity by adding or removing degrees of freedom. The symbolic treatment allows for analytical gradients and linearized equations of motion. The linear and nonlinear equations can be exported to Python code or dedicated software. There are multiple applications, such as time domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, and digital twins. In this article, we describe the method we used to systematically generate the equations of motion of multibody systems and present the implementation of the framework using the Python package SymPy. We apply the framework to generate illustrative land-based and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. The Python implementation is provided as an open-source project.
Stochastic control of infinite dimensional systems in Hilbert space: A factorization perspective
A factorization perspective on problems of optimal causal estimation and optimal causal control of linear stochastic systems defined on an infinite-dimensional Hilbert space is presented. A separation principle is derived for the case in which the system input/output map is generated by an abstract evolution operator. The factorization formalism allows for an essentially algebraic approach to these problems.