scholarly journals A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: without direct estimation for an inverse of the linearized operator

2020 ◽  
Vol 146 (4) ◽  
pp. 907-926
Author(s):  
Kouta Sekine ◽  
Mitsuhiro T. Nakao ◽  
Shin’ichi Oishi
Keyword(s):  
Gaussian Elimination ◽  
Inverse Operator ◽  
Schur Complement ◽  
Elliptic Pdes ◽  
Computer Assisted ◽  
Operator Matrix ◽  
Infinite Dimensional ◽  
New Formulation ◽  
Matrix Problems ◽  
Linearized Operator

AbstractInfinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite-dimensional Newton-type fixed point equation $$w = - {\mathcal {L}}^{-1} {\mathcal {F}}(\hat{u}) + {\mathcal {L}}^{-1} {\mathcal {G}}(w)$$ w = - L - 1 F ( u ^ ) + L - 1 G ( w ) , where $${\mathcal {L}}$$ L is a linearized operator, $${\mathcal {F}}(\hat{u})$$ F ( u ^ ) is a residual, and $${\mathcal {G}}(w)$$ G ( w ) is a nonlinear term. Therefore, the estimations of $$\Vert {\mathcal {L}}^{-1} {\mathcal {F}}(\hat{u}) \Vert $$ ‖ L - 1 F ( u ^ ) ‖ and $$\Vert {\mathcal {L}}^{-1}{\mathcal {G}}(w) \Vert $$ ‖ L - 1 G ( w ) ‖ play major roles in the verification procedures . In this paper, using a similar concept to block Gaussian elimination and its corresponding ‘Schur complement’ for matrix problems, we represent the inverse operator $${\mathcal {L}}^{-1}$$ L - 1 as an infinite-dimensional operator matrix that can be decomposed into two parts: finite-dimensional and infinite-dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, which enables a more efficient verification procedure compared with existing Nakao’s methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as $${\mathcal {L}}^{-1}$$ L - 1 are presented in the “Appendix”.

scholarly journals MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

2021 ◽  
Author(s):  
Dong T.P. Nguyen ◽  
Dirk Nuyens
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Monte Carlo ◽  
Convergence Rates ◽  
Higher Order ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Quasi Monte Carlo ◽  
Higher Order Convergence ◽  
Element Method

We introduce the \emph{multivariate decomposition finite element method} (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\bsy) = \sum_{j \ge 1} y_j \, \phi_j$ with $y_j \sim \calN(0,1)$ and a given sequence of functions $\{\phi_j\}_{j \ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the \emph{multivariate decomposition method} (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using \emph{quasi-Monte Carlo} (QMC) methods, and for which we use the \emph{finite element method} (FEM) to solve different instances of the PDE.   We develop higher-order quasi-Monte Carlo rules for integration over the finite-di\-men\-si\-onal Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of \emph{anchored Gaussian Sobolev spaces} while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm.   Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-\dd/\lambda}) = O(\epsilon^{-(p^* + \dd/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-\dd/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $\dd = d \, (1+\ddelta)$ for some $\ddelta \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2 + \dd/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j \ge 1}$.

Generalized Inverse of Upper Triangular Infinite Dimensional Hamiltonian Operators

2013 ◽  
Vol 20 (03) ◽  
pp. 395-402
Author(s):  
Junjie Huang ◽  
Xiang Guo ◽  
Yonggang Huang ◽  
Alatancang
Keyword(s):  
Explicit Expression ◽  
Generalized Inverse ◽  
Hamiltonian Operator ◽  
Operator Matrix ◽  
Infinite Dimensional ◽  
Symplectic Spaces ◽  
Hamiltonian Operators

In this paper, we deal with the generalized inverse of upper triangular infinite dimensional Hamiltonian operators. Based on the structure operator matrix J in infinite dimensional symplectic spaces, it is shown that the generalized inverse of an infinite dimensional Hamiltonian operator is also Hamiltonian. Further, using the decomposition of spaces, an upper triangular Hamiltonian operator can be written as a new operator matrix of order 3, and then an explicit expression of the generalized inverse is given.

scholarly journals Risk-averse optimal control of semilinear elliptic PDEs

2020 ◽  
Vol 26 ◽  
pp. 53 ◽  
Author(s):  
D.P. Kouri ◽  
T.M. Surowiec
Keyword(s):  
Optimal Control ◽  
Stochastic Optimization ◽  
Optimization Problems ◽  
Risk Measures ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Continuity Properties ◽  
Semilinear Elliptic ◽  
Solution Regularity ◽  
Semilinear Elliptic Pdes

In this paper, we consider the optimal control of semilinear elliptic PDEs with random inputs. These problems are often nonconvex, infinite-dimensional stochastic optimization problems for which we employ risk measures to quantify the implicit uncertainty in the objective function. In contrast to previous works in uncertainty quantification and stochastic optimization, we provide a rigorous mathematical analysis demonstrating higher solution regularity (in stochastic state space), continuity and differentiability of the control-to-state map, and existence, regularity and continuity properties of the control-to-adjoint map. Our proofs make use of existing techniques from PDE-constrained optimization as well as concepts from the theory of measurable multifunctions. We illustrate our theoretical results with two numerical examples motivated by the optimal doping of semiconductor devices.

scholarly journals Generalized Probabilistic Theories in a New Light

2021 ◽  
Author(s):  
Raed Shaiia
Keyword(s):  
Quantum Mechanics ◽  
Measurement Problem ◽  
Quantum Mechanical ◽  
Infinite Dimensional ◽  
Classical Systems ◽  
Generalized Probabilistic Theories ◽  
Deterministic Description ◽  
New Formulation ◽  
The Universe ◽  
Probabilistic Theories

Abstract In this paper we will present a modified formulation of generalized probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, and give the guidelines to how to extend this work to infinite dimensional Hilbert spaces. Moreover, this new formulation which we will call extended operational-probabilistic theories, applies not only to quantum systems, but also equally well to classical systems, without violating Bell’s theorem, and at the same time solves the measurement problem. This is why we will see that the question of why our universe is quantum mechanical rather than classical is misplaced. The only difference that exists between a classical universe and a quantum mechanical one lies merely in which observables are compatible and which are not. Besides, this extended probability theory which we present in this paper shows that it is non-determinacy, or to be more precise, the non-deterministic description of the universe, that makes the laws of physics the way they are. In addition, this paper shows us that what used to be considered as purely classical systems and to be treated that way are in fact able to be manipulated according to the rules of quantum mechanics –with this new understanding of these rules- and that there is still a possibility that there might be a deterministic level from which our universe emerges, which if understood correctly, may open the door wide to applications in areas such as quantum computing. In addition to all that, this paper shows that without the use of complex vector spaces, we cannot have any kind of continuous evolution of the states of any system.

scholarly journals The residual spectrum and the continuous spectrum of upper triangular operator matrices

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 65-71 ◽  
Author(s):  
Guojun Hai ◽  
Alatancang Chen
Keyword(s):  
Continuous Spectrum ◽  
Hilbert Spaces ◽  
Operator Matrix ◽  
Operator Matrices ◽  
Residual Spectrum ◽  
Infinite Dimensional ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Upper Triangular Operator Matrix

Let H and K be separable infinite dimensional Hilbert spaces. We denote by MC the 2x2 upper triangular operator matrix acting on H ? K of the form MC = (A C/0 B ). For given operators A ? B(H) and B ? B(K), the sets C?B?(K,H) ?r(MC) and C?B?(K,H) ?c(MC) are characterized, where ?r(?) and ?c(?) denote the residual spectrum and the continuous spectrum, respectively

scholarly journals Efficient Implementations of Rainbow and UOV using AVX2

2021 ◽  
pp. 245-269
Author(s):  
Kyung-Ah Shim ◽  
Sangyub Lee ◽  
Namhun Koo
Keyword(s):  
Linear Systems ◽  
Digital Signature ◽  
Gaussian Elimination ◽  
Schur Complement ◽  
Matrix Inversion ◽  
Instruction Set ◽  
Signature Scheme ◽  
Secret Key ◽  
Quadratic Equations ◽  
Post Quantum Cryptography

A signature scheme based on multivariate quadratic equations, Rainbow, was selected as one of digital signature finalists for NIST Post-Quantum Cryptography Standardization Round 3. In this paper, we provide efficient implementations of Rainbow and UOV using the AVX2 instruction set. These efficient implementations include several optimizations for signing to accelerate solving linear systems and the Vinegar value substitution. We propose a new block matrix inversion (BMI) method using the Lower-Diagonal-Upper decomposition of blocks matrices based on the Schur complement that accelerates solving linear systems. Compared to UOV implemented with Gaussian elimination, our implementations with the BMI result in speedups of 12.36%, 24.3%, and 34% for signing at security categories I, III, and V, respectively. Compared to Rainbow implemented with Gaussian elimination, our implementations with the BMI result in speedups of 16.13% and 20.73% at the security categories III and V, respectively. We show that precomputation for the Vinegar value substitution and solving linear systems dramatically improve their signing. UOV with precomputation is 16.9 times, 35.5 times, and 62.8 times faster than UOV without precomputation at the three security categories, respectively. Rainbow with precomputation is 2.1 times, 2.2 times, and 2.8 times faster than Rainbow without precomputation at the three security categories, respectively. We then investigate resilience against leakage or reuse of the precomputed values in UOV and Rainbow to use the precomputation securely: leakage or reuse of the precomputed values leads to their full secret key recoveries in polynomial-time.

ε-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs

2018 ◽  
Vol 46 ◽  
pp. 66-89 ◽  
Author(s):  
Dinh Dũng ◽  
Michael Griebel ◽  
Vu Nhat Huy ◽  
Christian Rieger
Keyword(s):  
Elliptic Pdes ◽  
Hyperbolic Cross ◽  
Infinite Dimensional

scholarly journals Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

2016 ◽  
Vol 51 (1) ◽  
pp. 341-363 ◽  
Author(s):  
Markus Bachmayr ◽  
Albert Cohen ◽  
Ronald DeVore ◽  
Giovanni Migliorati
Keyword(s):  
Convergence Rates ◽  
Hermite Polynomials ◽  
Sufficient Conditions ◽  
Elliptic Pdes ◽  
Linear Elliptic Equation ◽  
Solution Map ◽  
Infinite Dimensional ◽  
Sparse Polynomial ◽  
The Mean ◽  
Linear Pdes

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = ∑ j ≥ 1yjψj where y = (yj) ∈ ℝNare i.i.d. standard scalar Gaussian variables and (ψj)j ≥ 1 is a given sequence of functions in L∞(D). We study the summability properties of Hermite-type expansions of the solution map y → u(y) ∈ V := H01(D) , that is, expansions of the form u(y) = ∑ ν ∈ ℱuνHν(y), where Hν(y) = ∏j≥1Hνj(yj) are the tensorized Hermite polynomials indexed by the set ℱ of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0 <p ≤ 1, the ℓp summability of the sequence (j ∥ψj ∥L∞)j ≥ 1 implies ℓp summability of the sequence (∥ uν∥V)ν ∈ ℱ. Such results ensure convergence rates n− s with s = (1/p)−(1/2) of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L2(ℝN,V,γ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓp summability of (∥uν∥V)ν ∈ ℱ expressed in terms of the pointwise summability properties of the sequence (|ψj|)j ≥ 1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the ℓp summability of (∥uν∥V)ν ∈ ℱfollows from the weaker assumption that (∥ψj∥L∞)j ≥ 1is ℓq summable for q := 2p/(2−p) . In the case of arbitrary supports, our results imply that the ℓp summability of (∥uν∥V)ν ∈ ℱ follows from the ℓp summability of (jβ∥ψj∥L∞)j ≥ 1 for some β>1/2 , which still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥uν∥V)ν ∈ ℱ. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

scholarly journals On Gaussian elimination and determinant formulas for matrices with chordal inverses

1992 ◽  
Vol 46 (3) ◽  
pp. 435-440 ◽  
Author(s):  
Mihály Bakonyi
Keyword(s):  
Gaussian Elimination ◽  
Chordal Graph ◽  
Invertible Operator ◽  
Chordal Graphs ◽  
Operator Matrix ◽  
Determinant Formula ◽  
Algorithmic Method ◽  
Finite Dimensional ◽  
Vertex Separators ◽  
Finite Dimensional Case

In this paper a formula is obtained for the entries of the diagonal factor in the U D L factorisation of an invertible operator matrix in the case when its inverse has a chordal graph. As a consequence, in the finite dimensional case a determinant formula is obtained in terms of some key principal minors. After a cancellation process this formula leads to a determinant formula from an earlier paper by W.W. Barrett and C.R. Johnson, deriving in this way a different and shorter proof of their result. Finally, an algorithmic method of constructing minimal vertex separators of chordal graphs is presented.

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