A new algorithm of proper generalized decomposition for parametric symmetric elliptic problems
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Citations
Advanced separated spatial representations for hardly separable domains
Low rank approximation of multidimensional data
Enhanced parametric shape descriptions in PGD-based space separated representations
A phase field model for the solid-state sintering with parametric proper generalized decomposition
On the computation of Proper Generalized Decomposition modes of parametric elliptic problems
References
Optimal Design of Heterogeneous Materials
Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces
On the Convergence of a Greedy Rank-One Update Algorithm for a Class of Linear Systems
Related Papers (5)
Frequently Asked Questions (11)
Q2. What are the future works in "A new algorithm of proper generalized decomposition for parametric symmetric elliptic problems" ?
In a future work the authors will consider the non-symmetric case.
Q3. What is the purpose of this paper?
The present paper is aimed at the direct determination of a variety of reduced dimension for the solution of parameterized symmetric elliptic PDEs.
Q4. What is the way to design a material?
the optimal design of heterogeneous materials with linear behavior law fits into the framework considered, as the parameters model the structural configuration of the various materials (cf. [29, 33]).
Q5. What is the correct definition of R(v) for a function v L?
For a function v ∈ L2(Γ, H; dµ), the authors denote by R(v) the closure of the vectorial space spanned by v(γ) when γ belongs to Γ; more exactly, taking into account that v is only defined up to sets of zero measure, the correct definition of R(v) is given byR(v) = ⋂µ(N)=0Span { v(γ) : γ ∈ Γ \\N } . (14)The following result proves that in (14) the intersection can be replaced a single closed spanned space corresponding to a single set M ∈ B.
Q6. What is the way to solve a parabolic problem?
Galerkin-POD strategies are well suited to solve parabolic problems, where the POD basis is obtained from the previous solution of the underlying elliptic operator (see [19, 26]).
Q7. how can i get a wn of f 60?
The existence of such a wn can be obtained by reasoning as in the proof of Theorem 3.3 or just using Weierstrass theorem because the dimension of Hn is finite.
Q8. What is the convergence of the PGD for the optimization problem?
The work [9] proves the convergence of the PGD for the optimization problem: Find u ∈ L2(Ω, H1(I)) such that u ∈ arg minv∈L2(Ω,H1(I)) E(v), where E is a strongly convex functional, with Lipschitzgradient on bounded sets.
Q9. What is the convergence of the PGD algorithm for the Laplace problem?
in [8] the authors prove the convergence of the PGD algorithm applied to the Laplace problem in a tensor product domain,−∆u = f in Ωx × Ωy, u|∂Ωx×Ωy = 0,where Ωx ⊂ R and Ωy ⊂ R are two bounded domains.
Q10. What is the connection between the PGD method and the standard POD method?
In particular it is strongly related to the PGD method in the sense that the standard formulation of the PGD method actually provides the optimality conditions of the minimization problem satisfied by the optimal 1D sub-spaces.
Q11. what is the recursive approximation of wn?
The authors build recursive approximations on finite-dimensional optimal subspaces by minimizing the mean parametric error of the current residual, similar to the one introduced in [11].