The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction... more The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei and is of fundamental importance for our understanding of atoms and molecules. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging, and it is conventionally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. We indicate why this conventional wisdom need not to be ironclad: the unexpectedly high regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and their antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The approximation schemes are based on multi-level decompositions of the corresponding function spaces, similar to those used in sparse grid methods. It is even possible to reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.
Communications on Pure and Applied Mathematics, 2006
Dual-Primal FETI methods are nonoverlapping domain decomposition methods where some of the contin... more Dual-Primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one-level FETI methods. The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained, which are independent of even large changes in the stiffnesses of the subdomains across the interface between them. A theoretical analysis is provided and condition number bounds are established which are uniform with respect to arbitrarily large jumps in the Young's modulus of the material and otherwise only depend polylogarithmically on the number of unknowns of a single subdomain.
Siam Journal on Scientific and Statistical Computing, 1989
Under the \timely communications" policy for the SIAM Journal on Scienti c and Statistical Comput... more Under the \timely communications" policy for the SIAM Journal on Scienti c and Statistical Computing, papers that have signi cant timely content and do not exceed ve pages automatically will be considered for a separate section of the journal with an accelerated reviewing process. It will be possible for the note to appear approximately six months after the date of acceptance.
In this note, we propose Steklov-Poincar iterative algorithms (mutuated from the analogy with het... more In this note, we propose Steklov-Poincar iterative algorithms (mutuated from the analogy with heterogeneous domain decomposition) to solve fluidstructure interaction problems. Although our framework is very general, the driving application is concerned with the interaction of blood flow and vessel walls in large arteries.
YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain ... more YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal ...
In this paper, we give a progress report on the development of a new fam-ily of domain decomposit... more In this paper, we give a progress report on the development of a new fam-ily of domain decomposition methods for the solution of Helmholtz's equation. We present three algorithms based on overlapping Schwarz methods; in our fa-vorite method we proceed to the continuous finite element ...
Abstract. Numerical experiments have shown that two-level Schwarz methods often perform very well... more Abstract. Numerical experiments have shown that two-level Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz ...
SIAM Journal on Scientific and Statistical Computing, 1992
Iterative methods for linear systems of algebraic equations arising from the finite element discr... more Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for pos-itive definite, symmetric problems are extended to certain nonsymmetric ...
MULTIPLICATIVE SCHWARZ ALGORITHMS FOR SOME NONSYMMETRIC AND INDEFINITE PROBLEMS* XIAO-CHUAN CAIt ... more MULTIPLICATIVE SCHWARZ ALGORITHMS FOR SOME NONSYMMETRIC AND INDEFINITE PROBLEMS* XIAO-CHUAN CAIt AND OLOF B. WIDLUNDt Abstract. The classical Schwarz alternating method has recently been generalized in several directions. This effort has resulted in a ...
. The iterative substructuring methods, also known as Schur complement methods,form one of two im... more . The iterative substructuring methods, also known as Schur complement methods,form one of two important families of domain decomposition algorithms. They are based on a partitioningof a given region, on which the partial dierential equation is dened, into non-overlappingsubstructures. The preconditioners of these conjugate gradient methods are then dened in termsof local problems dened on individual substructures and pairs of
YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain ... more YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal ...
Abstract. Finite element problems can often naturally be divided into subproblems which correspon... more Abstract. Finite element problems can often naturally be divided into subproblems which correspond to subregions into which the region has been partitioned or from which it was originally assembled. A class of iterative methods is discussed in which these subproblems are ...
The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction... more The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei and is of fundamental importance for our understanding of atoms and molecules. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging, and it is conventionally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. We indicate why this conventional wisdom need not to be ironclad: the unexpectedly high regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and their antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The approximation schemes are based on multi-level decompositions of the corresponding function spaces, similar to those used in sparse grid methods. It is even possible to reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.
Communications on Pure and Applied Mathematics, 2006
Dual-Primal FETI methods are nonoverlapping domain decomposition methods where some of the contin... more Dual-Primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one-level FETI methods. The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained, which are independent of even large changes in the stiffnesses of the subdomains across the interface between them. A theoretical analysis is provided and condition number bounds are established which are uniform with respect to arbitrarily large jumps in the Young's modulus of the material and otherwise only depend polylogarithmically on the number of unknowns of a single subdomain.
Siam Journal on Scientific and Statistical Computing, 1989
Under the \timely communications" policy for the SIAM Journal on Scienti c and Statistical Comput... more Under the \timely communications" policy for the SIAM Journal on Scienti c and Statistical Computing, papers that have signi cant timely content and do not exceed ve pages automatically will be considered for a separate section of the journal with an accelerated reviewing process. It will be possible for the note to appear approximately six months after the date of acceptance.
In this note, we propose Steklov-Poincar iterative algorithms (mutuated from the analogy with het... more In this note, we propose Steklov-Poincar iterative algorithms (mutuated from the analogy with heterogeneous domain decomposition) to solve fluidstructure interaction problems. Although our framework is very general, the driving application is concerned with the interaction of blood flow and vessel walls in large arteries.
YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain ... more YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal ...
In this paper, we give a progress report on the development of a new fam-ily of domain decomposit... more In this paper, we give a progress report on the development of a new fam-ily of domain decomposition methods for the solution of Helmholtz's equation. We present three algorithms based on overlapping Schwarz methods; in our fa-vorite method we proceed to the continuous finite element ...
Abstract. Numerical experiments have shown that two-level Schwarz methods often perform very well... more Abstract. Numerical experiments have shown that two-level Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz ...
SIAM Journal on Scientific and Statistical Computing, 1992
Iterative methods for linear systems of algebraic equations arising from the finite element discr... more Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for pos-itive definite, symmetric problems are extended to certain nonsymmetric ...
MULTIPLICATIVE SCHWARZ ALGORITHMS FOR SOME NONSYMMETRIC AND INDEFINITE PROBLEMS* XIAO-CHUAN CAIt ... more MULTIPLICATIVE SCHWARZ ALGORITHMS FOR SOME NONSYMMETRIC AND INDEFINITE PROBLEMS* XIAO-CHUAN CAIt AND OLOF B. WIDLUNDt Abstract. The classical Schwarz alternating method has recently been generalized in several directions. This effort has resulted in a ...
. The iterative substructuring methods, also known as Schur complement methods,form one of two im... more . The iterative substructuring methods, also known as Schur complement methods,form one of two important families of domain decomposition algorithms. They are based on a partitioningof a given region, on which the partial dierential equation is dened, into non-overlappingsubstructures. The preconditioners of these conjugate gradient methods are then dened in termsof local problems dened on individual substructures and pairs of
YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain ... more YVES ACHDOUt, YVON MADAYt, AND OLOF B. WIDLUND? Abstract. The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal ...
Abstract. Finite element problems can often naturally be divided into subproblems which correspon... more Abstract. Finite element problems can often naturally be divided into subproblems which correspond to subregions into which the region has been partitioned or from which it was originally assembled. A class of iterative methods is discussed in which these subproblems are ...
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