Interpolation of scattered data: Distance matrices and conditionally positive definite functions

Micchelli, Charles A.

doi:10.1007/bf01893414

Cited by 1,193 publications

(455 citation statements)

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“…That is, the matrix −D (2) is conditionally positive semidefinite on V = e ⊥ . Another important piece of contribution is by Micchelli [29] on connections between EDMs and reproducing kernels, which play an essential role in machine learning research (e.g., support vector machines). We hope to gather examples in this field in future.…”

Section: Classical Contributionsmentioning

confidence: 99%

“…The matrices in K n + are known as almost positive matrices [12, Chp. XV] or almost positive definite matrices [29,Def. 2.2].…”

Section: Nearest Edm Problemsmentioning

confidence: 99%

“…If there exist a point Y 0 such that A(Y 0 ) = b and Y 0 ∈ int K n + (V) (known as generalized Slater condition), then there is no duality gap between the primal problem (27) and the dual problem (29). Suppose y is an optimal solution of (29), then 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64is the optimal solution of (27).…”

Section: The Lagrangian Dual Problemmentioning

confidence: 99%

“…We omit the details. Theorem 4.6 Let y be an optimal solution of the dual problem (29). Let A := Π K n + (V) (A + A * (y)).…”

Section: Nonsingularity Of ∂F (Y)mentioning

confidence: 99%

“…Reams [38] used it for V = e ⊥ , the subspace orthogonal to the vector e of all ones. Earlier, Chabrillac and Crouzeix in their survey paper [7] call it the semidefiniteness of the restricted quadratic form to the space V. Micchelli [29] (see also Baxter [3]) used a slightly different name for V = e ⊥ . They referred it as the almost positive definiteness, whose origin can be traced back to the seminar work [42] of Schoenberg on the Euclidean distance embeddability in the (real) Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

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Conditional Quadratic Semidefinite Programming: Examples and Methods

2014

J. Oper. Res. Soc. China

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Abstract:The conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem.For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy.We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP. Response to Reviewers:The author would like to thank the referees and the associate editor for their efforts in their speedy review. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Conditional Quadratic Semidefinite Programming: Examples and MethodsHou-Duo Qi * April 14, 2014; Revised May 3, 2014 AbstractThe conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem. For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy. We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP.

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Section: Classical Contributionsmentioning

confidence: 99%

“…The matrices in K n + are known as almost positive matrices [12, Chp. XV] or almost positive definite matrices [29,Def. 2.2].…”

Section: Nearest Edm Problemsmentioning

confidence: 99%

Section: The Lagrangian Dual Problemmentioning

confidence: 99%

“…We omit the details. Theorem 4.6 Let y be an optimal solution of the dual problem (29). Let A := Π K n + (V) (A + A * (y)).…”

Section: Nonsingularity Of ∂F (Y)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Conditional Quadratic Semidefinite Programming: Examples and Methods

2014

J. Oper. Res. Soc. China

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Solving Nonlinear Differential Equations of Mechanics With the Boundary Element Method and Radial Basis Functions

Pollandt

1997

Int. J. Numer. Meth. Engng.

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SUMMARYThe Boundary Element Method is a very e ective method for solving linear di erential equations. To use it also in the consideration of non-linear problems some di erent procedures were developed, among them the dual reciprocity method and the particular integral method. Both procedures use interpolation conditions for the approximation with radial basis functions. In this paper a method is presented which avoids problems connected with interpolation by means of quasi-interpolation. It is possible to solve di erential equations of the kind m u = p(u) approximately; the application to two non-linear problems of plate theory yield good results. Hints to a theoretical examination of the method including su cient conditions for feasibility and convergence are given.KEY WORDS: boundary element method; dual reciprocity method; radial basis functions; non-linear di erential equations

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An indirect Boundary Element Method for solving low Reynolds number Navier-Stokes equations in a three-dimensional cavity

Power

Botte

1998

Int. J. Numer. Meth. Engng.

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In the present work, we propose an indirect boundary-only integral equation approach for the numerical solution of the Navier-Stokes system of equations in a three-dimensional ow cavity. The formulation is based on an indirect integral representational formula for the permanent Stokes equations, and the use of a particular solution of a nonhomogeneous Stokes system of equations in order to obtain in an iterative way the corresponding complete solution of the problem. Previous boundary-only integral equation approaches to the present problem, using direct boundary elements formulations, result in a series of matrix multiplications that make these approaches computationally costly. Due to the use of an indirect formulation, the present approach is free from those matrix multiplications. ? 1998 John Wiley & Sons, Ltd.

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Interpolation of scattered data: Distance matrices and conditionally positive definite functions

Abstract: Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.

Cited by 1,193 publications

References 23 publications

Conditional Quadratic Semidefinite Programming: Examples and Methods

Conditional Quadratic Semidefinite Programming: Examples and Methods

Solving Nonlinear Differential Equations of Mechanics With the Boundary Element Method and Radial Basis Functions

An indirect Boundary Element Method for solving low Reynolds number Navier-Stokes equations in a three-dimensional cavity

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