124,943 research outputs found
Least-squares solutions of multi-valued linear operator equations in Hilbert spaces
AbstractLet M be a linear manifold in H1 ⊕ H2, where H1, and H2 are Hilbert spaces. Two notions of least-squares solutions for the multi-valued linear operator equation (inclusion) y ϵ M(x) are introduced and investigated. The main results include (i) equivalent conditions for least-squares solvability, (ii) properties of a least-squares solution, (iii) characterizations of the set of all least-squares solutions in terms of algebraic operator parts and generalized inverses of linear manifolds, and (iv) best approximation properties of generalized inverses and operator parts of multi-valued linear operators. The principal tools in this investigation are an abstract adjoint theory, orthogonal operator parts, and orthogonal generalized inverses of linear manifolds in Hilbert spaces
Adaptive least-squares space-time finite element methods
We consider the numerical solution of an abstract operator equation by
using a least-squares approach. We assume that is an
isomorphism, and that implies a norm in , where and
are Hilbert spaces. The minimizer of the least-squares functional , i.e., the solution of the operator equation, is then
characterized by the gradient equation with an elliptic and
self-adjoint operator . When introducing the
adjoint we end up with a saddle point formulation to be
solved numerically by using a mixed finite element method. Based on a discrete
inf-sup stability condition we derive related a priori error estimates. While
the adjoint is zero by construction, its approximation serves as a
posteriori error indicator to drive an adaptive scheme when discretized
appropriately. While this approach can be applied to rather general equations,
here we consider second order linear partial differential equations, including
the Poisson equation, the heat equation, and the wave equation, in order to
demonstrate its potential, which allows to use almost arbitrary space-time
finite element methods for the adaptive solution of time-dependent partial
differential equations
Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: General Discontinuous Interface
We studied the least-squares ReLU neural network method (LSNN) for solving
linear advection-reaction equation with discontinuous solution in [Cai,
Zhiqiang, Jingshuang Chen, and Min Liu. ``Least-squares ReLU neural network
(LSNN) method for linear advection-reaction equation.'' Journal of
Computational Physics 443 (2021), 110514]. The method is based on a
least-squares formulation and uses a new class of approximating functions: ReLU
neural network (NN) functions. A critical and additional component of the LSNN
method, differing from other NN-based methods, is the introduction of a proper
designed discrete differential operator.
In this paper, we study the LSNN method for problems with arbitrary
discontinuous interfaces. First, we show that ReLU NN functions with depth
can approximate any -dimensional step function
on arbitrary discontinuous interfaces with any prescribed accuracy. By
decomposing the solution into continuous and discontinuous parts, we prove
theoretically that discretization error of the LSNN method using ReLU NN
functions with depth is mainly determined by the
continuous part of the solution provided that the solution jump is constant.
Numerical results for both two and three dimensional problems with various
discontinuous interfaces show that the LSNN method with enough layers is
accurate and does not exhibit the common Gibbs phenomena along the
discontinuous interface.Comment: 24 page
Adaptive Boundary and Point Control of Linear Stochastic Distributed Parameter Systems
This is the published version, also available here: http://dx.doi.org/10.1137/S0363012992228726.An adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system is formulated and solved in this paper. The distributed parameter system is modeled by an evolution equation with an infinitesimal generator for an analytic semigroup. Since there is boundary or point control, the linear transformation for the control in the state equation is also an unbounded operator. The unknown parameters in the model appear affinely in both the infinitesimal generator of the semigroup and the linear transformation of the control. Strong consistency is verified for a family of least squares estimates of the unknown parameters. An Itô formula is established for smooth functions of the solution of this linear stochastic distributed parameter system with boundary or point control. The certainty equivalence adaptive control is shown to be self-tuning by using the continuity of th solution of a stationary Riccati equation as a function of parameters in a uniform operator topology. For a quadratic cost functional of the state and the control, the certainty equivalence control is shown to be self-optimizing; that is, the family of average costs converges to the optimal ergodic cost. Some examples of stochastic parabolic problems with boundary control and a structurally damped plate with random loading and point control are described that satisfy the assumptions for the adaptive control problem solved in this paper
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