In other words, there exist positive constants and , such that.
The Equations 5 - 9 has a unique solution, and the solution is. Proof: From Theorem 2. Then the result follows from Lax-Milgram theorem. In principle, the LSMFE approach simply consists of minimizing 12 in finite-dimensional subspaces and. Suitable spaces are based on a triangulation of and consist of piecewise polynomials with sufficient continuity conditions. Now we consider the Ciarlet-Raviart mixed finite element form.
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Let and. Minimizing the functional 14 is equivalent to the following variational problem: find and such that. The discrete bilinear form is defined as follows:. Theorem 3. The bilinear is continuous and coercive, i. The proof is the same as the Theorem 2. One of the main motivations for using least-squares finite element approaches is the fact that the element-wise evaluation of the functional serves as an a posteriori error estimator.
A posteriori estimate attempt to provide quantitatively accurate measures of the discretization error through the socalled a posteriori error estimators which are derived by using the information obtained during the solution process. In recent years, the use of a posteriori error estimators has become an efficient tool for assessing and controlling computational errors in adaptive computations . Theorem 4. The least-squares functional constitutes an a posteriori error estimator. In other words, for. The positive constants and , this completes the proof.
Remark: The mesh is adapted and based on a posteriori error estimate of the fourth order elliptic problems. Based on the computed a posteriori error estimator , we use a mesh optimization procedure to compute the size of elements in the new mesh.
- Least-Squares Fitting.
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Adaptive refinement strategies consist in refining those triangles with the largest values of. We now briefly introduce the main idea of adaptive leastsquares mixed finite element methods through local refinement.
Given an initial triangulation , we shall generate a sequence of nested conforming triangulations using the following loop :. More precisely to get from we first solve 5 - 9 to get on. The error is estimated using and to mark a set of that are to be refined.
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- Error estimates for least-squares mixed finite elements.
Triangles are refined in such a way that the triangulation is still shape regular and conforming. The a posteriori error estimator is usually split into local error indicators and they are then employed to make local modifications by dividing the elements whose error indicator is large and possibly coarsening the elements whose error indicator is small.
The convergence of local refinement algorithms based on the repetition of loop is established by the error reduction type result. Let be a shape regular triangulation of and is a refinement of such that. Let and be the finite element approximation of in and , respectively.
Weighted least-squares finite elements based on particle imaging velocimetry data
We shall use the following results in the proof of the convergence :. Let be an initial shape regular triangulation, let be a solution of 5 - 9 in the loop. We have the following theorem:. Theorem 5. Let be a solution obtained in the loop in the algorithm, then there exists a constants depending the shape regularity of such that:. Proof: In the step we select such that. We describe an adaptive least-squares mixed finite element procedure for solving the fourth order parabolic problems in this paper, and the procedure uses a leastsquares mixed finite element formulation and adaptive refinement based on a posteriori error estimate.
The methods were applied to study the continuous and coercivity of the fourth order parabolic problems. In this paper, we applied relatively standard a posteriori error estimation techniques to adaptively solve the fourth order parabolic problems and shown the convergence of the adaptive least-squares mixed finite element method. Gu, D. Yang, S. Sui and X. Gu and X. Cai, B. Lee and P. There are no requirements on how to compute value and derivatives. The DerivativeStructure class may be useful to compute analytically derivatives in difficult cases, but this class is not mandated by the API which only expects the derivatives as a Jacobian matrix containing primitive double entries.
One non-obvious feature provided by both the builder and the factory is lazy evaluation. This feature allows to defer calls to the model functions until they are really needed by the engine. This can save some calls for engines that evaluate the value and the Jacobians in different loops this is the case for Levenberg-Marquardt.
However, lazy evaluation is possible only if the model functions are themselves separated, i. Setting up the lazyEvaluation flag to true in the builder or factory and setting up the model function as one MultivariateJacobianFunction instance at the same time will trigger an illegal state exception telling that the model function misses required functionality.
Least-Squares Fitting - MATLAB & Simulink
In some cases, the model function requires parameters to lie within a specific domain. The least square solvers available in Apache Commons Math currently don't allow to set up constraints on the parameters. This is a known missing feature. There are two ways to circumvent this. Both ways are achieved by setting up a ParameterValidator instance.
The input of the value and jacobian model functions will always be the output of the parameter validator if one exists. One way to constrain parameters is to use a continuous mapping between the parameters that the least squares solver will handle and the real parameters of the mathematical model. Using mapping functions like logit and sigmoid , one can map a finite range to the infinite real line. Using mapping functions based on log and exp , one can map a semi-infinite range to the infinite real line. It is possible to use such a mapping so that the engine will always see unbounded parameters, whereas on the other side of the mapping the mathematical model will always see parameters mapped correctly to the expected range.
Care must be taken with derivatives as one must remember that the parameters have been mapped. Care must also be taken with convergence status. This may be tricky. Another way to constrain parameters is to simply truncate the parameters back to the domain when one search point escapes from it and not care about derivatives. This works only if the solution is expected to be inside the domain and not at the boundary, as points out of the domain will only be temporary test points with a cost function higher than the real solution and will soon be dropped by the underlying engine.
As a rule of thumb, these conditions are met only when the domain boundaries correspond to unrealistic values that will never be achieved null distances, negative masses, Among the elements to be provided to the least squares problem builder or factory are some tuning parameters for the solver. The maximum number of iterations refers to the engine algorithm main loop, whereas the maximum number of iterations refers to the number of calls to the model method. Some algorithms like Levenberg-Marquardt have two embedded loops, with iteration number being incremented at outer loop level, but a new evaluation being done at each inner loop.