Set-Valued Approximation—Revisited and Improved
We address the problem of approximating a set-valued function <i>F</i>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>:</mo><mrow><mo>[...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/7/1194 |
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| Summary: | We address the problem of approximating a set-valued function <i>F</i>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>:</mo><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>→</mo><mi>K</mi><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> given its samples <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><mi>F</mi><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>i</mi><mi>h</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></msubsup></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>=</mo><mo>(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>)</mo><mo>/</mo><mi>N</mi></mrow></semantics></math></inline-formula>. We revisit an existing method that approximates set-valued functions by interpolating signed-distance functions. This method provides a high-order approximation for general topologies but loses accuracy near points where <i>F</i> undergoes topological changes. To address this, we introduce new techniques that enhance efficiency and maintain high-order accuracy across <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></semantics></math></inline-formula>. Building on the foundation of previous publication, we introduce new techniques to improve the method’s efficiency and extend its high-order approximation accuracy throughout the entire interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></semantics></math></inline-formula>. Particular focus is placed on identifying and analyzing the behavior of <i>F</i> near topological transition points. To address this, two algorithms are introduced. The first algorithm employs signed-distance quasi-interpolation, incorporating specialized adjustments to effectively handle singularities at points of topological change. The second algorithm leverages an implicit function representation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>r</mi><mi>a</mi><mi>p</mi><mi>h</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula>, offering an alternative and robust approach to its approximation. These enhancements improve accuracy and stability in handling set-valued functions with changing topologies. |
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| ISSN: | 2227-7390 |